College Algebra, 8th Edition

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College Algebra

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Barnett, Ziegler, and Byleen’s Precalculus Series College Algebra, Eighth Edition This book is the same as Precalculus without the three chapters on trigonometry. ISBN 0-07-286738-8, ISBN 978-0-07-286738-1 Precalculus, Sixth Edition This book is the same as College Algebra with three chapters of trigonometry added. The trigonometry functions are introduced by a unit circle approach. ISBN 0-07-286739-6, ISBN 978-0-07-286739-8 College Algebra with Trigonometry, Eighth Edition This book differs from Precalculus in that College Algebra with Trigonometry uses right triangle trigonometry to introduce the trigonometric functions. ISBN 0-07-331264-9, ISBN 978-0-07-331264-4 College Algebra: Graphs and Models, Second Edition This book is the same as Precalculus: Graphs and Models without the three chapters on trigonometry. This text assumes the use of a graphing calculator. ISBN 0-07-242428-1, ISBN 978-0-07-242428-7 Precalculus: Graphs and Models, Second Edition This book is the same as College Algebra: Graphs and Models with three additional chapters on trigonometry. The trigonometric functions are introduced by a unit circle approach. This text assumes the use of a graphing calculator. ISBN 0-07-242430-3, ISBN 978-0-07-242430-0 College Algebra with Trigonometry: Graphs and Models This book is the same as Precalculus: Graphs and Models except that the trigonometric functions are introduced by right triangle trigonometry. This text assumes the use of a graphing calculator. ISBN 0-07-291699-0, ISBN 978-0-07-291699-7

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EIGHTH EDITION

College Algebra Raymond A. Barnett Merritt College

Michael R. Ziegler Marquette University

Karl E. Byleen Marquette University

Boston Burr Ridge, IL Dubuque, IA New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

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COLLEGE ALGEBRA, EIGHTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 9 8 7 6 ISBN 978–0–07–286738–1 MHID 0–07–286738–8 ISBN 978–0–07–330360–4 (Annotated Instructor’s Edition) MHID 0–07–330360–7 Publisher: Elizabeth J. Haefele Sponsoring Editor: Dawn R. Bercier Developmental Editor: Lindsay Roth Lead Project Manager: Peggy J. Selle Senior Production Supervisor: Kara Kudronowicz Lead Media Producer: Jeff Huettman Senior Media Project Manager: Sandra M. Schnee Senior Designer: David W. Hash

Cover/Interior Designer: Asylum Studios; Mona Grigaliunas and Tom J. Nemoda Senior Photo Research Coordinator: Lori Hancock Photo Research: LouAnn K. Wilson Supplement Producer: Melissa M. Leick Compositor: Techbooks Typeface:10/12 Times Roman Printer: R. R. Donnelley Willard, OH

Chapter R Opener: © CORBIS/Royalty Free; p. 26: © Vol. 34/Getty Images; p. 46: © McGraw-Hill Companies, Inc., John Thoeming, photographer Chapter 1 Opener: © CORBIS/Royalty Free; p. 84: © Vol. 49/Getty Images; p. 91: © Vol. 71/Getty Images; p. 138: © BigStock Photo Chapter 2 Opener: © Vol. 88/Getty Images; p. 203: © BigStock Photo; p. 209: © CORBIS/Royalty Free; p. 215: © Vol. 112/Getty Images; p. 219: © Jupiter Images/Getty Images RF Chapter 3 Opener: © Vol. 76/Getty Images; p. 266: © Vol. 88/Getty Images; p. 306: © CORBIS/Royalty Free; p. 345: © The McGraw-Hill Companies, Inc., Andrew Resek, photographer Chapter 4 Opener: © BigStock Photo; p. 376: © CORBIS/Royalty Free; p. 377: © Vol. 4/Getty Images; p. 435: © Vol. 121/Digital Stock/CORBIS Chapter 5 Opener: © RoyaltyFree/Getty Images; p. 462: © Vol. 68/Getty Images; p. 478: © CORBIS/Royalty Free; p. 505: © Digital Vision/SuperStock RF; p. 510: Courtesy Ron Nichols, USDA Chapter 6 Opener: © Brand X/ SuperStock Royalty-Free; p. 548: © California Institute of Technology Chapter 7 Opener: © BigStock Photo; p. 590: Courtesy Bill Tapenning, USDA; p. 592: © Vol. 5/Getty Images; p. 626: © Vol. 48/Getty Images Chapter 8 Opener: © Vol. 6/Getty Images; p. 745: © ThinkStock/PictureQuest RF; p. 760: © CORBIS/ Royalty Free; p. 776: © Vol. 76/Getty Images Library of Congress Cataloging-in-Publication Data Barnett, Raymond A. College algebra. – 8th ed. / Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen. p. cm. – (Barnett, Ziegler & Byleen's precalculus series) Includes indexes. ISBN 978–0–07–286738–1 — ISBN 0–07–286738–8 (acid-free paper) 1. Algebra–Textbooks. I. Ziegler, Michael R. II. Byleen, Karl. III. Title. QA154.3.B365 2008 512.9–dc22 2006033538

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About the Authors

Raymond A. Barnett, a native of and educated in California, received his B.A. in mathematical statistics from the University of California at Berkeley and his M.A. in mathematics from the University of Southern California. He has been a member of the Merritt College Mathematics Department and was chairman of the department for four years. Associated with four different publishers, Raymond Barnett has authored or co-authored 18 textbooks in mathematics, most of which are still in use. In addition to international English editions, a number of the books have been translated into Spanish. Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University; Charles Burke, City College of San Francisco; John Fujii, Merritt College; and Karl Byleen, Marquette University. Michael R. Ziegler received his B.S. from Shippensburg State College and his M.S. and Ph.D. from the University of Delaware. After completing postdoctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he currently holds the rank of Professor in the Department of Mathematics, Statistics, and Computer Science. Dr. Ziegler has published more than a dozen research articles in complex analysis and has co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Karl Byleen. Karl E. Byleen received his B.S., M.A., and Ph.D. degrees in mathematics from the University of Nebraska. He is currently an Associate Professor in the Department of Mathematics, Statistics, and Computer Science of Marquette University. He has published a dozen research articles on the algebraic theory of semigroups and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler.

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Brief Contents

Preface ix Features xvi Application Index xxi

R CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6 CHAPTER 7 CHAPTER 8

CHAPTER

Basic Algebraic Operations 1 Equations and Inequalities 75 Graphs 163 Functions 231 Polynomial and Rational Functions 359 Exponential and Logarithmic Functions 453 Additional Topics in Analytic Geometry 529 Systems of Equations and Inequalities; Matrices 575 Sequences, Induction, and Probability 711 Appendix A Special Topics A-1 Appendix B Geometric Formulas A-21 Student Answers SA-1 Subject Index I-1

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Contents Preface ix Features xvi Applications Index xxi

CHAPTER R-1 R-2 R-3 R-4 R-5 R-6

Basic Algebraic Operations 1 Algebra and Real Numbers 2 Exponents 13 Radicals 27 Polynomials: Basic Operations 36 Polynomials: Factoring 47 Rational Expressions: Basic Operations 58 Chapter R Review 68 Chapter R Review Exercises 70 Chapter R Group Activity: Rational and Irrational Numbers 74

CHAPTER 1-1 1-2 1-3 1-4 1-5 1-6

1

Equations and Inequalities 75 Linear Equations and Applications 76 Linear Inequalities 92 Absolute Value in Equations and Inequalities 103 Complex Numbers 115 Quadratic Equations and Applications 128 Additional Equation-Solving Techniques 145 Chapter 1 Review 155 Chapter 1 Review Exercises 157 Chapter 1 Group Activity: Solving a Cubic Equation 159

CHAPTER 2-1 2-2 2-3 2-4

R

2

Graphs 163 Cartesian Coordinate System 164 Distance in the Plane 180 Equation of a Line 192 Linear Equations and Models 210 Chapter 2 Review 225 Chapter 2 Review Exercises 226 Chapter 2 Group Activity: Rates of Change 228

CHAPTER 3-1 3-2 3-3 3-4 3-5 3-6

3

Functions 231 Functions 232 Graphing Functions 250 Transformations of Functions 267 Quadratic Functions 284 Operations on Functions; Composition 310 Inverse Functions 325 Chapter 3 Review 345 Chapter 3 Review Exercises 348 Chapter 3 Group Activity: Mathematical Modeling: Choosing a LongDistance Calling Plan 353

Cumulative Review Exercises Chapters 1–3 354

CHAPTER 4-1 4-2 4-3 4-4 4-5

Polynomial and Rational Functions 359 Polynomial Functions and Models 361 Real Zeros and Polynomial Inequalities 385 Complex Zeros and Rational Zeros of Polynomials 399 Rational Functions and Inequalities 413 Variation and Modeling 436 Chapter 4 Review 444 Chapter 4 Review Exercises 446 Chapter 4 Group Activity: Interpolating Polynomials 450

CHAPTER 5-1 5-2 5-3 5-4 5-5

4

5

Exponential and Logarithmic Functions 453 Exponential Functions 454 Exponential Models 470 Logarithmic Functions 487 Logarithmic Models 502 Exponential and Logarithmic Equations 511 Chapter 5 Review 519 Chapter 5 Review Exercises 520 Chapter 5 Group Activity: Comparing Regression Models 523

Cumulative Review Exercises Chapters 4 and 5 525

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CHAPTER 6-1 6-2 6-3

Additional Topics in Analytic Geometry 529 Conic Sections; Parabola 530 Ellipse 541 Hyperbola 553 Chapter 6 Review 569 Chapter 6 Review Exercises 571 Chapter 6 Group Activity: Focal Chords 573

CHAPTER 7-1 7-2 7-3 7-4 7-5 7-6 7-7 7-8

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7

Systems of Equations and Inequalities; Matrices 575 Systems of Linear Equations in Two Variables 576 Systems of Linear Equations in Three Variables 594 Systems of Linear Equations: Gauss–Jordan Elimination 607 Matrix Operations 626 Systems of Linear Equations: Matrix Inverse Methods 644 Systems of Nonlinear Equations 667 Systems of Linear Inequalities in Two Variables 677 Linear Programming 690 Chapter 7 Review 702 Chapter 7 Review Exercises 706 Chapter 7 Group Activity: Modeling with Systems of Linear Equations 709

CHAPTER 8-1 8-2 8-3 8-4 8-5 8-6

8

Sequences, Induction, and Probability 711 Sequences and Series 712 Mathematical Induction 722 Arithmetic and Geometric Sequences 731 Multiplication Principle, Permutations, and Combinations 746 Sample Spaces and Probability 760 Binomial Formula 779 Chapter 8 Review 787 Chapter 8 Review Exercises 789 Chapter 8 Group Activity: Sequences Specified by Recursion Formulas 791

Cumulative Review Exercises Chapters 6–8 792

Appendix A Special Topics A-1 Scientific Notation and Significant Digits A-1 Partial Fractions A-2 Parametric Equations A-3 Appendix B Geometric Formulas A-21 Student Answers SA-1 Subject Index I-1

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Preface Enhancing a Tradition of Success This eighth edition of College Algebra represents a substantial step forward in student accessibility. Every aspect of the revision of this classic text focuses on making the text more accessible to students, while retaining the precise presentation of the mathematics for which the Barnett name is renowned. Specifically, we concentrated on the areas of writing, exercises, worked examples, design, and technology. Based on numerous reviews, advice from expert consultants, and direct correspondence with the many users of previous editions, we feel that this edition is more relevant and accessible than ever before. We hope you will agree. Writing Without sacrificing breadth or depth or coverage, we have rewritten explanations to make them clearer and more direct. As in previous editions, the text emphasizes computational skills, essential ideas, and problem solving rather than theory. Exercises With an eye to improving student performance and to make the book more useful for instructors, we have extensively revised the exercise sets. We added exercises at the easy to moderate level and expanded the variety of problem types to ensure a gradual increase in difficulty level throughout each exercise set. Examples In the new edition, we added expanded color annotations to many examples, explaining the solution steps in words. Each example is then followed by a similar matched problem for the student to solve. Answers to the matched problems are located at the end of each section for easy reference. This active involvement in learning while reading helps students develop a more thorough understanding of concepts and processes. Technology Instructors who use technology to teach college algebra, whether it be exploring mathematics with a graphing calculator or assigning homework and quizzes online, will find the eighth edition to be much improved. New “Technology Connections” boxes included at appropriate points in the text illustrate how problems previously introduced in an algebraic context may be solved using a graphing calculator. Exercise sets include calculator-based exercises marked with a calculator icon. Note, however, that the use of graphing technology is completely optional with this text. We understand that at many colleges a single text must serve the purposes of teachers with widely divergent views on the proper use of graphing and scientific calculators in college algebra, and this edition is quite flexible regarding the degree of calculator integration. See page xii for more information on McGraw-Hill’s MathZone, the Web-based homework and course management system. The publisher has greatly increased the amount and variety of algorithmic exercises and other assignable content for this edition of the text.

A Central Theme In the Barnett series, the function concept serves as a unifying theme. A brief look at the table of contents reveals this emphasis. A major objective of this book is the development of a library of elementary functions, including their important properties

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and uses. Employing this library as a basic working tool, students will be able to proceed through this book with greater confidence and understanding. Reflecting trends in the way college algebra is taught, the eighth edition emphasizes modeling more strongly than previous editions. In more examples and problems, functions are used to model the real world. In some cases, data are provided, and the student is asked to produce an approximate corresponding function using regression on a graphing calculator (in this edition as in past editions, use of a graphing calculator is optional, and such examples and exercises are easily omitted without loss of continuity).

Design: A New Book with a New Look In short, the eighth edition of College Algebra presents the subject in the precise and straightforward way that users have come to rely on in Barnett textbooks, now updated for students in the twenty-first century. The changes to the text are manifested visually in a new design. We think the pages of this edition offer a more contemporary and inviting visual backdrop for the mathematics. We are confident that the design and other changes to the text described here will help to improve your students’ enjoyment of and success in the course.

Features New to the Eighth Edition More balanced exercise sets give instructors maximum flexibility in assigning homework. We added exercises at the easy to moderate level and expanded the variety of problem types to ensure a gradual increase in difficulty level throughout each exercise set (the A, B, and C designations for the level of problems are now removed from the student edition, though instructors will see them in the Annotated Instructor’s Edition). Extensive annotation of algebraic steps in solutions to example problems will help students to follow the authors’ reasoning and to improve their own problemsolving strategies. A greater emphasis on mathematical modeling is evident in new section titles such as “Linear Equations and Models” and “Exponential Models.” These titles reflect a focus on the relationship between functions and real-world phenomena, especially in examples and exercises. Modeling problems vary from those where only the function model is given (e.g., when the model is a physical law such as F  ma), through problems where a table of data and the function are provided, to cases where the student is asked to approximate a function from data using the regression function of a calculator or computer. A new full-color design gives the book a more contemporary feel and will appeal to students who are accustomed to high production values in books, magazines, and nonprint media. New Technology Connections boxes employ graphing calculators to show graphical and numerical alternatives to pencil-and-paper symbolic methods for problem solving—but the algebraic methods are not omitted. Screen shots are from the TI-84 Plus calculator, but the Technology Connections will interest users of any automated graphing utility. As in previous editions, the use of graphing calculators with College Algebra, eighth edition, is optional.

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The division of exercise sets into A (routine, easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) is no longer explicit in the student edition of the text: The letter designations appear only in the Annotated Instructor’s Edition. This change was made in order to avoid fueling students’ anxiety about challenging exercises. As in previous editions, students at all levels can be challenged by the exercises in this text.

Features Retained Matched problems following worked examples encourage students to practice problem solving immediately after reading through a solution. Answers to the matched problems are located at the end of each section for easy reference. This book gives the student substantial experience in modeling and solving applied problems. Over 500 application exercises help convince even the most skeptical student that mathematics is relevant to life outside the classroom. The most difficult application problems are marked with two stars (★★), the moderately difficult application problems with one star (★), and the easier application problems are not marked. An Applications Index is included following the Guided Tour to help locate particular applications. Interspersed throughout each section, Explore-Discuss boxes foster conceptual understanding by asking students to think about a relationship or process before a result is stated. Verbalization of mathematical concepts, results, and processes is strongly encouraged in these explanations and activities. Many Explore-Discuss boxes are appropriate for group work. Think boxes (color dashed boxes) are used to enclose steps that, with some experience, many students will be able to perform mentally. A Group Activity is located at the end of each chapter and involves many of the concepts discussed in that chapter. These activities require students to discuss and write about mathematical concepts in a complex, real-world context. Most exercise sets include calculator-based exercises that are clearly marked with a calculator icon. These exercises may use real or realistic data, making them computationally heavy, or they may employ the calculator to explore mathematics in a way that would be impractical with paper and pencil. As many students will use this book to prepare for a calculus course, examples and exercises that are especially pertinent to calculus are marked with an icon.

Content Changes Chapter R, “Basic Algebraic Operations,” is reorganized to achieve a more natural flow of topics. Integer exponents and rational exponents are now covered in a single section (R-2). Exponents and radicals now precede the sections on polynomials. Chapter 1, “Equations and Inequalities,” also is streamlined. Two sections of the seventh edition are eliminated by moving topics to more natural locations later in the book. Section 7-1 now provides full treatment of systems of linear equations and their applications, eliminating some redundancy in the previous edition. Polynomial and rational inequalities are treated in Chapter 4 on polynomial and rational functions, after the discussion of zeros of polynomials.

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Chapter 2 “Graphs,” and Chapter 3, “Functions,” are separate chapters in the eighth edition, placing greater emphasis on the distinction between graphs of equations and graphs of functions. A new section (2-4) presents a thorough treatment of linear models, with the optional use of linear regression on a graphing calculator. In a new separate section on quadratic functions (3-4) the authors stress mathematical modeling with quadratic regression (use of a graphing calculator is optional, as always). Chapter 4, “Polynomial and Rational Functions,” is reorganized. Techniques for approximating the easily visualized real zeros of polynomials with real coefficients now precede the more technical algebraic methods for finding complex and rational zeros. The section on partial fractions has been moved to the appendix and replaced with a new section on direct, inverse, joint, and combined variation. This new section supports the book’s emphasis on mathematical modeling. Chapter 5, “Exponential and Logarithmic Functions,” also better emphasizes mathematical modeling. Separate sections, 5-2 and 5-4, treat exponential and logarithmic models, respectively. Chapter 6, “Additional Topics in Analytic Geometry,” is somewhat less technical and theoretical, with several of the lengthier derivations of formulas moved to exercises. Chapter 7, “Systems of Equations and Inequalities; Matrices,” combines two chapters from the seventh edition. A new section (7-2) treats systems of three linear equations in three variables, with applications, before the general Gauss–Jordan algorithm is introduced. Three sections from the seventh edition on determinants and Cramer’s rule do not appear in the new edition.

Supplements MathZone McGraw-Hill’s MathZone is an electronic homework and course management system that is designed for greater flexibility, power, and ease of use than any other system. Whether you are looking for a “ready-to-use, straight-out-of-the-box” system or one you can customize to fit your specific course needs, MathZone is your smart solution for • Assigning and grading homework specific to this textbook or other McGraw-Hill titles. • Creating your study plan using the ALEKS® assessment results. • Providing unlimited practice for students. Students and instructors can go to www.mathzone.com to learn more and register! ALEKS® ALEKS® (Assessment and Learning in Knowledge Spaces) is an inexpensive, customizable, Web-based system that utilizes artificial intelligence to assess your students’ current state of mathematical knowledge and provide individualized assessment and learning. ALEKS has a proven record of success: Over the past seven years, more than 1000 schools implemented ALEKS in their classrooms to help over 1 million students reach their full potential in mathematics. For students, ALEKS can • Determine current math knowledge. • Teach them what they are most ready to learn. • Analyze answers to problems and respond with specific advice when mistakes are made.

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• Provide text-specific assets, such as videos, multimedia tutorials, and textbook PDF pages. • Provide value with no setup or site license fees. For instructors, ALEKS can • Save time by providing comprehensive reporting of individual student and classwide assessment. • Make your job simpler by providing access utilizing a single sign-on to ALEKS® accounts via recent versions of WebCT and Blackboard so you can automatically import results directly into your gradebook. Students and instructors can go to www.ALEKS.com to learn more and to register! Student’s Solutions Manual Prepared by Fred Safier of City College of San Francisco, the Student’s Solutions Manual provides complete worked-out solutions to oddnumbered exercises from the text. The procedures followed in the solutions in the manual match exactly those shown in worked examples in the text. Video Lectures on Digital Video Disk (DVD) The video series is based on exercises from the textbook. J. D. Herdlick of St. Louis Community Colleges introduces essential definitions, theorems, formulas, and problem-solving procedures. Professor Herdlick then works through selected problems from the textbook, following the solution methodologies employed by the authors. The video series is available on DVD or online as part of MathZone. The DVDs are closed-captioned for the hearing impaired, subtitled in Spanish, and meet the Americans with Disabilities Act Standards for Accessible Design. NetTutor Available through MathZone, NetTutor is a revolutionary system that enables students to interact with a live tutor over the web. NetTutor’s Web-based, graphical chat capabilities enable students and tutors to use mathematical notation and even to draw graphs as they work through a problem together. Students can also submit questions and receive answers, browse previously answered questions, and view previous sessions. Tutors are familiar with the textbook’s objectives and problemsolving styles. Instructor’s Testing and Resource CD This cross-platform CD-ROM provides a wealth of resources for the instructor. Supplements featured on this CD-ROM include a computerized test bank utilizing Brownstone Diploma® testing software to quickly create customized exams. This user-friendly program allows instructors to search for questions by topic, format, or difficulty level; edit existing questions or add new ones; and scramble questions and answer keys for multiple versions of the same test. Instructor’s Solutions Manual Prepared by Fred Safier of City College of San Francisco, this supplement provides detailed solutions to exercises in the text. The methods used to solve the problems in the manual are the same as those used to solve the examples in the textbook.

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Acknowledgments In addition to the authors, many others are involved in the successful publication of a book. We wish to thank personally the following people who reviewed the eighth edition manuscript and offered invaluable advice for improvements: Marwan Abu-Sawwa, Florida Community College at Jacksonville Gerardo Aladro, Florida International University Eugene Allevato, Woodbury University Joy Becker, University of Wisconsin–Stout Susan Bradley, Angelina College Ellen Brook, Cuyahoga Community College, Eastern Campus Kelly Brooks, Pierce College Denise Brown, Collin County Community College Cheryl Davids, Central Carolina Technical College Timothy Delworth, Purdue University Marcial Echenique, Broward Community College Gay Ellis, Missouri State University Jackie English, Northern Oklahoma College Enid Mike Everett, Santa Ana College Nicki Feldman, Pulaski Technical College James Fightmaster, Virginia Western Community College Perry Gillespie, University of North Carolina at Fayetteville Vanetta Grier-Felix, Seminole Community College David Gurney, Southeastern Louisiana University Celeste Hernandez, Richland College Fredrick Hoffman, Florida Atlantic University Syed Hossain, University of Nebraska at Kearney Glenn Jablonski, Triton College Sarah Jackson, Pratt Community College Charles Johnson, South Georgia College Larry Johnson, Metropolitan State College of Denver Cheryl Kane, University of Nebraska Lincoln Raja Khoury, Collin County Community College Betty Larson, South Dakota State University Owen Mertens, Southwest Missouri State University Dana Nimic, Southeast Community College, Lincoln Campus Lyn Noble, Florida Community College at Jacksonville Luke Papademas, DeVry University, DeVry Chicago Campus David Phillips, Georgia State University Margaret Rosen, Mercer County Community College Patty Schovanec, Texas Tech University Eleanor Storey, Front Range Community College, Westminster Campus Linda Sundbye, Metropolitan State College of Denver Cynthia Woodburn, Pittsburg State University Martha Zimmerman, University of Louisville We also wish to thank Hossein Hamedani for providing a careful and thorough check of all the mathematical calculations in the book (a tedious but extremely important job).

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Fred Safier for developing the supplemental manuals that are so important to the success of a text. Jeanne Wallace for accurately and efficiently producing most of the manuals that supplement the text. Mitchel Levy (Broward Community College) for scrutinizing our exercises in the manuscript and making recommendations that helped us to build balanced exercise sets. Tony Palermino for providing excellent guidance in making the writing more direct and accessible to students. Pat Steele for carefully editing and correcting the manuscript. Amy Gembala for organizing the revision process, determining the objectives for the new edition, and supervising the preparation of the manuscript. Peggy Selle for guiding the book smoothly through all publication details. All the people at McGraw-Hill who contributed their efforts to the production of this book, especially Steve Stembridge. Producing this new edition with the help of all these extremely competent people has been a most satisfying experience.

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Features Examples and Matched Problems Integrated throughout the text, completely worked examples and practice problems are used to introduce concepts and demonstrate problem-solving techniques—algebraic, graphical, and numerical. Each example is followed by a similar Matched Problem for the student to work EXAMPLE 2 Using the Distance Formula through while reading the Find the distance between the points (3, 5) and (2, 8).* material. Answers to the matched problems are SOLUTION Let (x , y )  (ⴚ3, 5) and (x , y )  (ⴚ2, ⴚ8). Then, located at the end of d  2 [(ⴚ2)  (ⴚ3)]  [(ⴚ8)  5 ] each section for easy  2(2  3)  (8  5)  21  (13)  21  169  2170 reference. This active Notice that if we choose (x , y )  (2, 8) and (x , y )  (3, 5), then involvement in the learnd  2 [(3)  (2) ]  [5  (8)]  21  169  2170 ing process helps students develop a more thorough so it doesn’t matter which point we designate as P or P . understanding of algebraic concepts and MATCHED PROBLEM 2 processes. 1

1

2

2

2

2

2

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2

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Find the distance between the points (6, 3) and (7, 5).

Exploration and Discussion Interspersed at appropriate places in every section, Explore-Discuss boxes encourage students to think critically about mathematics and to explore key concepts in more detail. Verbalization of mathematical concepts, results, and processes is encouraged in these ExploreDiscuss boxes, as well as in ZZZ EXPLORE-DISCUSS 1 some matched problems, and To graph the equation y  x  2x, we use point-by-point plotting to obtain the graph in Figure 5. in problems marked with (A) Do you think this is the correct graph of the equation? If so, why? If y not, why? color numerals in almost (B) Add points on the graph for x  2, 0.5, 0.5, and 2. every exercise set. Explorex y (C) Now, what do you think the graph looks like? Sketch your version of Discuss material can be used 1 1 the graph, adding more points as necessary. 0 0 1 1 in class or in an out-of-class (D) Write a short statement explaining any conclusions you might draw from parts A, B, and C. activity. 3

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Applications One of the primary objectives of this book is to give the student substantial experience in modeling and solving real-world problems. Over 500 application exercises help convince even the most skeptical student that mathematics is relevant to everyday In Problems 65–76, use a graphing utility to approximate to life. The most difficult applitwo decimal places any solutions of the equation in the interval 0 ⱕ x ⱕ 1. None of these equations can be solved exactly cation problems are marked using any step-by-step algebraic process. 65. 2 ⫺ 2x ⫽ 0 66. 3 ⫺ 3x ⫽ 0 with two stars (★★), the 67. x3 ⫺ 1 ⫽ 0 68. x2 ⫺ 1 ⫽ 0 moderately difficult applica69. e ⫺ x ⫽ 0 70. xe ⫺ 1 ⫽ 0 71. xe ⫺ 2 ⫽ 0 72. e ⫺ 2x ⫽ 0 tion problems with one star 73. ln x ⫹ 2x ⫽ 0 74. ln x ⫹ x ⫽ 0 (★), and the easier applica75. ln x ⫹ e ⫽ 0 76. ln x ⫹ x ⫽ 0 tion problems are not APPLICATIONS marked. An Applications 77. How many years, to the nearest year, will it take a sum of money to double if it is invested at 7% Index is included following compounded annually? the features to help locate 78. How many years, to the nearest year, will it take money to quadruple if it is invested at 6% comparticular applications. pounded annually? ⫺x

⫺x

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2x

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COMPOUND INTEREST

COMPOUND INTEREST

0

where A is the amount after t years and A0 is the amount when t ⫽ 0. Estimate the age of a skull uncovered in an archaeological site if 10% of the original amount of carbon-14 is still present. Compute the answer to three significant digits. ★84. ARCHAEOLOGY—CARBON-14

DATING Refer to Problem 83. What is the half-life of carbon-14? That is, how long will it take for half of a sample of carbon-14 to decay? Compute the answer to three significant digits.

★85.

PHOTOGRAPHY An electronic flash unit for a camera is activated when a capacitor is discharged through a filament of wire. After the flash is triggered and the capacitor is discharged, the circuit (see the figure) is connected and the battery pack generates a current to recharge the capacitor. The time it takes for the capacitor to recharge is called the recycle time. For a particular flash unit using a 12-volt battery pack, the charge q, in coulombs, on the capacitor t seconds after recharging has started is given by

q ⫽ 0.0009(1 ⫺ e⫺0.2t ) How many seconds will it take the capacitor to reach a charge of 0.0007 coulomb? Compute the answer to three significant digits.

79. COMPOUND INTEREST At what annual rate compounded continuously will $1,000 have to be invested to amount to $2,500 in 10 years? Compute the answer to three significant digits. 80. COMPOUND INTEREST How many years will it take $5,000 to amount to $8,000 if it is invested at an annual rate of 9% compounded continuously? Compute the answer to three significant digits. 81. WORLD POPULATION A mathematical model for world population growth over short periods is given by P ⫽ P0ert where P is the population after t years, P0 is the population at t ⫽ 0, and the population is assumed to grow continuously at the annual rate r. How many years, to the nearest year, will it take the world population to double if it grows continuously at an annual rate of 1.14%?

R I

C

V

S

★86.

ADVERTISING A company is trying to expose as many people as possible to a new product through television advertising in a large metropolitan area with 2 million possible viewers. A model for the number of people N, in millions, who are aware of the product after t days of advertising was found to be

N ⫽ 2(1 – e⫺0.037t ) How many days, to the nearest day, will the advertising campaign have to last so that 80% of the possible viewers will be aware of the product?

Technology Connections Figure 1 shows the details of constructing the loga-

rithmic model of Example 5 on a graphing calculator. New Technology Connections boxes integrated Technology Connections at appropriate points in the text illustrate how concepts previously introduced in an algebraic context may be approached using a graphing In Problems 93–104, use a graphing calculator to find local extrema, y intercepts, and x intercepts. Investigate the behavior calculator. Students always as x S ⬁ and as x ⫺⬁ and identify and horizontal asymptotes. learn the algebraic methods Round any approximate values to two decimal places. Z Figure 1 93. f(x) ⫽ 2 ⫹ e 94. g(x) ⫽ ⫺3 ⫹ e first so that they develop 95. m(x) ⫽ e 96. n(x) ⫽ e a solid grasp of these 97. s(x) ⫽ e 98. r(x) ⫽ e methods and do not become 100 200 99. F(x) ⫽ 100. G(x) ⫽ 1 ⫹ 3e 1⫹e calculator-dependent. The exercise sets contain calculator-based exercises 102. h(x) ⫽ 3x(2 ) ⫺ 1 101. m(x) ⫽ 2x(3 ) ⫹ 2 that are clearly marked with a calculator icon. The use of technology is 2 ⫹2 3 ⫹3 103. f (x) ⫽ 104. g(x) ⫽ completely optional with this text. All technology features and exercises 2 2 105. Use a graphing calculator to investigate the behavior of may be omitted without sacrificing content coverage. f (x) ⫽ (1 ⫹ x) as x approaches 0. 100

0

120

0

(a) Entering the data

(b) Finding the model

(c) Graphing the data and the model

x⫺2

冟x冟

1⫹x

⫺冟x冟

⫺x2

x2

⫺x

⫺x

⫺x

x

⫺x

⫺x

x

⫺x

1兾x

106. Use a graphing calculator to investigate the behavior of f (x) ⫽ (1 ⫹ x)1兾x as x approaches ⬁.

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Group Activities A Group Activity is located at the end of each chapter and involves many of the concepts discussed in that chapter. These activities strongly encourage the verbalization of mathematical concepts, CHAPTER 5 results, and processes. All of these special activities are highlighted to emphasize ZZZ GROUP ACTIVITY Comparing Regression Models their importance. We have used polynomial, exponential, and logarithmic regression models to fit curves to data sets. How can we determine which equation provides the best fit for a given set of data? There are two principal ways to select models. The first is to use information about the type of data to help make a choice. For example, we expect the weight of a fish to be related to the cube of its length. And we expect most populations to grow exponentially, at least over the short term. The second method for choosing among equations involves developing a measure of how closely an equation fits a given data set. This is best introduced through an example. Consider the data set in Figure 1, where L1 represents the x coordinates and L2 represents the y coordinates. The graph of this data set is shown in Figure 2. Suppose we arbitrarily choose the equation y1  0.6x  1 to model these data (Fig. 3). 10

10

0

10

0

10

0

Z Figure 1

0

Z Figure 3 y1  0.6x  1.

Z Figure 2

To measure how well the graph of y1 fits these data, we examine the difference between the y coordinates in the data set and the corresponding y coordinates on the graph of y1 (L3 in Figs. 4 and 5). 10

0

10

0

Z Figure 5 Here  is L2 and 䊐 is L3.

Z Figure 4

Foundation for Calculus As many students will use this book to prepare for a calculus course, examples and exercises that are especially pertinent to calculus are marked with an icon. EXAMPLE

6

Evaluating and Simplifying a Difference Quotient For f(x)  x2  4x  5, find and simplify: (A) f(x  h)

(B) f(x  h)  f (x)

(C)

f (x  h)  f (x) ,h0 h

SOLUTIONS

(A) To find f (x  h), we replace x with x  h everywhere it appears in the equation that defines f and simplify: f (x ⴙ h)  (x ⴙ h)2  4(x ⴙ h)  5  x2  2xh  h2  4x  4h  5 (B) Using the result of part A, we get f (x ⴙ h)  f (x)  x2 ⴙ 2xh ⴙ h2 ⴙ 4x ⴙ 4h ⴙ 5  (x2 ⴙ 4x ⴙ 5)  x2  2xh  h2  4x  4h  5  x2  4x  5  2xh  h2  4h

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Student Aids

The domain of f is (, 52 ) 傼 (52, ). A rational expression is 0 if and only if the numerator is zero:

Annotation of examples and explanations, in small colored type, is found throughout the text to help students through critical stages. Think Boxes are dashed boxes used to enclose steps that are usually performed mentally.

Screen Boxes are used to highlight important definitions, theorems, results, and step-by-step processes.

4  3x  0 3x  4 x  43

Subtract 4 from both sides. Divide both sides by 3.

The x intercept of f is 43. 4  3(0)

The y intercept is f (0) 

2(0)  5

4  . 5



Z COMPOUND INTEREST If a principal P is invested at an annual rate r compounded m times a year, then the amount A in the account at the end of n compounding periods is given by A  P a1 

r n b m

The annual rate r is expressed in decimal form. Z DEFINITION 1 Increasing, Decreasing, and Constant Functions Let I be an interval in the domain of a function f. Then: 1. f is increasing on I if f(b) f(a) whenever b a in I. 2. f is decreasing on I if f (b)  f(a) whenever b a in I. 3. f is constant on I if f (a)  f (b) for all a and b in I. Z THEOREM 1 Tests for Symmetry

Caution Boxes appear throughout the text to indicate where student errors often occur.

Symmetry with respect to the:

Equation is equivalent when:

y axis

x is replaced with x

x axis

y is replaced with y

Origin

x and y are replaced with x and y

ZZZ

CAUTION ZZZ

1. If f is a function, then the symbol f(x  h) represents the value of f at the number x  h and must be evaluated by replacing the independent variable in the equation that defines f with the expression x  h, as we did in Example 6. Do not confuse this notation with the familiar algebraic notation for multiplication: f (x  h)  fx  fh 4(x  h)  4x  4h

f(x  h) is function notation. 4(x  h) is algebraic multiplication notation.

2. There is another common incorrect interpretation of the symbol f (x  h). If f is an arbitrary function, then f (x  h)  f(x)  f(h) It is possible to find some particular functions for which f(x  h)  f(x)  f(h) is a true statement, but in general these two expressions are not equal.

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Chapter Review sections are provided at the end of each chapter and include a thorough review of all the important terms and symbols. This recap is followed by a comprehensive set of review exercises.

CHAPTER 4-1

4

Review

Polynomial Functions and Models

A function that can be written in the form P(x)  anxn  an1xn1  . . .  a1x  a0, an  0, is a polynomial function of degree n. In this chapter, when not specified otherwise, the coefficients an, an1, . . . , a1, a0 are complex numbers and the domain of P is the set of complex numbers. A number r is said to be a zero (or root) of a function P(x) if P(r)  0. The zeros of P(x) are thus the solutions of the equation P(x)  0. The real zeros of P(x) are just the x intercepts of the graph of P(x). A point on a continuous graph that separates an increasing portion from a decreasing portion, or vice versa, is called a turning point. If P(x) is a polynomial of degree n 7 0 with real coefficients, then the graph of P(x): 1. Is continuous for all real numbers 2. Has no sharp corners 3. Has at most n real zeros 4. Has at most n –1 turning points 5. Increases or decreases without bound as x S  and as x S  The left and right behavior of such a polynomial P(x) is determined by its highest degree or leading term: As x S , both an xn and P(x) approach , with the sign depending on n and the sign of an. Let P(x) be a polynomial of degree n 7 0 and r a complex number. Then we have the following important results: Division Algorithm P(x)  (x  r)Q(x)  R where the quotient Q(x) and remainder R are unique; x – r is the divisor. Remainder Theorem P(r)  R

A Cumulative Review Exercise set is provided after every second or third chapter, for additional reinforcement.

CHAPTERS

4–5

Zeros Theorem P(x) has at most n zeros. Synthetic division is an efficient method for dividing polynomials by linear terms of the form x – r that is well suited to calculator use.

4-2

Real Zeros and Polynomial Inequalities

The following theorems are useful in locating and approximating all real zeros of a polynomial P(x) of degree n 7 0 with real coefficients, an 7 0: Upper and Lower Bound Theorem 1. Upper bound: A number r 7 0 is an upper bound for the real zeros of P(x) if, when P(x) is divided by x – r by synthetic division, all numbers in the quotient row, including the remainder, are nonnegative. 2. Lower bound: A number r 6 0 is a lower bound for the real zeros of P(x) if, when P(x) is divided by x – r by synthetic division, all numbers in the quotient row, including the remainder, alternate in sign. Location Theorem Suppose that a function f is continuous on an interval I that contains numbers a and b. If f (a) and f (b) have opposite signs, then the graph of f has at least one x intercept between a and b. The bisection method uses the location theorem repeatedly to approximate real zeros to any desired accuracy. Polynomial inequalities can be solved by finding the zeros and inspecting the graph of an appropriate polynomial with real coefficients.

Cumulative Review Exercises

Work through all the problems in this cumulative review and check answers in the back of the book. Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that type of problem is discussed. Where weaknesses show up, review appropriate sections in the text. 1. Let P(x) be the polynomial whose graph is shown in the figure. (A) Assuming that P(x) has integer zeros and leading coefficient 1, find the lowest-degree equation that could produce this graph. (B) Describe the left and right behavior of P(x). P(x)

6. Let P(x)  x3  x2  10x  8. Find all rational zeros for P(x). 7. Solve for x. (A) y  10 x

(B) y  ln x

8. Simplify. x 3

(A) (2e )

(B)

e3x e2x

9. Solve for x exactly. Do not use a calculator or a table. (A) log3 x  2 (B) log3 81  x (C) logx 4  2 10. Solve for x to three significant digits. (A) 10 x  2.35 (B) ex  87,500 (C) log x  1.25 (D) ln x  2.75

5

5

Factor Theorem x – r is a factor of P(x) if and only if R  0.

5

In Problems 11 and 12, translate each statement into an equation using k as the constant of proportionality.

x

11. E varies directly as p and inversely as the cube of x. 5

12. F is jointly proportional to q1 and q2 and inversely proportional to the square of r.

2. Match each equation with the graph of f, g, m, or n in the figure. (A) y  (34)x (B) y  (43)x (C) y  (34)x  (43)x (D) y  (43)x  (34)x mn

13. Explain why the graph in the figure is not the graph of a polynomial function. y

3 5

g

4.5

4.5

x

5

f

5

3 5

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CA

APPLICATION INDEX Advertising, 449, 518 Aeronautical engineering, 484, 552 Agriculture, 510, 690, 697–699 AIDS epidemic, 481, 485 Air temperature, 208, 744 Airfreight, 642 Airspeed, 588–589, 592 Alcohol consumption, 308 Animal nutrition, 709 Anthropology, 443 Approximation, 115 Archaeology, 518 Architecture, 144, 191, 228, 249, 307, 448, 552, 567, 796 Astronomy, 509–510, 528, 540, 744 Atmospheric pressure, 512–513, 745 Atomic clock, 23 Average cost, 435 Bacterial growth, 471, 472–473, 484, 744 Biology, 26, 376–377, 443, 484–485, 709, 744 Body surface area, 211–212, 227 Body weight, 222, 344–345 Boiling point of water, 208 Braking distance, 26 Break-even analysis, 158, 297–299, 309, 356, 592 Business, 90, 102, 115, 178, 485, 592, 708, 743, 753–754, 755–756 Business markup policy, 212–213, 222, 345, 352 Car rental, 208, 249, 265 Car sales, 219–220 Carbon-14 dating, 475–476, 518, 522 Card games, 757, 770 Cell division, 744 Chemistry, 90–91, 99–100, 115, 158, 509, 592, 625 Cigarette smoking, 222 Circuit analysis, 666 City planning, 210 Code words, 748–749 Coin problems, 46 Coin toss, 772–773 Committee selection, 755–756, 770–771 Communications, 572 Compound interest, 463–465, 512, 518, 527, 791

Computer science, 26, 261, 266, 353, 357 Computer-generated tests, 748 Construction, 58, 73, 144, 145, 154, 191, 244, 249, 306, 352–353, 356, 383–384, 395–396, 398, 413, 435, 448, 676 Continuous compound interest, 464–465 Cost analysis, 158, 203–204, 209, 221, 227, 641 Cost functions, 248–249, 283 Counting code words, 748–749 Counting serial numbers, 757–758 Cryptography, 660–662, 665, 709 Cylindrical shell volume, 44 Data analysis, 228, 602–604 Delivery charges, 266, 626 Demand, 140 Demographics, 209–210 Depreciation, 222, 227, 357, 486 Design, 154, 158–159, 552, 670–671, 676 Diamond prices, 216–218 Dice rolls, 763–764 Diet, 587–588, 666, 702, 708, 796 Distance-rate-time problems, 84–86, 138–139 Divorce, 384 Dominance relation, 643 Earth science, 26, 90, 102, 484, 594 Earthquakes, 504–505, 509, 516, 522, 527 Ecology, 509 Economics, 26, 73, 90, 102, 743, 791, 796 Economy stimulation, 740–741 Electrical circuits, 73 Electricity, 443 Elliptical design, 548–549, 552 Employee training, 434, 485 Energy, 102 Engineering, 144, 191, 306–307, 443, 540, 552, 572, 744, 796 Epidemics, 477–478 Evaporation, 325 Fabrication, 413 Falling object, 306, 352

XXI

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APPLICATION INDEX

Family of curves, 284 Finance, 356, 469, 486, 522, 592, 606, 744, 796 Flight conditions, 222 Fluid flow, 325 Food chain, 744 Forestry, 221–222, 227 Gaming, 484 Genealogy, 744 Genetics, 443 Geometry, 90, 398, 444, 449, 527, 605–606, 625, 666, 676, 708, 745, 796 Half-life, 473–474 Health care, 384 Heat conduction, 710 Home ownership rates, 507 Hooke’s law, 209 Hydroelectric power, 377–378 Illumination, 443 Income, 352 Income tax, 259–260, 266, 353 Infectious diseases, 479–480 Influenza epidemic, 477–478 Insecticides, 484 Installation charges, 208–209, 249 Insurance company data, 774–775 Interest, 463–465, 512, 518, 527, 791 Inventory value, 642 Investment allocation, 658–660 Labor costs, 637–639, 641–642, 708–709 Law of cooling (Newton), 485, 519 Learning curve, 476–477 Learning theory, 435 Licensed drivers, 223–224 Life expectancy, 528, 606 Life science, 90 Linear depreciation, 227 Lithotripsy, 548–549 Loan repayment, 791 Logistic growth, 477–478 Manufacturing, 154, 249, 384, 398, 449, 689–690, 796 Marine biology, 484–485, 519, 522 Market analysis, 779, 791 Market research, 324, 352 Markup policy, 212–213, 222, 345, 352, 641

Marriage, 384 Maximizing revenue, 308–309 Maximum area, 292–293 Medicare, 522 Medicinal lithotripsy, 548–549 Medicine, 228, 472–473, 477–478, 481, 485, 522, 548 Mixture problems, 87–88, 213–214, 356 Money growth, 469, 522 Music, 91, 443, 444, 745 Naval architecture, 552 Navigation, 144, 222, 562–563 New car sales, 219–220 Newton’s law of cooling, 485, 519 Nuclear power, 486, 568 Numbers, 89, 143, 158, 356, 676, 757–758 Nutrition, 593, 625–626, 643, 690, 709 Oceanography, 209 Officer selection, 753–754 Oil production, 430–431 Olympic games, 224 Operational costs, 250 Ozone levels, 168–169 Packaging, 47, 413 Parabolic reflector, 537–538, 540 Parachutes, 223 Pendulum, 35 Photography, 443, 485, 518, 745 Physics, 144, 179, 209, 249, 357, 442–443, 449, 528, 593, 744 Physiology, 435 Plant nutrition, 690, 702 Political science, 228, 796 Politics, 643 Pollution, 324, 701–702 Population growth, 470–471, 484, 518, 521–522, 527, 606, 743 Present value, 469, 522 Price and demand, 141, 143–144, 178–179, 345, 352, 356, 357 Price and supply, 178–179, 345 Pricing plan, 356 Prize money, 737 Production, 592–593 Production costs, 283 Production scheduling, 600–602, 625, 633, 666, 685–688, 691–694 Profit analysis, 308, 321–322, 383 Profit and loss, 297–299, 357

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CA APPLICATION INDEX

Projectile flight, 306 Projectile motion, 294, A-19 Psychology, 91, 102, 443, 690, 701 Purchasing, 620–623, 701, 796 Puzzle, 91, 144, 709, 745 Quality control, 791 Quantity-rate-time problems, 86–87, 139–140

XXIII

Sports, 191 Sports medicine, 158, 228 Spring stretch, 179 State income tax, 266, 353 Statistics, 115 Stopping distance, 300–301, 307–308, 345, 352, 357 Storage, 413 Subcommittee selection, 755–756 Supply and demand, 224, 589–591, 593

Radioactive decay, 473–474 Radioactive tracers, 484 Rate of descent, 223 Rate-time problems, 83–87, 91, 158, 249, 356, 592 Real estate, 223 Relativistic mass, 35 Replacement time, 435 Resource allocation, 606, 626, 665, 689–690, 701, 708, 709 Retention, 435 Revenue, 309, 335–336, 383 Rocket flight, 505–506

Telephone charges, 266 Temperature, 179, 208 Test averaging, 796 Tests, 748 Timber harvesting, 283 Time measurement, 23 Traffic flow, 710 Training, 434, 485 Transportation, 144–145, 676, 701, 791

Safety research, 283 Sales commissions, 265–266, 631 Serial numbers, 757–758 Service charges, 266 Shipping, 357, 398, 527 Signal light, 540 Significant digits, 115 Simple interest, 449 Smoking, 222 Sociology, 626, 690, 702 Sound, 503, 509, 522, 527 Space science, 443, 484, 540, 568, 572 Space vehicles, 509 Speed of sound, 222

Velocity, 319–320 Volume of a cylindrical shell, 44

Underwater pressure, 215–216

Weather, 250 Weather balloon, 324 Weight of fish, 376–377 Wildlife management, 485, 522 Women in the workforce, 449 Work, 449 Workforce, 449 Zeno’s paradox, 745

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Page 1

Basic Algebraic Operations

ALGEBRA is “generalized arithmetic.” In arithmetic we add, subtract, multiply, and divide specific numbers. In algebra we use all that we know about arithmetic, but, in addition, we work with symbols that represent one or more numbers. In this chapter we review some important basic algebraic operations usually studied in earlier courses.

R

SECTIONS R-1

Algebra and Real Numbers

R-2

Exponents

R-3

Radicals

R-4

Polynomials: Basic Operations

R-5

Polynomials: Factoring

R-6

Rational Expressions: Basic Operations Chapter R Review Chapter R Group Activity: Rational and Irrational Numbers

CHAPTER

C

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BASIC ALGEBRAIC OPERATIONS

Algebra and Real Numbers Z Sets Z The Set of Real Numbers Z The Real Number Line Z Addition and Multiplication of Real Numbers Z Further Operations and Properties

3 The numbers 14, 3, 0, 75, 12, and 1 6 are all examples of real numbers. Because the symbols we use in algebra often stand for real numbers, we will discuss important properties of the real number system. We first introduce some useful notions about sets.

Z Sets Georg Cantor (1845–1918) developed a theory of sets as an outgrowth of his studies on infinity. His work has become a milestone in the development of mathematics. Our use of the word “set” will not differ appreciably from the way it is used in everyday language. Words such as “set,” “collection,” “bunch,” and “flock” all convey the same idea. Thus, we think of a set as a collection of objects with the important property that we can tell whether any given object is or is not in the set. Each object in a set is called an element, or member, of the set. Symbolically, aA aA

means means

“a is an element of set A” “a is not an element of set A”

3  51, 3, 56

2  51, 3, 56

Capital letters are often used to represent sets and lowercase letters to represent elements of a set. A set is finite if the number of elements in the set can be counted and infinite if there is no end in counting its elements. A set is empty if it contains no elements. The empty set is also called the null set and is denoted by . It is important to observe that the empty set is not written as 56. A set is usually described in one of two ways—by listing the elements between braces, { }, or by enclosing within braces a rule that determines its elements. For example, if D is the set of all numbers x such that x2  4, then using the listing method we write D  52, 26

Listing method

or, using the rule method we write D  5x ƒ x2  46

Rule method

The notation used in the rule method is sometimes called set-builder notation; the vertical bar ƒ represents “such that,” and 5x ƒ x2  46 is read, “The set of all x such that x2  4.”

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S E C T I O N R–1

Algebra and Real Numbers

3

The letter x introduced in the rule method is a variable. In general, a variable is a symbol that is used as a placeholder for the elements of a set with two or more elements. This set is called the replacement set for the variable. A constant, on the other hand, is a symbol that names exactly one object. The symbol “8” is a constant, since it always names the number eight. If each element of set A is also an element of set B, we say that A is a subset of set B, and we write A ( B

51, 56 ( 51, 3, 56

Note that the definition of a subset allows a set to be a subset of itself. Since the empty set  has no elements, every element of  is also an element of any given set. Thus, the empty set is a subset of every set. For example,  ( 51, 3, 56

and

 ( 52, 4, 66

If two sets A and B have exactly the same elements, the sets are said to be equal, and we write AB

54, 2, 66  56, 4, 26

Notice that the order of listing elements in a set does not matter. We can now begin our discussion of the real number system. Additional set concepts will be introduced as needed.

Z The Set of Real Numbers The real number system is the number system you have used most of your life. Informally, a real number is any number that has a decimal representation. Table 1 describes the set of real numbers and some of its important subsets. Figure 1 on page 4 illustrates how these sets of numbers are related to each other. Table 1 The Set of Real Numbers Symbol

Name

Description

Examples

N

Natural numbers

Counting numbers (also called positive integers)

1, 2, 3, . . .

Z

Integers

Natural numbers, their negatives, and 0 (also called whole numbers)

. . . , 2, 1, 0, 1, 2, . . .

Q

Rational numbers

Numbers that can be represented as ab, where a and b are integers and b  0; decimal representations are repeating or terminating

I

Irrational numbers

Numbers that can be represented as nonrepeating and nonterminating decimal numbers

R

Real numbers

Rational numbers and irrational numbers

*The overbar indicates that the number (or block of numbers) repeats indefinitely. †Note that the ellipsis does not indicate that a number (or block of numbers) repeats indefinitely.

2 4, 0, 1, 25, 3 5 , 3 , 3.67, 0.333,* 5.272727

3 12, , 1 7, 1.414213 . . . ,† 2.71828182 . . .†

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BASIC ALGEBRAIC OPERATIONS

Z Figure 1

Real numbers (R )

Real numbers and important subsets.

NZQR Rational numbers (Q ) Irrational numbers (I )

Integers (Z )

Natural numbers (N )

Zero

Noninteger ratios of integers

Negatives of natural numbers

Z The Real Number Line A one-to-one correspondence exists between the set of real numbers and the set of points on a line. That is, each real number corresponds to exactly one point, and each point to exactly one real number. A line with a real number associated with each point, and vice versa, as in Figure 2, is called a real number line, or simply a real line. Each number associated with a point is called the coordinate of the point. The point with coordinate 0 is called the origin. The arrow on the right end of the line indicates a positive direction. The coordinates of all points to the right of the origin are called positive real numbers, and those to the left of the origin are called negative real numbers. The real number 0 is neither positive nor negative. Z Figure 2

27

A real number line. 10

5

4  3 Origin

␲

0

7.64

5

10

Z Addition and Multiplication of Real Numbers How do you add or multiply two real numbers that have nonrepeating and nonterminating decimal expansions? The answer to this difficult question relies on a solid understanding of the arithmetic of rational numbers. The rational numbers are numbers that can be written in the form ab, where a and b are integers and b  0 (see Table 1 on page 3). The numbers 75 and 23 are rational, and any integer a is rational because it can be written in the form a1. Two rational numbers ab and cd are equal if ad  bc; for example, 3510  72. Recall how the sum and product of rational numbers are defined. Z DEFINITION 1 Addition and Multiplication of Rationals For rational numbers ab and cd, where a, b, c, and d are integers and b  0, d  0: Addition: Multiplication:

a  b a  b

c ad  bc  d bd c ac  d bd

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S E C T I O N R–1

Algebra and Real Numbers

5

Addition and multiplication of rational numbers are commutative; changing the order in which two numbers are added or multiplied does not change the result. 3 5 5 3    2 7 7 2 3 5 5 3    2 7 7 2

Addition is commutative.

Multiplication is commutative.

Addition and multiplication of rational numbers is also associative; changing the grouping of three numbers that are added or multiplied does not change the result: 3 5 a  2 7 3 5 a  2 7

9 3 5 9 ba  b 4 2 7 4 9 3 5 9 ba  b 4 2 7 4

Addition is associative.

Multiplication is associative.

Furthermore, the operations of addition and multiplication are related in that multiplication distributes over addition: 3 5 9 3 5 3 9 a  b    2 7 4 2 7 2 4 5 9 3 5 3 9 3 a  b     7 4 2 7 2 4 2

Left distributive law

Right distributive law

The rational number 0 is an additive identity; adding 0 to a number does not change it. The rational number 1 is a multiplicative identity; multiplying a number by 1 does not change it. Every rational number r has an additive inverse, denoted r; the additive inverse of 45 is 45, and the additive inverse of 32 is 32. The sum of a number and its additive inverse is 0. Every nonzero rational number r has a multiplicative inverse, denoted r1; the multiplicative inverse of 45 is 54, and the multiplicative inverse of 32 is 23. The product of a number and its multiplicative inverse is 1. The rational number 0 has no multiplicative inverse.

EXAMPLE

1

Arithmetic of Rational Numbers Perform the indicated operations. (A)

1 6  3 5

(C) (179)1

(B)

8 5  3 4

(D) (6  92)1

SOLUTION

(A)

1 6 5  18 23    3 5 15 15

(B)

40 10 8 5    3 4 12 3

40 10  12 3

because

40  3  12  10

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(C) (179)1  917 (D) (6  9 2)1  a

6 9 1 12  9 1 3 1 2  b a b a b  1 2 2 2 3

MATCHED PROBLEM



1*

Perform the indicated operations. (A) (5 2  7 3) (C)

21 15  20 14

(B) (8 17)1 (D) 5  (1 2  1 3)



Rational numbers have decimal expansions that are repeating or terminating. For example, by long division, 2  0.666 3 22  3.142857 7 13  1.625 8

The number 6 repeats indefinitely.

The block 142857 repeats indefinitely.

Terminating expansion

Conversely, any decimal expansion that is repeating or terminating represents a rational number (see Problems 69 and 70 in Exercise R-1). The number 12 is irrational because it cannot be written in the form ab, where a and b are integers, b  0 (for an explanation, see Problem 83 in Section R-5 or the Chapter R Group Activity). Similarly, 13 is irrational. But 14, which is equal to 2, is a rational number. In fact, if n is a positive integer, then 1n is irrational unless n belongs to the sequence of perfect squares 1, 4, 9, 16, 25, . . . (see Problem 84 in Section R-5). We now return to our original question: how do you add or multiply two real numbers that have nonrepeating and nonterminating decimal expansions? Although we will not give a detailed answer to this question, the key idea is that every real number can be approximated to any desired precision by rational numbers. For example, the irrational number 12  1.414 213 562 . . . is approximated by the rational numbers 14  1.4 10 141  1.41 100 *Answers to matched problems in a given section are found near the end of the section, before the exercise set.

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Algebra and Real Numbers

7

1,414  1.414 1,000 14,142  1.4142 10,000 141,421  1.41421 100,000 .. . Using the idea of approximation by rational numbers, we can extend the definitions of rational number operations to include real number operations. The following box summarizes the basic properties of real number operations.

Z BASIC PROPERTIES OF THE SET OF REAL NUMBERS Let R be the set of real numbers, and let x, y, and z be arbitrary elements of R. Addition Properties Closure:

x  y is a unique element in R.

Associative:

(x  y)  z  x  (y  z)

Commutative:

xyyx

Identity:

0 is the additive identity; that is, 0  x  x  0  x for all x in R, and 0 is the only element in R with this property.

Inverse:

For each x in R, x is its unique additive inverse; that is, x  (x)  (x)  x  0, and x is the only element in R relative to x with this property.

Multiplication Properties Closure:

xy is a unique element in R.

Associative:

(xy)z  x( yz)

Commutative:

xy  yx

Identity:

1 is the multiplicative identity; that is, for all x in R, (1)x  x(1)  x, and 1 is the only element in R with this property.

Inverse:

For each x in R, x  0, x1 is its unique multiplicative inverse; that is, xx1  x1x  1, and x1 is the only element in R relative to x with this property.

Combined Property Distributive:

x( y  z)  xy  xz

(x  y)z  xz  yz

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2

Using Real Number Properties Which real number property justifies the indicated statement? Statement (A) (7x)y  7(xy) (B) a(b  c)  (b  c)a (C) (2x  3y)  5y  2x  (3y  5y) (D) (x  y)(a  b)  (x  y)a  (x  y)b (E) If a  b  0, then b  a.

MATCHED PROBLEM

Property Illustrated Associative () Commutative () Associative () Distributive Inverse ()



2

Which real number property justifies the indicated statement? (A) 4  (2  x)  (4  2)  x

(B) (a  b)  c  c  (a  b)

(C) 3x  7x  (3  7)x

(D) (2x  3y)  0  2x  3y

(E) If ab  1, then b  1a.



Z Further Operations and Properties Subtraction of real numbers can be defined in terms of addition and the additive inverse. If a and b are real numbers, then a  b is defined to be a  (b). Similarly, division can be defined in terms of multiplication and the multiplicative inverse. If a and b are real numbers and b  0, then a b (also denoted ab) is defined to be a  b1.

Z DEFINITION 2 Subtraction and Division of Real Numbers For all real numbers a and b: Subtraction:

a  b  a  (b)

Division:

a b  a  b1

It is important to remember that Division by 0 is never allowed.

5  3  5 (3)  2

b0

3  2  3  21  3 

1  1.5 2

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ZZZ EXPLORE-DISCUSS

Algebra and Real Numbers

9

1

(A) Give an example that shows that subtraction of real numbers is not commutative. (B) Give an example that shows that division of real numbers is not commutative.

The basic properties of the set of real numbers, together with the definitions of subtraction and division, imply the following properties of negatives and zero.

Z THEOREM 1 Properties of Negatives For all real numbers a and b: (a)  a (a)b  (ab)  a(b)  ab (a)(b)  ab (1)a  a a a a   b0 5. b b b a a a a    6. b0 b b b b 1. 2. 3. 4.

Z THEOREM 2 Zero Properties For all real numbers a and b: 1. a  0  0  a  0 2. ab  0 if and only if*

a  0 or b  0 or both

Note that if b  0, then 0b  0  b1  0 by Theorem 2. In particular, 03  0; but the expressions 30 and 00 are undefined. *Given statements P and Q, “P if and only if Q” stands for both “if P, then Q” and “if Q, then P.”

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3

Using Negative and Zero Properties Which real number property or definition justifies each statement? Statement (A) 3  (2)  3  [(2)]  5 (B) (2)  2 3 3 (C)   2 2 5 5 (D)  2 2 (E) If (x  3)(x  5)  0, then either x  3  0 or x  5  0.

MATCHED PROBLEM

Property or Definition Illustrated Subtraction (Definition 1 and Theorem 1, part 1) Negatives (Theorem 1, part 1) Negatives (Theorem 1, part 6) Negatives (Theorem 1, part 5) Zero (Theorem 2, part 2) 

3

Which real number property or definition justifies each statement? (A)

3 1  3a b 5 5

(B) (5)(2)  (5  2)

(D)

7 7  9 9

(E) If x  5  0, then (x  3)(x  5)  0.

ZZZ

(C) (1)3  3 

EXPLORE-DISCUSS 2

A set of numbers is closed under an operation if performing the operation on numbers of the set always produces another number in the set. For example, the set of odd integers is closed under multiplication, but is not closed under addition. (A) Give an example that shows that the set of irrational numbers is not closed under addition. (B) Explain why the set of irrational numbers is closed under taking multiplicative inverses.

If a and b are real numbers, b  0, the quotient a b, when written in the form ab, is called a fraction. The number a is the numerator, and b is the denominator. It can be shown that fractions satisfy the following properties. (Note that some of these properties, under the restriction that numerators and denominators be integers, were used earlier to define arithmetic operations on the rationals.)

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Algebra and Real Numbers

Z THEOREM 3 Fraction Properties For all real numbers a, b, c, d, and k (division by 0 excluded): 1.

a c  b d 4 6  6 9

2.

if and only if since

4966

ka a  kb b

3.

73 3  75 5

5.

a c ac   b b b 3 4 3 4 7   6 6 6 6

ANSWERS

ad  bc

a c ac   b d bd

4.

3 7 37 21    5 8 58 40

6.

a c ac   b b b 7 2 72 5    8 8 8 8

a c a d   b d b c 2 5 2 7 14     3 7 3 5 15

7.

a c ad  bc   b d bd 2 1 25 31 13   3 5 35 15

TO MATCHED PROBLEMS

1. (A) 296 (B) 17 8 (C) 98 (D) 256 2. (A) Associative () (B) Commutative () (C) Distributive (D) Identity () (E) Inverse () 3. (A) Division (Definition 1) (B) Negatives (Theorem 1, part 2) (C) Negatives (Theorem 1, part 4) (D) Negatives (Theorem 1, part 5) (E) Zero (Theorem 2, part 1)

R-1

Exercises

In Problems 1–12, indicate true (T) or false (F). 1. 4  53, 4, 56

2. 6  52, 4, 66

3. 3  53, 4, 56

4. 7  52, 4, 66

5. 51, 26 ( 51, 3, 56

7. 57, 3, 56 ( 53, 5, 76 9.   5A, B, C6

6. 52, 66 ( 52, 4, 66

8. 57, 3, 56  53, 5, 76

10.  ( 5A, B, C, D6

11. 5A, L, G, E, B, R, A6  5G, A, R, B, L, E6 12. 5A, B, C, D, E6 ( 5A, C, E6

In Problems 13–20, perform the indicated operations, if defined, and express the result as an integer. 13.

3 5 14. a b a b 5 3

4 9  9 4

15. 100 0

16. 0 0

17. 0 (9  8)

18.

19. 0  a100 

1 b 100

4 6 a3  b 7 2

20. 3(0  5) 204 1

11

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In Problems 21–26, perform the indicated operations and express the result in the form ab, where a and b are integers. 4 12 21.  9 5 23. a

4 1  b 100 25

3 1 25. a b  21 8

17 2 22.  8 7

50. Indicate which of the following are true: (A) All integers are natural numbers. (B) All rational numbers are real numbers. (C) All natural numbers are rational numbers.

26. (1  31)

In Problems 27–38, each statement illustrates the use of one of the following properties or definitions. Indicate which one.

27. x  ym  x  my

Inverse () Inverse () Subtraction Division Negatives (Theorem 1) Zero (Theorem 2)

1 31. (2)(2 )1

28. 7(3m)  (7  3)m u u 30.   v v 32. 8  12  8  (12)

33. w  (w)  0

34. 5 (6) 

29. 7u  9u  (7  9)u

1 5(6 )

35. 3(xy  z)  0  3(xy  z) 36. ab(c  d )  abc  abd x x 37.  y y

48. If ab  1, does either a or b have to be 1? 49. Indicate which of the following are true: (A) All natural numbers are integers. (B) All real numbers are irrational. (C) All rational numbers are real numbers.

5 14 1 24. a  b 2 25

Commutative () Commutative () Associative () Associative () Distributive Identity () Identity ()

47. If ab  0, does either a or b have to be 0?

38. (x  y)  0  0

Write each set in Problems 39–44 using the listing method; that is, list the elements between braces. If the set is empty, write . 39. { x | x is an even integer between 3 and 5} 40. { x | x is an odd integer between 4 and 6} 41. { x | x is a letter in “status”} 42. { x | x is a letter in “consensus”} 43. { x | x is a month starting with B} 44. { x | x is a month with 32 days} 45. The set S1  5a6 has only two subsets, S1 and . How many subsets does each of the following sets have? (A) S2  5a, b6 (B) S3  5a, b, c6 (C) S4  5a, b, c, d6 46. Based on the results in Problem 45, how many subsets do you think a set with n elements will have?

51. Give an example of a rational number that is not an integer. 52. Give an example of a real number that is not a rational number. In Problems 53 and 54, list the subset of S consisting of (A) natural numbers, (B) integers, (C) rational numbers, and (D) irrational numbers. 53. S  53, 23, 0, 1, 13, 95, 11446

54. S  515, 1, 12, 2, 17, 6, 1625 9, 6 In Problems 55 and 56, use a calculator* to express each number in decimal form. Classify each decimal number as terminating, repeating, or nonrepeating and nonterminating. Identify the pattern of repeated digits in any repeating decimal numbers. 55. (A) 98

(B) 113

56. (A) 136

(B) 121

(C) 15 (C) 167

(D) 118 29 (D) 111

57. Indicate true (T) or false (F), and for each false statement find real number replacements for a and b that will provide a counterexample. For all real numbers a and b: (A) a  b  b  a (B) a  b  b  a (C) ab  ba (D) a b  b a 58. Indicate true (T) or false (F), and for each false statement find real number replacements for a, b, and c that will provide a counterexample. For all real numbers a, b, and c: (A) (a  b)  c  a  (b  c) (B) (a  b)  c  a  (b  c) (C) a(bc)  (ab)c (D) (a b) c  a (b c)

*Later in the book you will encounter optional exercises that require a graphing calculator. If you have such a calculator, you can certainly use it here. Otherwise, any scientific calculator will be sufficient for the problems in this chapter.

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In Problems 59–66, indicate true (T) or false (F), and for each false statement give a specific counterexample.

69. If c  0.151515 . . . , then 100c  15.1515 . . . and 100c  c  15.1515 . . .  0.151515 . . .

59. The difference of any two natural numbers is a natural number.

99c  15 c  15 99 

60. The quotient of any two nonzero integers is an integer.

63. The product of any two irrational numbers is an irrational number. 64. The product of any two rational numbers is a rational number.

70. Repeat Problem 69 for 0.181818. . . . 71. To see how the distributive property is behind the mechanics of long multiplication, compute each of the following and compare: Long Multiplication 23

12

65. The multiplicative inverse of any irrational number is an irrational number. 66. The multiplicative inverse of any nonzero rational number is a rational number. 67. If A  51, 2, 3, 46 and B  52, 4, 66, find: (A) 5x ƒ x  A or x  B6 (B) 5x ƒ x  A and x  B6

Use of the Distributive Property 23  12  23(2  10)  23  2  23  10 

72. For a and b real numbers, justify each step using a property in this section. Statement 1. (a  b)  (a)  (a)  (a  b) 2.  3(a)  a 4  b 3. 0b 4. b

68. If F  52, 0, 26 and G  51, 0, 1, 26, find: (A) 5x ƒ x  F or x  G6 (B) 5x ƒ x  F and x  G6

R-2

5 33

Proceeding similarly, convert the repeating decimal 0.090909 . . . into a fraction. (All repeating decimals are rational numbers, and all rational numbers have repeating decimal representations.)

61. The sum of any two rational numbers is a rational number. 62. The sum of any two irrational numbers is an irrational number.

13

Exponents

Reason 1. 2. 3. 4.

Exponents Z Integer Exponents Z Roots of Real Numbers Z Rational Exponents Z Scientific Notation

The French philosopher/mathematician René Descartes (1596–1650) is generally credited with the introduction of the very useful exponent notation “x n.” This notation as well as other improvements in algebra may be found in his Geometry, published in 1637. If n is a natural number, x n denotes the product of n factors, each equal to x. In this section the meaning of x n will be expanded to allow the exponent n to be any rational number. Each of the following expressions will then represent a unique real number: 75

54

3.140

612

1453

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Z Integer Exponents Definition 1 generalizes exponent notation to include 0 and negative integer exponents.

n Z DEFINITION 1 a , n an integer and a a real number

1. For n a positive integer: an  a  a  . . .  a

35  3  3  3  3  3

n factors of a

2. For n  0: a0  1 a0 00 is not defined

1320  1

3. For n a negative integer: an 

1

a0

n

a

73



*

1 7



(3)

1 73

Note: It can be shown that for any a  0 and for all integers n an 

EXAMPLE

1

1 an

a5 

1 a5

a(3) 

1 a3

Using the Definition of Integer Exponents Write each part as a decimal fraction or using positive exponents. Assume all variables represent nonzero real numbers. (A) (u3v2)0  1 (C) x8 

1 x8

(B) 103 

(D)

x3 y5

MATCHED PROBLEM

1 1   0.001 3 1,000 10



5 x3 1 1 y  5  3  1 1 y x



y5 x3



1

Write parts (A) – (D) as decimal fractions and parts (E) and (F) with positive exponents. Assume all variables represent nonzero real numbers. (A) 6360 (D)

1 103

(B) (x2)0 (E)

1 x4

(C) 105 (F)

u7 v3

*Throughout the book, dashed boxes—called think boxes—are used to represent steps that may be performed mentally.



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Exponents

15

The basic properties of integer exponents are summarized in Theorem 1. The proof of this theorem involves mathematical induction, which is discussed in Chapter 8.

Z THEOREM 1 Properties of Integer Exponents For n and m integers and a and b real numbers: 1. aman  amn

a5a 7

 a5(7)

2. (an)m  amn 3. (ab)m  ambm a m am 4. a b  m b b

(a3)2

 a(2)3

amn a 5. n  • 1 a anm

 a6

(ab)3  a3b3 a 4 a4 a b  4 b b

b0

a3

m

ZZZ

 a2

a2 a3

a0

a2

 a3(2)  a5 

1 a23



1 a5

EXPLORE-DISCUSS 1

Property 1 in Theorem 1 can be expressed verbally as follows: To find the product of two exponential forms with the same base, add the exponents and use the same base. Express the other properties in Theorem 1 verbally. Decide which you find easier to remember, a formula or a verbal description.

EXAMPLE

2

Using Exponent Properties Simplify using exponent properties, and express answers using positive exponents only.* (A) (3a5)(2a3)

(B)

6x2 8x5



 (3  2)(a5a3)

3x2(5) 4



 6a2

3x3 4

*By “simplify” we mean eliminate common factors from numerators and denominators and reduce to a minimum the number of times a given constant or variable appears in an expression. We ask that answers be expressed using positive exponents only in order to have a definite form for an answer. Later (in this section and elsewhere) we will encounter situations where we will want negative exponents in a final answer.

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(C) 4y3  (4y)3  4y3  (4)3y3

 4y3  (64)y3

 4y3  64y3  60y3

MATCHED PROBLEM



2

Simplify using exponent properties, and express answers using positive exponents only. (A) (5x3)(3x4)

ZZZ

(B)

9y7

(C) 2x4  (2x)4

6y4



CAUTION ZZZ

Be careful when using the relationship an  ab1 

1 ab

ab1 

1  a1  b1 ab

a b

and

1  (a  b)1 ab

1 : an (ab)1 

and

1 ab 1 1   a1  b1 a b

Do not confuse properties 1 and 2 in Theorem 1: a3a4  a34 1a3 2 4  a34

a3a4  a34  a7

property 1, Theorem 1

(a )  a

property 2, Theorem 1

3 4

34

a

12

From the definition of negative exponents and the five properties of exponents, we can easily establish the following properties, which are used very frequently when dealing with exponent forms.

Z THEOREM 2 Further Exponent Properties For a and b any real numbers and m, n, and p any integers (division by 0 excluded): 1. (ambn) p  a pmb pn 3.

an bm m  n b a

2. a

a m p a pm b  pn bn b

a n b n 4. a b  a b a b

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17

We prove properties 1 and 4 in Theorem 2 and leave the proofs of 2 and 3

PROOF

to you. 1. (ambn) p  (am) p(bn) p  a pmb pn n

a 4. a b b

EXAMPLE

3

property 2, Theorem 1

n

a bn bn  n a b n a b a 

property 3, Theorem 1

property 4, Theorem 1

property 3, Theorem 2

property 4, Theorem 1

Using Exponent Properties Simplify using exponent properties, and express answers using positive exponents only. (A) (2a3b2)2  22a6b4  (B) a (C)

a3 2 a6 b10  10  6 5b b b a

4x3y5 4 3

6x y

(D) a



a

or

a3 2 b5 2 b10  a 3b  6 5b b a a

2x3(4) 2x  8 3y 3y3(5)

m3m3 2 b n2

(E) (x  y)3 

a6 4b4

a

m33 2 m0 2  a 2 b 2 b n n

a

n

1 (x  y)3

2

2 b

1



1 n4 

MATCHED PROBLEM

3

Simplify using exponent properties, and express answers using positive exponents only. (A) (3x4y3)2 (D) a

x3 3 b y4y4

(B) a (E)

x2 3 b y4

1 (a  b)2

(C)

6m2n3 15m1n2 

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In simplifying exponent forms there is often more than one sequence of steps that will lead to the same result (see Example 3B). Use whichever sequence of steps makes sense to you.

Z Roots of Real Numbers Perhaps you recall that a square root of a number b is a number c such that c2  b, and a cube root of a number b is a number d such that d 3  b. What are the square roots of 9? 3 is a square root of 9, since 32  9. 3 is a square root of 9, since (3)2  9. Thus, 9 has two real square roots, one the negative of the other. What are the cube roots of 8? 2 is a cube root of 8, since 23  8. And 2 is the only real number with this property. In general:

Z DEFINITION 2 Definition of an nth Root For a natural number n and a and b real numbers: a is an nth root of b if an  b

3 is a fourth root of 81, since 34  81.

How many real square roots of 4 exist? Of 5? Of 9? How many real fourth roots of 5 exist? Of 5? How many real cube roots of 27 are there? Of 27? The following important theorem (which we state without proof ) answers these questions.

Z THEOREM 3 Number of Real nth Roots of a Real Number b* n even b positive

n odd

Two real nth roots

One real nth root

3 and 3 are both fourth roots

2 is the only real cube root of 8.

of 81.

b negative

No real nth root

One real nth root

9 has no real square roots.

2 is the only real cube root of 8.

*In this section we limit our discussion to real roots of real numbers. After the real numbers are extended to the complex numbers (see Section 1-4), additional roots may be considered. For example, it turns out that 1 has three cube roots: in addition to the real number 1, there are two other cube roots of 1 in the complex number system.

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Exponents

19

Thus, 4 and 5 have two real square roots each, and 9 has none. There are two real fourth roots of 5 and none for 5. And 27 and 27 have one real cube root each. What symbols do we use to represent these roots? We turn to this question now.

Z Rational Exponents If all exponent properties are to continue to hold even if some of the exponents are rational numbers, then (513)3  533  5

and

(712)2  722  7

Since Theorem 3 states that the number 5 has one real cube root, it seems reasonable to use the symbol 513 to represent this root. On the other hand, Theorem 3 states that 7 has two real square roots. Which real square root of 7 does 712 represent? We answer this question in the following definition.

1/n Z DEFINITION 3 b , Principal nth Root

For n a natural number and b a real number, b1n is the principal nth root of b defined as follows: 1. If n is even and b is positive, then b1n represents the positive nth root of b. 1612  4 16

12

not 4 and 4. 1612 and (16)12 are not the same.

 4

2. If n is even and b is negative, then b1n does not represent a real number. (More will be said about this case later.) (16)12 is not real.

3. If n is odd, then b1n represents the real nth root of b (there is only one). 3215  2

4. 01n  0

EXAMPLE

4

019  0

(32)1 5  2

016  0

Principal nth Roots (A) 912  3 (B) 912  3 12

(C) (9)

Compare parts (B) and (C).

is not a real number.

(E) (27)13  3

(D) 2713  3 (F) 017  0



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MATCHED PROBLEM

4

Find each of the following: (A) 412

(B) 412

(C) (4)12

(D) 813

(E) (8)13

(F) 018



How should a symbol such as 723 be defined? If the properties of exponents are to hold for rational exponents, then 723  (713)2; that is, 723 must represent the square of the cube root of 7. This leads to the following general definition: mn and bmn, Rational Number Exponent Z DEFINITION 4 b

For m and n natural numbers and b any real number (except b cannot be negative when n is even): bmn  (b1n)m 4

32

12 3

 (4

) 2 8 3

4

32

1

bmn 

and 1

1  32 8 4 

mn

b 3 2

(4)

is not real

(32)35  [(32)15 ] 3  (2)3  8

We have now discussed bmn for all rational numbers m n and real numbers b. It can be shown, though we will not do so, that all five properties of exponents listed in Theorem 1 continue to hold for rational exponents as long as we avoid even roots of negative numbers. With the latter restriction in effect, the following useful relationship is an immediate consequence of the exponent properties:

Z THEOREM 4 Rational Exponent Property For m and n natural numbers and b any real number (except b cannot be negative when n is even): bmn  e

(b1n)m (bm)1n

82  3  e

(813)2 (82)13

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ZZZ

21

Exponents

EXPLORE-DISCUSS 2

Find the contradiction in the following chain of equations: 1  (1)22  3(1)2 4 12  112  1

(1)

Where did we try to use Theorem 4? Why was this not correct?

The three exponential forms in Theorem 4 are equal as long as only real numbers are involved. But if b is negative and n is even, then b1n is not a real number and Theorem 4 does not necessarily hold, as illustrated in Explore-Discuss 2. One way to avoid this difficulty is to assume that m and n have no common factors.

EXAMPLE

5

Using Rational Exponents Simplify, and express answers using positive exponents only. All letters represent positive real numbers. (A) 823  (813)2  22  4

823  (82)13  6413  4

or

(B) (8)53  3(8)13 4 5  (2)5  32 (C) (3x13)(2x12)  6x1312  6x56 (D) a

4x13 12 412x16 2 2   1416  112 12 b 14 x x x x

(E) (u12  2v12)(3u12  v12)  3u  5u12v12  2v

MATCHED PROBLEM



5

Simplify, and express answers using positive exponents only. All letters represent positive real numbers. (A) 932 (E) a

8x12 13 b x23

(B) (27)43

(C) (5y34)(2y13)

(F) (2x12  y12)(x12  3y12)

(D) (2x34y14)4 

Z Scientific Notation Scientific work often involves the use of very large numbers or very small numbers. For example, the average cell contains about 200,000,000,000,000 molecules, and the diameter of an electron is about 0.000 000 000 0004 centimeter. It is generally troublesome to write and work with numbers of this type in standard decimal form. The

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two numbers written here cannot even be entered into most calculators as they are written. However, each can be expressed as the product of a number between 1 and 10 and an integer power of 10: 200,000,000,000,000  2  1014 0.000 000 000 0004  4  1013 In fact any positive number written in decimal form can be expressed in scientific notation, that is, in the form a  10n

EXAMPLE

6

1  a 6 10, n an integer, a in decimal form

Scientific Notation Each number is written in scientific notation: 7 720  6,430  5,350,000 

7  100 7.2  102 6.43  103 5.35  106

0.5 0.08 0.000 32 0.000 000 0738

   

5  101 8  102 3.2  104 7.38  108

Can you discover a rule relating the number of decimal places the decimal point is moved to the power of 10 that is used? 7,320,000

 7.320 000.  106

 7.32  106

6 places left Positive exponent

0.000 000 54

 0.000 000 5.4  107

 5.4  107

7 places right Negative exponent

MATCHED PROBLEM



6

(A) Write each number in scientific notation: 430; 23,000; 345,000,000; 0.3; 0.0031; 0.000 000 683 (B) Write in standard decimal form: 4  103; 5.3  105; 2.53  102; 7.42  106



Most calculators express very large and very small numbers in scientific notation. (Later in the book you will encounter optional exercises that require a graphing calculator. If you have such a calculator, you can certainly use it here. Otherwise, any scientific calculator will be sufficient for the problems in this chapter.) Consult the manual for your calculator to see how numbers in scientific notation are entered in

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Exponents

23

your calculator. Some common methods for displaying scientific notation on a calculator are shown here.

EXAMPLE

7

Number Represented

Typical Scientific Calculator Display

Typical Graphing Calculator Display

5.427 493  1017

5.427493  17

5.427493E17

2.359 779  1012

2.359779 12

2.359779E12

Using Scientific Notation on a Calculator Write each number in scientific notation; then carry out the computations using your calculator. (Refer to the user’s manual accompanying your calculator for the procedure.) Express the answer to three significant digits* in scientific notation. 325,100,000,000 3.251  1011  0.000 000 000 000 0871 8.71  1014  3.732491389E24

Calculator display

 3.73  1024

To three significant digits

Figure 1 shows two solutions to this problem on a graphing calculator. In the first solution we entered the numbers in scientific notation, and in the second we used standard decimal notation. Although the multiple-line screen display on a graphing calculator allows us to enter very long standard decimals, scientific notation is usually more efficient and less prone to errors in data entry. Furthermore, as Figure 1 shows, the calculator uses scientific notation to display the answer, regardless of the manner in which the numbers are entered.  Z Figure 1

MATCHED PROBLEM

7

Repeat Example 7 for: 0.000 000 006 932 62,600,000,000

EXAMPLE

8



Measuring Time with an Atomic Clock An atomic clock that counts the radioactive emissions of cesium is used to provide a precise definition of a second. One second is defined to be the time it takes cesium to emit 9,192,631,770 cycles of radiation. How many of these cycles will occur in 1 hour? Express the answer to five significant digits in scientific notation. SOLUTION

(9,192,631,770)(602)  3.309347437E13  3.3093  1013 *For those not familiar with the meaning of significant digits, see Appendix A for a brief discussion of this concept.



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MATCHED PROBLEM

8

Refer to Example 8. How many of these cycles will occur in 1 year? Express the answer to five significant digits in scientific notation. 

EXAMPLE

9

Evaluating Rational Exponential Forms with a Calculator Evaluate to four significant digits using a calculator. (Refer to the instruction book for your particular calculator to see how exponential forms are evaluated.) (A) 1134

(B) 3.104623

(C) (0.000 000 008 437)311

SOLUTIONS

(A) First change calculator.

3 4

to the standard decimal form 0.75; then evaluate 110.75 using a 1134  6.040

(B) 3.104623  0.4699 (C) (0.000 000 008 437)311  (8.437  109)311  0.006 281

MATCHED PROBLEM



9

Evaluate to four significant digits using a calculator. (A) 238

(B) 57.2856

ANSWERS

(C) (83,240,000,000)53

TO MATCHED PROBLEMS

(A) 1 (B) 1 (C) 0.000 01 (D) 1,000 (E) x4 (F) v3u7 3 4 (A) 15x (B) 3(2y ) (C) 14x (A) y6 (9x8) (B) y12 x6 (C) 2n5(5m) (D) x9 (E) (a  b)2 (A) 2 (B) 2 (C) Not real (D) 2 (E) 2 (F) 0 (A) 27 (B) 81 (C) 10y1312 (D) 16yx3 (E) 2x118 (F) 2x  5x12y12  3y 6. (A) 4.3  102; 2.3  104; 3.45  108; 3  101; 3.1  103; 6.83  107 (B) 4,000; 530,000; 0.0253; 0.000 007 42 7. 1.11  1019 8. 2.8990  1017 9. (A) 1.297 (B) 0.034 28 (C) 1.587  1018

1. 2. 3. 4. 5.



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R-2

25

Exponents

Exercises

All variables are restricted to prevent division by 0. In Problems 1–26, evaluate each expression that results in a rational number. 1. 28

2. 38

2 4 3. a b 3

3 3 4. a b 5

5. 44

6. 26

7. (5)4

8. (2)5

9. (3)1

10. (3)2

11. 72

12. 70

Simplify Problems 51–62, and write the answers using positive exponents only. 51. a

x4y1 2 3

x y

b

2

2

3

52. a

m2n3 2 b m4n1

53. a

2x3y2

55. a

3 112

56. a

m23 6 b n12

54. a

6mn b 3m1n2

a b b4

57. a

4x2 12 b y4

58. a

w4 12 b 9x2

59. a

8a4b3 13 b 27a2b3

60. a

25x5y1 3 5

16x y

1

4xy

b

2

12

b

1 0 13. a b 3

1 1 14. a b 5

15. 10012

16. 36112

17. 12513

18. 2723

63. What is the result of entering 23 on a calculator?

19. 932

20. 212

1 12 21. a b 2

22. 6443

23. 13  31

24. 41  42

64. Refer to Problem 63. What is the difference between 2(3 ) 3 2 32 and (2 ) ? Which agrees with the value of 2 obtained with a calculator?

4

*61. 3(x  3) (3x ) 3

2

62. 2(x2  3x)3(2x  3) 2

2

8 82

65. If n  0, then property 1 in Theorem 1 implies that ama0  am0  am. Explain how this helps motivate the definition of a0.

Simplify Problems 27–38 and express answers using positive exponents only.

66. If m  n, then property 1 in Theorem 1 implies that anan  a0  1. Explain how this helps motivate the definition of an.

25.

5 54

26.

27. x5x2

28. y6y8

29. (2y)(3y2)(5y4)

30. (6x3)(4x7)(x5) 31. (a2b3)5

32. (2c4d2)3

33. u13u53

34. v15v65

35. (x3)16

36. ( y23)94

37. (49a4b2)12

38. (125a9b12)13

Write the numbers in Problems 39–44 in scientific notation. 39. 45,320,000

Evaluate Problems 67–70, to three significant digits using scientific notation and a calculator. 67.

(32.7)(0.000 000 008 42) (0.0513)(80,700,000,000)

68.

(4,320)(0.000 000 000 704) (835)(635,000,000,000)

69.

(5,760,000,000) (527)(0.000 007 09)

70.

0.000 000 007 23 (0.0933)(43,700,000,000)

40. 3,670

41. 0.066

42. 0.029

43. 0.000 000 084

44. 0.000 497

In Problems 45–50, write each number in standard decimal form. 45. 9  105

46. 3  103

47. 3.48  106

48. 8.63  108

9

7

49. 4.2  10

50. 1.6  10

*The symbol

denotes problems that are related to calculus.

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In Problems 71–78, evaluate to four significant digits using a calculator. (Refer to the instruction book for your calculator to see how exponential forms are evaluated.) 71. 1554

72. 2232

73. 10334

74. 82738

75. 2.87685

76. 37.0973

77. (0.000 000 077 35)27

78. (491,300,000,000)74

Problems 79–82, illustrate common errors involving rational exponents. In each case, find numerical examples that show that the left side is not always equal to the right side. 79. (x  y)12  x12  y12 81. (x  y)13 

1 (x  y)3

80. (x3  y3)13  x  y 82. (x  y)12 

1 (x  y)2

In Problems 83–86, m and n represent positive integers. Simplify and express answers using positive exponents. 83. (a3nb3m)13

84. (an2bn3)1n

85. (xm4yn3)12

86. (am3bn2)6

87. If possible, find a real value of x such that: (A) (x2)12  x (B) (x2)12  x (C) (x3)13  x 88. If possible, find a real value of x such that: (A) (x2)12  x (B) (x2)12  x (C) (x3)13  x 89. If n is even and b is negative, then b1n is not real. If m is odd, n is even, and b is negative, is (bm)1n real? 90. If we assume that m is odd and n is even, is it possible that one of (b1n)m and (bm)1n is real and the other is not?

APPLICATIONS 91. EARTH SCIENCE If the mass of Earth is approximately 6.1  1027 grams and each gram is 2.2  103 pound, what is the mass of Earth in pounds? Express the answer to two significant digits in scientific notation.

is produced by life alone. If 1 gram is approximately 2.2  103 pound, what is the weight of the free oxygen in pounds? Express the answer to two significant digits in scientific notation. 93. COMPUTER SCIENCE If a computer can perform a single operation in 1010 second, how many operations can it perform in 1 second? In 1 minute? ★94. COMPUTER SCIENCE If electricity travels in a computer circuit at the speed of light (1.86  105 miles per second), how far will electricity travel in the superconducting computer (see Problem 93) in the time it takes it to perform one operation? (Size of circuits is a critical problem in computer design.) Give the answer in miles, feet, and inches (1 mile  5,280 feet). Compute answers to three significant digits.

95. ECONOMICS If in the United States in 2003 the national debt was about $6,760,000,000,000 and the population was about 291,000,000, estimate to three significant digits each individual’s share of the national debt. Write your answer in scientific notation and in standard decimal form. 96. ECONOMICS If in the United States in 2003 the gross domestic product (GDP) was about $10,990,000,000,000 and the population was about 291,000,000, estimate to three significant digits the GDP per person. Write your answer in scientific notation and in standard decimal form. 97. ECONOMICS The number of units N of a finished product produced from the use of x units of labor and y units of capital for a particular Third World country is approximated by N  10x34y14

Cobb-Douglas equation

Estimate how many units of a finished product will be produced using 256 units of labor and 81 units of capital. 98. ECONOMICS The number of units N of a finished product produced by a particular automobile company where x units of labor and y units of capital are used is approximated by N  50x12y12

Cobb-Douglas equation

Estimate how many units will be produced using 256 units of labor and 144 units of capital. 99. BRAKING DISTANCE R. A. Moyer of Iowa State College found, in comprehensive tests carried out on 41 wet pavements, that the braking distance d (in feet) for a particular automobile traveling at v miles per hour was given approximately by d  0.0212v73 Approximate the braking distance to the nearest foot for the car traveling on wet pavement at 70 miles per hour.

92. BIOLOGY In 1929 Vernadsky, a biologist, estimated that all the free oxygen of the earth weighs 1.5  1021 grams and that it

100. BRAKING DISTANCE Approximately how many feet would it take the car in Problem 99 to stop on wet pavement if it were traveling at 50 miles per hour? (Compute answer to the nearest foot.)

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Radicals

Radicals Z From Rational Exponents to Radicals, and Vice Versa Z Properties of Radicals Z Simplifying Radicals Z Rationalizing Denominators

What do the following algebraic expressions have in common? 212

2x23

12

3 2 22 x

1 x  y1  2 1 1x  1y 12

Each vertical pair represents the same quantity, one in rational exponent form and the other in radical form. There are occasions when it is more convenient to work with radicals than with rational exponents, or vice versa. In this section we see how the two forms are related and investigate some basic operations on radicals.

Z From Rational Exponents to Radicals, and Vice Versa We start this discussion by defining an nth-root radical: n

Z DEFINITION 1 2b, nth-Root Radical n

For n a natural number greater than 1 and b a real number, we define 2b to be the principal nth root of b (see Definition 3 in Section R-2); that is, n

2b  b1/n The symbol 1 is called a radical, n is called the index, and b is called the radicand. 2 If n  2, we write 1b in place of 2 b. 125 125

 2512

5

 2512

125 is not real

 5

5 2 32

5 2 32

 3215

2

 (32) 15

4 2 0  01  4  0

 2

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As already stated, it is often an advantage to be able to shift back and forth between rational exponent forms and radical forms. The following relationships, which are direct consequences of Definition 4 and Theorem 4 in Section R-2, are useful in this regard:

Z RATIONAL EXPONENT/RADICAL CONVERSIONS For m and n positive integers (n 7 1), and b not negative when n is even,

bm/n 



(bm)1/n (b1/n)m

n

bm  

22  3  e

n

 (b)m

3 2 22 3 (2 2)2

Note: Unless stated to the contrary, all variables in the rest of the discussion represent positive real numbers.

ZZZ

EXPLORE-DISCUSS 1

In each of the following, evaluate both radical forms. 1632  2163  ( 216)3 3 3 2723  2 272  ( 2 27)2 Which radical conversion form is easier to use if you are performing the calculations by hand?

EXAMPLE

1

Rational Exponent/Radical Conversions Change from rational exponent form to radical form. 7 (A) x17  2 x 5 (B) (3u2v3)35  2 (3u2v3)3

(C) y23 

1 23

y



1 2y 3

2

or or

5 (2 3u2v3)3 3 2 2 y

or

The first is usually preferred.

1 B y2 3

Change from radical form to rational exponent form. 5 (D) 2 6  615

3 2 (E) 2 x  x23

(F) 2x2  y2  (x2  y2)12



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MATCHED PROBLEM

Radicals

29

1

Change from rational exponent form to radical form. (A) u15

(B) (6x2y5)29

(C) (3xy)35

Change from radical form to rational exponent form. 4 (D) 2 9u

7 (E) 2 (2x)4

3 3 (F) 2 x  y3



Z Properties of Radicals The exponent properties considered earlier imply the following properties of radicals.

Z THEOREM 1 Properties of Radicals For n a natural number greater than 1, and x and y positive real numbers: n

1. 2xn  x n

3 3 2 x x

n

n

2. 2xy  2x2y n

3.

EXAMPLE

2

x 2x  n By 2y

5 5 5 2 xy  2 x2 y 4 1 x 4 x  4 Ay 1y

n

Simplifying Radicals Simplify: 5 (A) 2 (3x2y)5  3x2y

(B) 11015  150  125  2  12512  512 (C)

3 3 x 1 x 1 x  3  A 27 3 1 27 3

or

MATCHED PROBLEM

1 3 1x 3



2

Simplify: 7 (A) 2 (u2  v2)7

(B) 1612

(C)

x2 B8 3



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ZZZ

CAUTION ZZZ

In general, properties of radicals can be used to simplify terms raised to powers, not sums of terms raised to powers. Thus, for x and y positive real numbers, 2x2  y2  2x2  2y2  x  y but 2x2  2xy  y2  2(x  y)2  x  y

Z Simplifying Radicals The properties of radicals provide us with the means of changing algebraic expressions containing radicals to a variety of equivalent forms. One form that is often useful is a simplified form. An algebraic expression that contains radicals is said to be in simplified form if all four of the conditions listed in the following definition are satisfied.

Z DEFINITION 2 Simplified (Radical) Form 1. No radicand (the expression within the radical sign) contains a factor to a power greater than or equal to the index of the radical. For example, 2x5 violates this condition.

2. No power of the radicand and the index of the radical have a common factor other than 1. 6 4 For example, 2 x violates this condition.

3. No radical appears in a denominator. For example, y/ 1x violates this condition.

4. No fraction appears within a radical. For example, 235 violates this condition.

EXAMPLE

3

Finding Simplified Form Express radicals in simplified form. (A) 212x3y5z2  2(4x2y4z2)(3xy)

x pmy pn  (x my n) p

 2(2xy2z)2(3xy)

2xy  2x 2y

 2(2xy2z)2 23xy

2 xn  x

 2xy2z13xy

n

n

n

n

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3 3 (B) 2 6x2y2 4x5y2

3  2 (6x2y)(4x5y2)

Simplify radicand.

3  2 24x7y3

Condition 1 is not met.

3  2 (8x6y3)(3x)

x pmy pn  (x m y n) p

3  2 (2x2y)3(3x)

2xy  2x 2y

3 3  2 (2x2y)3 2 3x

2 xn  x

n

n

Radicals

31

n

n

3  2x2y2 3x

6 (C) 2 16x4y2  3(4x2y)2 4 16

Condition 2 is not met.

 (4x2y)26

Note the convenience of using rational exponents.

 (4x2y)13

Write as a radical.

 24x y 3

2

(D) 2 127  3(33)12 4 13 3

 (33)16  336

(xm)n  xmn Write as a radical.

 312  13

MATCHED PROBLEM



3

Express radicals in simplified form. (A) 218x5y2z3

4 4 (B) 2 27a3b3 2 3a5b3

9 (C) 2 8x6y3

3 (D) 21 4



Algebraic expressions involving radicals often can be simplified by adding and subtracting terms that contain exactly the same radical expressions. The distributive property of real numbers plays a central role in this process.

EXAMPLE

4

Combining Like Terms Combine as many terms as possible: (A) 513  413

 (5  4)13

3 3 (B) 22 xy2  72 xy2

 913

3  (2  7)2 xy2

3 3 (C) 32xy  22 xy  42xy  72 xy

3  52 xy2

3 3  32xy  42xy  22 xy  72 xy 3  7 2xy  92 xy



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MATCHED PROBLEM

4

Combine as many terms as possible: (A) 612  212

5 5 (B) 32 2x2y3  82 2x2y3

3 3 (C) 52 mn2  32mn  22 mn2  72mn

EXAMPLE

5



Multiplication with Radical Forms Multiply and simplify: (A) 12(110  3)  12110  12  3  120  312  215  312 (B) (12  3)(12  5)  1212  312  512  15  2  212  15  212  13 (C) (1x  3)( 1x  5)  1x1x  31x  51x  15  x  21x  15 3 3 3 3 3 3 (D) (2m  2n )( 2m  2 n)  2 m3  2 m2n2  2 mn  2 n 3

3

2

3

2

3 3 m 2 mn  2 m2n2  n

MATCHED PROBLEM



5

Multiply and simplify: (A) 13(16  4)

(B) (13  2)(13  4)

(C) (1y  2)( 1y  4)

3 2 3 2 3 3 (D) (2 x  2 y )( 2 x 2 y)



Z Rationalizing Denominators Eliminating a radical from a denominator is referred to as rationalizing the denominator. To rationalize the denominator, we multiply the numerator and denominator by a suitable factor that will rationalize the denominator—that is, will leave the denominator free of radicals. This factor is called a rationalizing factor. The following special products are of use in finding some rationalizing factors (see Examples 6C, D): (a  b)(a  b)  a2  b2 (a  b)(a2  ab  b2)  a3  b3 (a  b)(a2  ab  b2)  a3  b3

(1) (2) (3)

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Radicals

33

EXPLORE-DISCUSS 2

Use the preceding special products (1) to (3) to find a rationalizing factor for each of the following: (A) 1a  1b

EXAMPLE

6

(B) 1a  1b

3 3 (C) 2 a 2 b

3 3 (D) 2 a 2 b

Rationalizing Denominators Rationalize denominators. (A)

3 15

(B)

2a2 B 3b2 3

(C)

1x  1y 31x  21y

(D)

1 2m  2 3

SOLUTIONS

(A) 25 is a rationalizing factor for 25, since 2525  252  5. Thus, we multiply the numerator and denominator by 25 to rationalize the denominator: 3 315 315   5 15 1515 (B)

3 3 3 2 2a2 2 2a2 2 2a2 2 3b   3 3 2 2 3 2 B 3b2 23b 23b 23 b 3



3 2 2  32a2b 3 3 3 2 3b



3 2 18a2b 3b

(C) Special product (1) suggests that if we multiply the denominator 31x  21y by 31x  21y, we will obtain the difference of two squares and the denominator will be rationalized. 1x  1y 31x  21y





1 1x  1y2131x  21y2

131x  21y2131x  21y2 32x2  22xy  32xy  22y2 132x2 2  122y2 2

Expand.

Combine like terms.

3x  51xy  2y 9x  4y 3 (D) Special product (3) suggests that if we multiply the denominator 2 m2 3 3 2 2 by (2m)  22m  2 , we will obtain the sum of two cubes and the denominator will be rationalized. 

1 3 2 m2







3 3 13 1 2 m2 2  22 m  22 4

3 3 3 12 m  22 3 1 2 m2 2  22 m  22 4 3 3 2 m2  22 m4 3 12 m2 3  23

3 3 2 m2  22 m4 m8

Expand.

n

(2x)n  x



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MATCHED PROBLEM

6

Rationalize denominators. (A)

6 12x

(B)

10x3 24x 3

ANSWERS

(C)

1x  2 21x  3

(D)

1

TO MATCHED PROBLEMS

5 9 9 5 1. (A) 2 u (B) 2 (6x2y5)2 or (2 6x2y5)2 (C) 1  2 (3xy)3 47 3 3 13 (E) (2x) (F) (x  y ) 3 2 3 2 2. (A) u2  v2 (B) 213 (C) (2 x ) 2 or 12 2 x

3. (A) 3x2yz 12xz (B) 3a2b2b2  3a2b1b (C) 22x2y 5 3 2 3 2 4. (A) 8 12 (B) 522x y (C) 32mn  41mn 5. (A) 3 12  413 (B) 213  5 (C) y  21y  8 4

3 2 3 (D) x  2 xy 2 xy2  y 312x 3 6. (A) (B) 5x2 2 2x2 x

R-3

In Problems 1–8, change to radical form. Do not simplify.

2

2. 62534 37

4. (3x y) 13

7. x

3. 8x23

12

25

5. 4x

6. 32y

13

8. (x  y)13

y

In Problems 9–14, change to rational exponent form. Do not simplify. 9. 1361 4 12. 2 7x3y2

3

(C)

2x  1x  6 4x  9

(D)

(D) (9u)14

(D) 22 3

3 3 1  2y  2y2 1y

Exercises

Unless stated to the contrary, all variables represent positive real numbers 1. 3215



3 1 2 y

3 10. 2 172

5 3 11. 4x 2 y

3 2 13. 2 x  y2

3 2 3 2 14. 2 x 2 y

In Problems 15–34, write in simplified form. 3 15. 2 8

3 16. 2 27

17. 1128

18.  1125

19. 127  5 13

20. 218  118

3 3 3 21. 2 5 2 25  2 625

3 3 22. 220  2 40  2 5

3 3 23. 2 25 2 10

24. 16114

25. 29x8y4

26. 216m4y8

4 27. 216m4n8

5 28. 232a15b10

4 29. 2 m2

10 6 30.  n

5 3 31. 2 1 xy

4 32. 2 15x

3 3 33. 29x2 29x

34. 12x18xy

In Problems 35–46, rationalize denominators, and write in simplified form. 35.

38.

1 215 10 25 4

36.

1 317

37.

39.

2 144

40.

41.

6x 13x

42.

44.

4 16  2

45.

12y

2

16y 12 16  2

1 17 3

3 254 3

43.

2 12  1

46.

12 110  2

In Problems 47–54, write in simplified form. 5 6 7 11 47. x 2 3xy

50.

5 2 32u12v8 uv

4 2 32m7n9 2mn

3 48. 2a2 8a8b13

49.

3 4 9 3 ab 51. 2 1

52. 21 x8 y6 6

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4 4 54. 2 4m5n2 6m3n4

81.

In Problems 55–60, rationalize denominators and write in simplified form. 55.

58.

12m15 120m

56.

5 1x 3  21x

59.

1618c 118c

57.

2 15  3 12 5 15  2 12

60.

31y 21y  3 3 12  213 313  212

1 1x  1y  1z

1t  1x 61. tx 63.

1x  h  1x h

62.

64.

1x  1y 1x  1y 12  h  12 h

65. 10.032 965

66. 1419.763

67. 245.0218

4 68. 2 0.098 553

69. 25.477  109

70. 24.892  1016

5 5 71. 2 9 1 9

3 3 72. 2 2 1 2

3 8

7

For what real numbers are Problems 73–76 true? 73. 2x2  x

74. 2x2  x

3 3 75. 2 x x

3 3 76. 2 x  x

In Problems 77 and 78, evaluate each expression on a calculator and determine which pairs have the same value. Verify these results algebraically. 77. (A) 13  15 (C) 1  13 (E) 28  160

(B) 22  13  22  13 3 (D) 2 10  6 13 (F) 16

3 78. (A) 2 2 2  15 (C) 13  17 (E) 210  184

(B) 18 (D) 23  18  23  18 (F) 1  15

In Problems 79–82, rationalize denominators. 79.

1 1a  1b 3

3

80.

1 3 1m  1 n 3

1 1x  1y  1z

82.

Problems 83 and 84 are calculus-related. Rationalize numerators. 83.

3 3 2 xh 2 x h

3 3 2 t 2 x tx

84.

85. Show that xkm  2xm for k, m, and n natural numbers greater than 1. kn

n

86. Show that 2 1x   x for m and n natural numbers greater than 1. n

mn

APPLICATIONS 87. PHYSICS—RELATIVISTIC MASS The mass M of an object moving at a velocity v is given by M0

M A

In Problems 65–72, evaluate to four significant digits using a calculator. (Read the instruction booklet accompanying n your calculator for the process required to evaluate 1x.)

35

[Hint for Problem 81: Start by multiplying numerator and denominator by (1x  1y)  1z.]

m

Problems 61–64 are calculus-related. Rationalize the numerators; that is, perform operations on the fractions that eliminate radicals from the numerators. (This is a particularly useful operation in some problems in calculus.)

Radicals

1

v2 c2

where M0  mass at rest and c  velocity of light. The mass of an object increases with velocity and tends to infinity as the velocity approaches the speed of light. Show that M can be written in the form M

M0c 2c2  v2 c2  v2

88. PHYSICS—PENDULUM A simple pendulum is formed by hanging a bob of mass M on a string of length L from a fixed support (see the figure). The time it takes the bob to swing from right to left and back again is called the period T and is given by T  2

L Ag

where g is the gravitational constant. Show that T can be written in the form T

2 1gL g

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Polynomials: Basic Operations Z Polynomials Z Combining Like Terms Z Addition and Subtraction Z Multiplication Z Application

In this section we review the basic operations on polynomials. Polynomials are expressions such as x4  5x2  1 or 3xy  2x  5y  6 that are built from constants and variables using only addition, subtraction, and multiplication (the power x4 is the product x  x  x  x). Polynomials are used throughout mathematics to describe and approximate mathematical relationships.

Z Polynomials Algebraic expressions are formed by using constants and variables and the algebraic operations of addition, subtraction, multiplication, division, raising to powers, and taking roots. Some examples are 3 3 2 x 5 xy7 x5 2 x  2x  5

5x4  2x2  7 (2x  y)2 1 1 1 1 x

An algebraic expression involving only the operations of addition, subtraction, multiplication, and raising to natural number powers is called a polynomial. (Note that raising to a natural number power is repeated multiplication.) Some examples are 2x  3 x  2y 5 0

4x2  3x  7 5x3  2x2  7x  9 x2  3xy  4y2 x3  3x2y  xy2  2y7

In a polynomial, a variable cannot appear in a denominator, as an exponent, or within a radical. Accordingly, a polynomial in one variable x is constructed by adding or subtracting constants and terms of the form axn, where a is a real number and n is a natural number. A polynomial in two variables x and y is constructed by adding and subtracting constants and terms of the form axmyn, where a is a real number and m and n are natural numbers. Polynomials in three or more variables are defined in a similar manner.

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Polynomials can be classified according to their degree. If a term in a polynomial has only one variable as a factor, then the degree of that term is the power of the variable. If two or more variables are present in a term as factors, then the degree of the term is the sum of the powers of the variables. The degree of a polynomial is the degree of the nonzero term with the highest degree in the polynomial. Any nonzero constant is defined to be a polynomial of degree 0. The number 0 is also a polynomial but is not assigned a degree.

EXAMPLE

1

Polynomials and Nonpolynomials (A) Polynomials in one variable: x2  3x  2

6x3  12x 

1 3

(B) Polynomials in several variables: 3x2  2xy  y2

4x3y2  13xy2z5

(C) Nonpolynomials: 12x 

3 5 x

x2  3x  2 x3

2x2  3x  1

(D) The degree of the first term in 6x3  12x  13 is 3, the degree of the second term is 1, the degree of the third term is 0, and the degree of the whole polynomial is 3. (E) The degree of the first term in 4x3y2  13xy2 is 5, the degree of the second term is 3, and the degree of the whole polynomial is 5. 

MATCHED PROBLEM

1

(A) Which of the following are polynomials? 3x2  2x  1

1x  3

x2  2xy  y2

x1 x2  2

(B) Given the polynomial 3x5  6x3  5, what is the degree of the first term? The second term? The whole polynomial? (C) Given the polynomial 6x4y2  3xy3, what is the degree of the first term? The second term? The whole polynomial? 

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In addition to classifying polynomials by degree, we also call a single-term polynomial a monomial, a two-term polynomial a binomial, and a three-term polynomial a trinomial. 5 2 3 2x y 3

Monomial

x  4.7 x4  12x2  9

Binomial Trinomial

Z Combining Like Terms We start with a word about coefficients. A constant in a term of a polynomial, including the sign that precedes it, is called the numerical coefficient, or simply, the coefficient, of the term. If a constant doesn’t appear, or only a  sign appears, the coefficient is understood to be 1. If only a  sign appears, the coefficient is understood to be 1. Thus, given the polynomial 2x4  4x3  x2  x  5

2x4  (4)x3  1x2  (1)x  5

the coefficient of the first term is 2, the coefficient of the second term is 4, the coefficient of the third term is 1, the coefficient of the fourth term is 1, and the coefficient of the last term is 5. At this point, it is useful to state two additional distributive properties of real numbers that follow from the distributive properties stated in Section R-1. Z ADDITIONAL DISTRIBUTIVE PROPERTIES 1. a(b  c)  (b  c)a  ab  ac 2. a(b  c  # # #  f )  ab  ac  # # #  af

Two terms in a polynomial are called like terms if they have exactly the same variable factors to the same powers. The numerical coefficients may or may not be the same. Since constant terms involve no variables, all constant terms are like terms. If a polynomial contains two or more like terms, these terms can be combined into a single term by making use of distributive properties. Consider the following example: 5x3y  2xy  x3y  2x3y

 5x3y  x3y  2x3y  2xy  (5x3y  x3y  2 x3y)  2xy  (5  1  2) x3y  2xy

Group like terms. Factor out x3y. Simplify.

 2x3y  2xy It should be clear that free use has been made of the real number properties discussed earlier. The steps done in the dashed box are usually done mentally, and the process is quickly mechanized as follows: Like terms in a polynomial are combined by adding their numerical coefficients.

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EXAMPLE

2

Polynomials: Basic Operations

39

Simplifying Polynomials Remove parentheses and combine like terms: (A) 2(3x2  2x  5)  (x2  3x  7)  2(3x2  2x  5)  1(x2  3x  7)  6x2  4x  10  x2  3x  7  7x2  x  3

Distribute, remove parentheses.

Combine like terms.

(B) (x3  2x  6)  (2x3  x2  2x  3)  1(x3  2x  6)  (1)(2x3  x2  2x  3)  x3  2x  6  2x3  x2  2x  3  x3  x2  4x  3

Distribute, remove parentheses.

Combine like terms.

(C) 33x2  (2x  1)4  (x2  1)  33x2  2x  1 4  (x2  1)  3x2  2x  1  x2  1  2x2  2x

MATCHED PROBLEM

Remove parentheses. Combine like terms.



2

Remove parentheses and combine like terms: (A) 3(u2  2v2)  (u2  5v2) (B) (m3  3m2  m  1)  (2m3  m  3) (C) (x3  2)  32x3  (3x  4)4



Z Addition and Subtraction Addition and subtraction of polynomials can be thought of in terms of removing parentheses and combining like terms, as illustrated in Example 2. Horizontal and vertical arrangements are illustrated in the next two examples. You should be able to work either way, letting the situation dictate the choice.

EXAMPLE

3

Adding Polynomials Add: x4  3x3  x2,

x3  2x2  3x,

and

3x2  4x  5

SOLUTION

Add horizontally: (x4  3x3  x2)  (x3  2x2  3x)  (3x2  4x  5)  x4  3x3  x2  x3  2x2  3x  3x2  4x  5  x4  4x3  2x2  x  5

Remove parentheses. Combine like terms.

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Or vertically, by lining up like terms and adding their coefficients: x4  3x3  x2  x3  2x2  3x 3x2  4x  5 x4  4x3  2x2  x  5

MATCHED PROBLEM



3

Add horizontally and vertically: 3x4  2x3  4x2,

EXAMPLE

4

x3  2x2  5x,

and

x2  7x  2



Subtracting Polynomials 4x2  3x  5

Subtract:

from

x2  8

SOLUTION

(x2  8)  (4x2  3x  5)  x2  8  4x2  3x  5  3x2  3x  13

or

MATCHED PROBLEM

ZZZ

d Change signs and add.



4

2x2  5x  4

Subtract:

x2  8 2 4x  3x  5 3x2  3x  13

from

5x2  6



CAUTION ZZZ

When you use a horizontal arrangement to subtract a polynomial with more than one term, you must enclose the polynomial in parentheses. Thus, to subtract 2x  5 from 4x  11, you must write 4x  11  (2x  5)

and not

4x  11  2x  5

Z Multiplication Multiplication of algebraic expressions involves the extensive use of distributive properties for real numbers, as well as other real number properties.

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EXAMPLE

5

Polynomials: Basic Operations

41

Multiplying Polynomials (2x  3)(3x2  2x  3)

Multiply: SOLUTION

(2x  3)(3x2  2x  3)  2x(3x2  2x  3)  3(3x2  2x  3)  6x3  4x2  6x  9x2  6x  9  6x3  13x2  12x  9

Distribute, remove parentheses.

Combine like terms.

Or, using a vertical arrangement, 3x2  2x  3 2x  3 6x3  4x2  6x  9x2  6x  9 6x3  13x2  12x  9

MATCHED PROBLEM



5

Multiply: (2x  3)(2x2  3x  2)



Thus, to multiply two polynomials, multiply each term of one by each term of the other, and combine like terms.

EXAMPLE

6

Multiplying Binomials Multiply: (A) (2x  3y)(5x  2y)

 10x2  4xy  15xy  6y2

(B) (3a  2b)(3a  2b)

 (3a)2  (2b)2

(C) (5x  3)2

 (5x)2  2(5x)(3)  32

(D) (m  2n)2  m2  4mn  4n2

 10x2  11xy  6y2

 9a2  4b2  25x2  30x  9 

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MATCHED PROBLEM

6

Multiply: (A) (4u  3v)(2u  v)

(B) (2xy  3)(2xy  3)

(C) (m  4n)(m  4n)

(D) (2u  3v)2

(E) (6x  y)2



Products of certain binomial factors occur so frequently that it is useful to remember formulas for their products. The following formulas are easily verified by multiplying the factors on the left:

Z SPECIAL PRODUCTS 1. (a  b)(a  b)  a2  b2 2. (a  b)2  a2  2ab  b2 3. (a  b)2  a2  2ab  b2

ZZZ EXPLORE-DISCUSS

1

(A) Explain the relationship between special product formula 1 and the areas of the rectangles in the figures. (a  b)(a  b)



a2  b2 a

ab

a a

b

b b

(B) Construct similar figures to provide geometric interpretations for special product formulas 2 and 3.

The following conventions govern the order in which algebraic operations are performed.

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Z ORDER OF OPERATIONS 1. Simplify inside the innermost grouping first, then the next innermost, and so on. 233  1x  42 4  233  x  44  217  x2  14  2x 2. Unless grouping symbols indicate otherwise, apply exponents before multiplication or division is performed. 21x  22 2  21x2  4x  42  2x2  8x  8 3. Unless grouping symbols indicate otherwise, perform multiplication and division before addition and subtraction. In either case, proceed from left to right. 5  21x  32  5  2x  6  11  2x

EXAMPLE

7

Combined Operations Perform the indicated operations and simplify: (A) 3x  55  3[x  x(3  x)] 6  3x  55  3[x  3x  x2 ] 6  3x  55  3[2x  x2 ] 6  3x  55  6x  3x2 6  3x  5  6x  3x2  3x2  3x  5

Combine like terms. Distribute, remove square brackets. Distribute, remove braces. Combine like terms.

(B) (x  2y)(2x  3y)  (2x  y)2  2x2  3xy  4xy  6y2  (4x2  4xy  y2)  2x  xy  6y  4x  4xy  y  2x2  5xy  7y2 2

2

2

2

(C) (2m  3n)3  (2m  3n)(2m  3n)2  (2m  3n)(4m2  12mn  9n2)  8m3  24m2n  18mn2  12m2n  36mn2  27n3  8m3  36m2n  54mn2  27n3

Distribute, remove parentheses. Combine like terms.

Expand the square. Expand. Combine like terms.



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MATCHED PROBLEM

7

Perform the indicated operations and simplify: (A) 2t  57  23t  t(4  t)4 6

(B) (u  3v)2  (2u  v)(2u  v)

(C) (4x  y)3



Z Application EXAMPLE

8

Volume of a Cylindrical Shell A plastic water pipe with a hollow center is 100 inches long, 1 inch thick, and has an inner radius of x inches (see the figure). Write an algebraic expression in terms of x that represents the volume of the plastic used to construct the pipe. Simplify the expression. [Recall: The volume V of a right circular cylinder of radius r and height h is given by V  r2h.] SOLUTION

1 inch x inches

A right circular cylinder with a hollow center is called a cylindrical shell. The volume of the shell is equal to the volume of the cylinder minus the volume of the hole. Since the radius of the hole is x inches and the pipe is 1 inch thick, the radius of the cylinder is x  1 inches. Thus, we have a

100 inches

Volume of Volume of Volume of ba ba b shell cylinder hole

Volume  (x  1)2100  x2100  100(x2  2x  1)  100x2  100x2  200x  100  100x2  200x  100

MATCHED PROBLEM

Expand the square. Distribute, remove parentheses. Combine like terms.



8

A plastic water pipe is 200 inches long, 2 inches thick, and has an outer radius of x inches. Write an algebraic expression in terms of x that represents the volume of the plastic used to construct the pipe. Simplify the expression. 

ANSWERS

TO MATCHED PROBLEMS

(A) 3x2  2x  1, x2  2xy  y2 (B) 5, 3, 5 (C) 6, 4, 6 (A) 4u2  v2 (B) m3  3m2  2m  4 (C) x3  3x  2 4. 3x2  5x  10 5. 4x3  13x  6 3x4  x3  5x2  2x  2 (A) 8u2  2uv  3v2 (B) 4x2y2  9 (C) m2  16n2 (D) 4u2  12uv  9v2 (E) 36x2  12xy  y2 2 7. (A) 2t  4t  7 (B) 3u2  6uv  10v2 (C) 64x3  48x2y  12xy2  y3 2 2 8. Volume  200x  200(x  2)  800x  800 1. 2. 3. 6.

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Exercises

Problems 1–8 refer to the following polynomials: (a) x4  2x2  3 and (b) x3  1.

In Problems 37–50, perform the indicated operations and simplify. 37. 2x  35x  23x  (x  5)4  16

1. What is the degree of (a)? 2. What is the degree of (b)?

38. m  5m  3m  (m  1)4 6

3. What is the degree of the sum of (a) and (b)?

39. 2533a  4(1  a)4  (5  a)6

4. What is the degree of the product of (a) and (b)?

40. 5b  35  32  4(2b  1)4  2(2  3b)6

5. Multiply (a) and (b).

6. Add (a) and (b).

41. (2x2  3x  1)(x2  x  2)

7. Subtract (b) from (a).

8. Subtract (a) from (b).

42. (x2  3xy  y2)(x2  3xy  y2) 43. (x  2y)2(x  2y)2

In Problems 9–14, is the algebraic expression a polynomial? If so, give its degree. 9. 6  5x  x2

10. 3x2  5x4  8

11. x3  5x  4 1x

12. x5  7x3  12

13. x3  9x2  52

14. 2x4  3x1  10

44. (n2  4nm  m2)(n2  4nm  m2) 45. (3u  2v)2  (2u  3v)(2u  3v) 46. (2a  b)2  (a  2b)2 47. (z  2)(z 2  2z  3)  z  7 48. ( y  3)( y2  3y  1)  8y  1

In Problems 15–36, perform the indicated operations and simplify.

49. (2m  n)3

15. 2(x  1)  3(2x  3)  (4x  5)

Problems 51–58 are calculus-related. Perform the indicated operations and simplify.

16. 2(u  1)  (3u  2)  2(2u  3) 17. 2y  3y 3 4  2(y  1)4

53. 2(x  h)2  3(x  h)  (2x2  3x)

19. (m  n)(m  n)

54. 4(x  h)2  6(x  h)  (4x2  6x)

20. (a  b)(a  b) 21. (4t  3)(t  2)

22. (3x  5)(2x  1)

23. (5y  1)(3  2y)

24. (2  m)(6m  5)

25. (3x  2y)(x  3y)

26. (2x  3y)(x  2y)

27. (2m  7)(2m  7)

28. (3y  2)(3y  2)

29. (6x  4y)(5x  3y)

30. (3m  7n)(2m  5n)

31. (3x  2y)(3x  2y)

32. (4m  3n)(4m  3n)

33. (4x  y)

34. (3u  4v)2

35. (a  b)(a2  ab  b2) 36. (a  b)(a2  ab  b2)

51. 3(x  h)  7  (3x  7) 52. (x  h)2  x2

18. 4a  2a 3 5  3(a  2)4

2

50. (3a  2b)3

55. 2(x  h)2  4(x  h)  9  (2x2  4x  9) 56. 3(x  h)2  5(x  h)  7  (3x2  5x  7) 57. (x  h)3  2(x  h)2  (x3  2x2) 58. (x  h)3  3(x  h)  (x3  3x) 59. Subtract the sum of the first two polynomials from the sum of the last two: 3m2  2m  5, 4m2  m, 3m2  3m  2, m3  m2  2. 60. Subtract the sum of the last two polynomials from the sum of the first two: 2x2  4xy  y2, 3xy  y2, x2  2xy  y2, x2  3xy  2y2.

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61. Explain the relationship between the equation (a  b)(c  d )  ac  ad  bc  bd and the rectangles in the figure.

In Problems 65–68, perform the indicated operations and simplify. 65. 2(x  2)3  (x  2)2  3(x  2)  4 66. (2x  1)3  2(2x  1)2  3(2x  1)  7

a

67. 3x5x3x  x(2  x)4  (x  2)(x2  3)6

b

69. Show by example that, in general, (a  b)2  a2  b2. Discuss possible conditions on a and b that would make this a valid equation.

68. 25(x  3)(x2  2x  1)  x33  x(x  2)4 6 c

d

62. Explain the relationship between the distributive property of real numbers and the areas of the rectangles in the figure. a

70. Show by example that, in general, (a  b)2  a2  b2. Discuss possible conditions on a and b that would make this a valid equation. 71. If you are given two polynomials, one of degree m and the other of degree n, m 7 n, what is the degree of the sum?

b c

63. In the figure, the area of the large square is equal to the sum of the area of the small square and the areas of the four congruent triangles. Translate this verbal statement into a mathematical equation and simplify. What famous theorem have you proven?

72. What is the degree of the product of the two polynomials in Problem 71? 73. How does the answer to Problem 71 change if the two polynomials can have the same degree? 74. How does the answer to Problem 72 change if the two polynomials can have the same degree?

APPLICATIONS c

a

75. GEOMETRY The width of a rectangle is 5 centimeters less than its length. If x represents the length, write an algebraic expression in terms of x that represents the perimeter of the rectangle. Simplify the expression. 76. GEOMETRY The length of a rectangle is 8 meters more than its width. If x represents the width of the rectangle, write an algebraic expression in terms of x that represents its area. Change the expression to a form without parentheses.

b

★77.

64. In the figure, the area of the large square is equal to the sum of the area of the small square and the areas of the four congruent triangles. Translate this verbal statement into a mathematical equation and simplify. What famous theorem have you proven?

b

c

a

COIN PROBLEM A parking meter contains nickels, dimes, and quarters. There are 5 fewer dimes than nickels, and 2 more quarters than dimes. If x represents the number of nickels, write an algebraic expression in terms of x that represents the value of all the coins in the meter in cents. Simplify the expression.

★78.

COIN PROBLEM A vending machine contains dimes and quarters only. There are 4 more dimes than quarters. If x represents the number of quarters, write an algebraic expression in terms of x that represents the value of all the coins in the vending machine in cents. Simplify the expression.

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79. PACKAGING A spherical plastic container for designer wristwatches has an inner radius of x centimeters (see the figure). If the plastic shell is 0.3 centimeters thick, write an algebraic expression in terms of x that represents the volume of the plastic used to construct the container. Simplify the expression. [Recall: The volume V of a sphere of radius r is given by V  43r3.]

0.3 cm x cm

80. PACKAGING A cubical container for shipping computer components is formed by coating a metal mold with polystyrene. If the metal mold is a cube with sides x centimeters long and the polystyrene coating is 2 centimeters thick, write an algebraic expression in terms of x that represents the volume of the polystyrene used to construct the container. Simplify the expression. [Recall: The volume V of a cube with sides of length t is given by V  t3.]

R-5

47

Figure for 79

Polynomials: Factoring Z Factoring—What Does It Mean? Z Common Factors and Factoring by Grouping Z Factoring Second-Degree Polynomials Z More Factoring

Z Factoring—What Does It Mean? A factor of a number is one of two or more numbers whose product is the given number. Similarly, a factor of an algebraic expression is one of two or more algebraic expressions whose product is the given algebraic expression. For example, 30  2  3  5 x2  4  (x  2)(x  2)

2, 3, and 5 are each factors of 30. (x  2) and (x  2) are each factors of x2  4.

The process of writing a number or algebraic expression as the product of other numbers or algebraic expressions is called factoring. We start our discussion of factoring with the positive integers. An integer such as 30 can be represented in a factored form in many ways. The products 65

(12)(10)(6)

15  2

235

all yield 30. A particularly useful way of factoring positive integers greater than 1 is in terms of prime numbers.

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Z DEFINITION 1 Prime and Composite Numbers An integer greater than 1 is prime if its only positive integer factors are itself and 1. An integer greater than 1 that is not prime is called a composite number. The integer 1 is neither prime nor composite. Examples of prime numbers:

2, 3, 5, 7, 11, 13

Examples of composite numbers:

4, 6, 8, 9, 10, 12

ZZZ EXPLORE-DISCUSS

1

In the array below, cross out all multiples of 2, except 2 itself. Then cross out all multiples of 3, except 3 itself. Repeat this for each integer in the array that has not yet been crossed out. Describe the set of numbers that remains when this process is completed. 1 21 41 61 81

2 22 42 62 82

3 23 43 63 83

4 24 44 64 84

5 25 45 65 85

6 26 46 66 86

7 27 47 67 87

8 28 48 68 88

9 29 49 69 89

10 30 50 70 90

11 31 51 71 91

12 32 52 72 92

13 33 53 73 93

14 34 54 74 94

15 35 55 75 95

16 36 56 76 96

17 37 57 77 97

18 38 58 78 98

19 39 59 79 99

20 40 60 80 100

This process is referred to as the sieve of Eratosthenes. (Eratosthenes was a Greek mathematician and astronomer who was a contemporary of Archimedes, circa 200 B.C.)

A composite number is said to be factored completely if it is represented as a product of prime factors. The only factoring of 30 given on page 47 that meets this condition is 30  2  3  5.

EXAMPLE

1

Factoring a Composite Number Write 60 in completely factored form. SOLUTION

60  6  10  2  3  2  5  22  3  5 or 60  5  12  5  4  3  22  3  5 or 60  2  30  2  2  15  22  3  5



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MATCHED PROBLEM

Polynomials: Factoring

49

1 

Write 180 in completely factored form.

Notice in Example 1 that we end up with the same prime factors for 60 irrespective of how we progress through the factoring process. This illustrates an important property of integers:

Z THEOREM 1 The Fundamental Theorem of Arithmetic Each integer greater than 1 is either prime or can be expressed uniquely, except for the order of factors, as a product of prime factors.

We can also write polynomials in completely factored form. A polynomial such as 2x2  x  6 can be written in factored form in many ways. The products (2x  3)(x  2)

2(x2  12x  3)

2(x  32)(x  2)

all yield 2x2  x  6. A particularly useful way of factoring polynomials is in terms of prime polynomials.

Z DEFINITION 2 Prime Polynomials A polynomial of degree greater than 0 is said to be prime relative to a given set of numbers if: (1) all of its coefficients are from that set of numbers; and (2) it cannot be written as a product of two polynomials (excluding constant polynomials that are factors of 1) having coefficients from that set of numbers. Relative to the set of integers: x2  2 is prime x2  9 is not prime, since x2  9  (x  3)(x  3)

[Note: The set of numbers most frequently used in factoring polynomials is the set of integers.]

A nonprime polynomial is said to be factored completely relative to a given set of numbers if it is written as a product of prime polynomials relative to that set of numbers. Our objective in this section is to review some of the standard factoring techniques for polynomials with integer coefficients. In Chapter 3 we will treat in detail the topic of factoring polynomials of higher degree with arbitrary coefficients.

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Z Common Factors and Factoring by Grouping Example 2 illustrates the use of the distributive properties in factoring.

EXAMPLE

2

Factoring Out Common Factors Factor out, relative to the integers, all factors common to all terms: (A) 2x3y  8x2y2  6xy3

(B) 2x(3x  2)  7(3x  2)

SOLUTIONS

(A) 2x3y  8x2y2  6xy3

 (2xy)x2  (2xy)4xy  (2xy)3y2

Factor out 2xy.

 2xy(x2  4xy  3y2) (B) 2x(3x  2)  7(3x  2)

 2x(3x  2)  7(3x  2)

Factor out 3x  2.

 (2x  7)(3x  2)

MATCHED PROBLEM



2

Factor out, relative to the integers, all factors common to all terms: (A) 3x3y  6x2y2  3xy3

EXAMPLE

3

(B) 3y(2y  5)  2(2y  5)



Factoring Out Common Factors Factor completely, relative to the integers: 4(2x  7)(x  3)2  2(2x  7)2(x  3) SOLUTION

4(2x  7)(x  3)2  2(2x  7)2(x  3)  2(2x  7)(x  3) 32(x  3)  (2x  7)4  2(2x  7)(x  3)(2x  6  2x  7)  2(2x  7)(x  3)(4x  1)

MATCHED PROBLEM

Factor out 2(2x  7)(x  3). Distribute, remove parentheses. Combine like terms.



3

Factor completely, relative to the integers: 4(2x  5)(3x  1)2  6(2x  5)2(3x  1)



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51

Some polynomials can be factored by first grouping terms to obtain an algebraic expression that looks something like Example 2B. We can then complete the factoring by the method used in that example.

EXAMPLE

4

Factoring by Grouping Factor completely, relative to the integers, by grouping: (A) 3x2  6x  4x  8

(B) wy  wz  2xy  2xz

(C) 3ac  bd  3ad  bc SOLUTIONS

(A) 3x2  6x  4x  8

Group the first two and last two terms.

 (3x  6x)  (4x  8)  3x(x  2)  4(x  2)  (3x  4)(x  2) 2

Remove common factors from each group. Factor out the common factor (x  2).

(B) wy  wz  2xy  2xz  (wy  wz)  (2xy  2xz)  w( y  z)  2x(y  z)  (w  2x)( y  z)

Group the first two and last two terms—be careful of signs. Remove common factors from each group. Factor out the common factor ( y  z).

(C) 3ac  bd  3ad  bc In parts (A) and (B) the polynomials are arranged in such a way that grouping the first two terms and the last two terms leads to common factors. In this problem neither the first two terms nor the last two terms have a common factor. Sometimes rearranging terms will lead to a factoring by grouping. In this case, we interchange the second and fourth terms to obtain a problem comparable to part (B), which can be factored as follows: 3ac  bc  3ad  bd  (3ac  bc)  (3ad  bd)  c(3a  b)  d(3a  b)  (c  d)(3a  b)

MATCHED PROBLEM

Factor out c, d. Factor out 3a  b.



4

Factor completely, relative to the integers, by grouping: (A) 2x2  6x  5x  15 (C) 6wy  xz  2xy  3wz

(B) 2pr  ps  6qr  3qs 

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Z Factoring Second-Degree Polynomials We now turn our attention to factoring second-degree polynomials of the form 2x2  5x  3

2x2  3xy  2y2

and

into the product of two first-degree polynomials with integer coefficients. Examples will illustrate an approach to the problem.

EXAMPLE

5

Factoring Second-Degree Polynomials Factor each polynomial, if possible, using integer coefficients: (A) 2x2  3xy  2y2

(B) x2  3x  4

(C) 6x2  5xy  4y2

SOLUTIONS

(A) 2x2  3xy  2y2  (2x  y)(x  y) c

c

?

?

Put in what we know. Signs must be opposite. (We can reverse this choice if we get 3xy instead of 3xy for the middle term.)

Now, what are the factors of 2 (the coefficient of y2)? 2 12 21

(2x  y)(x  2y)  2x2  3xy  2y2 (2x  2y)(x  y)  2x2  2y2

The first choice gives us 3xy for the middle term—close, but not there—so we reverse our choice of signs to obtain 2x2  3xy  2y2  (2x  y)(x  2y) (B) x2  3x  4  (x  )(x  ) 4 22 14 41

Signs must be the same because the third term is positive and must be negative because the middle term is negative.

(x  2)(x  2)  x2  4x  4 (x  1)(x  4)  x2  5x  4 (x  4)(x  1)  x2  5x  4

No choice produces the middle term; hence x2  3x  4 is not factorable using integer coefficients. (C) 6x2  5xy  4y2  ( x  y)( x  y) c

c

c

c

?

?

?

?

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53

The signs must be opposite in the factors, because the third term is negative. We can reverse our choice of signs later if necessary. We now write all factors of 6 and of 4: 6 23 32 16 61

4 22 14 41

and try each choice on the left with each on the right—a total of 12 combinations that give us the first and last terms in the polynomial 6x2  5xy  4y2. The question is: Does any combination also give us the middle term, 5xy? After trial and error and, perhaps, some educated guessing among the choices, we find that 3  2 matched with 4  1 gives us the correct middle term. Thus, 6x2  5xy  4y2  (3x  4y)(2x  y) If none of the 24 combinations (including reversing our sign choice) had produced the middle term, then we would conclude that the polynomial is not factorable using integer coefficients.

MATCHED PROBLEM



5

Factor each polynomial, if possible, using integer coefficients: (A) x2  8x  12

(B) x2  2x  5

(C) 2x2  7xy  4y2

(D) 4x2  15xy  4y2



Z More Factoring The special factoring formulas listed here will enable us to factor certain polynomial forms that occur frequently.

Z SPECIAL FACTORING FORMULAS 1. u2  2uv  v2  (u  v)2

Perfect Square

2. u  2uv  v  (u  v)

Perfect Square

2

2

2

3. u2  v2  (u  v)(u  v)

Difference of Squares

4. u  v  (u  v)(u  uv  v )

Difference of Cubes

5. u3  v3  (u  v)(u2  uv  v2)

Sum of Cubes

3

3

2

2

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The formulas in the box can be established by multiplying the factors on the right.

ZZZ

CAUTION ZZZ

Note that we did not list a special factoring formula for the sum of two squares. In general, u2  v2  1au  bv21cu  dv2 for any choice of real number coefficients a, b, c, and d. In Chapter 1 we will see that u2  v2 can be factored using complex numbers.

EXAMPLE

6

Using Special Factoring Formulas Factor completely relative to the integers: (A) x2  6xy  9y2

(B) 9x2  4y2

(C) 8m3  1

(D) x3  y3z3

SOLUTIONS

(A) x2  6xy  9y2

 x2  2(x)(3y)  (3y)2

 (x  3y)2

(B) 9x2  4y2

 (3x)2  (2y)2

 (3x  2y)(3x  2y)

(C) 8m3  1

 (2m)3  13  (2m  1)3(2m)2  (2m)(1)  12 4

Perfect square

Difference of squares

Difference of cubes Simplify.

 (2m  1)(4m2  2m  1) (D) x3  y3z3

 x3  ( yz)3

Sum of cubes

 (x  yz)(x2  xyz  y2z2)

MATCHED PROBLEM



6

Factor completely relative to the integers: (A) 4m2  12mn  9n2

(B) x2  16y2

(C) z3  1

(D) m3  n3



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Polynomials: Factoring

55

EXPLORE-DISCUSS 2

(A) Verify the following factor formulas for u4  v4: u4  v4  (u  v)(u  v)(u2  v2)  (u  v)(u3  u2v  uv2  v3) (B) Discuss the pattern in the following formulas: u2  v2  (u  v)(u  v) u3  v3  (u  v)(u2  uv  v2) u4  v4  (u  v)(u3  u2v  uv2  v3) (C) Use the pattern you discovered in part (B) to write similar formulas for u5  v5 and u6  v6. Verify your formulas by multiplication.

We complete this section by considering factoring that involves combinations of the preceding techniques as well as a few additional ones. Generally speaking: When asked to factor a polynomial, we first take out all factors common to all terms, if they are present, and then proceed to factor until all factors are prime.

EXAMPLE

7

Combining Factoring Techniques Factor completely relative to the integers: (A) 18x3  8x

(B) x2  6x  9  y2

(D) 2t4  16t

(E) 2y4  5y2  12

(C) 4m3n  2m2n2  2mn3

SOLUTIONS

(A) 18x3  8x  2x(9x2  4)  2x(3x  2)(3x  2) (B) x2  6x  9  y2  (x2  6x  9)  y2  (x  3)2  y2  3(x  3)  y4 3 (x  3)  y4  (x  3  y)(x  3  y)

Group the first three terms. Factor x2  6x  9. Difference of squares Remove parentheses.

(C) 4m3n  2m2n2  2mn3  2mn(2m2  mn  n2)

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(D) 2t4  16t  2t(t3  8)  2t(t  2)(t 2  2t  4)

Difference of cubes

(E) 2y4  5y2  12  (2y2  3)( y2  4) Difference of squares 2  (2y  3)( y  2)( y  2)

MATCHED PROBLEM



7

Factor completely relative to the integers: (A) 3x3  48x

(B) x2  y2  4y  4

(C) 3u4  3u3v  9u2v2

(D) 3m4  24mn3

(E) 3x4  5x2  2

ANSWERS



TO MATCHED PROBLEMS

2 2  32  5 2. (A) 3xy(x2  2xy  y2) (B) (3y  2)(2y  5) 2(2x  5)(3x  1)(12x  17) (A) (2x  5)(x  3) (B) (p  3q)(2r  s) (C) (3w  x)(2y  z) (A) (x  2)(x  6) (B) Not factorable using integers (C) (2x  y)(x  4y) (D) (4x  y)(x  4y) 6. (A) (2m  3n)2 (B) (x  4y)(x  4y) (C) (z  1)(z2  z  1) 2 2 (D) (m  n)(m  mn  n ) 7. (A) 3x(x  4)(x  4) (B) (x  y  2)(x  y  2) (C) 3u2(u2  uv  3v2) (D) 3m(m  2n)(m2  2mn  4n2) (E) (3x2  2)(x  1)(x  1)

1. 3. 4. 5.

R-5

Exercises

In Problems 1–8, factor out, relative to the integers, all factors common to all terms. 1. 6x4  8x3  2x2

2. 6m4  9m3  3m2

3. 10x3y  20x2y2  15xy3

4. 8u3v  6u2v2  4uv3

5. 5x(x  1)  3(x  1)

6. 7m(2m  3)  5(2m  3)

7. 2w( y  2z)  x( y  2z)

8. a(3c  d )  4b(3c  d )

11. 6m2  10m  3m  5

14. 3a2  12ab  2ab  8b2 15. 8ac  3bd  6bc  4ad 16. 3pr  2qs  qr  6ps In Problems 17–30, factor completely, relative to the integers. If a polynomial is prime relative to the integers, say so. 17. 2x2  x  3

18. 3y2  8y  3

10. 2y2  6y  5y  15

19. x2  3x  8

20. u2  4uv  12v2

12. 5x2  40x  x  8

21. x2  5xy  14y2

22. m2  m  20

In Problems 9–16, factor completely, relative to the integers. 9. x2  2x  3x  6

13. 2x2  4xy  3xy  6y2

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23. 4a2  9b2

24. x2  4y2

25. 4x2  20x  25

26. a2b2  c2

27. a2b2  c2

28. 9x2  4

In Problems 61–78, factor completely, relative to the integers. In polynomials involving more than three terms, try grouping the terms as a first step. If a polynomial is prime relative to the integers, say so.

29. 4x2  9

30. 16x2  25

61. x3  3x2  9x  27

62. x3  x2  x  1

63. a3  2a2  a  2

64. t3  2t 2  t  2

In Problems 31–46, factor completely, relative to the integers. If a polynomial is prime relative to the integers, say so. 31. 6x2  48x  72

32. 3z2  28z  48

33. 2y3  22y2  48y

34. 2x4  24x3  40x2

35. 16x2y  8xy  y

36. 4xy2  12xy  9x

37. 6m2  mn  12n2

38. 6s2  7st  2t2

39. x3y  9xy3

40. 4u3v  uv3

41. 3m3  6m2  15m

42. 2x3  2x2  8x

43. m3  n3

44. r3  t3

45. 8x3  125

46. 27y3  1

Problems 47–54 are calculus-related. Factor completely, relative to the integers. 47. 2x(x  1)  4x (x  1) 4

2

3

48. (x  1)3  3x(x  1)2 49. 6(3x  5)(2x  3)2  4(3x  5)2(2x  3) 50. 2(x  3)(4x  7)2  8(x  3)2(4x  7) 51. 5x4(9  x)4  4x5(9  x)3 52. 3x (x  7)  4x (x  7) 4

2

3

3

53. 2(x  1)(x2  5)2  4x(x  1)2(x2  5) 54. 4(x  3)3(x2  2)3  6x(x  3)4(x2  2)2 In Problems 55–60, factor completely, relative to the integers. In polynomials involving more than three terms, try grouping the terms in various combinations as a first step. If a polynomial is prime relative to the integers, say so. 55. (a  b)2  4(c  d )2 56. (x  2)2  9 57. 2am  3an  2bm  3bn 58. 15ac  20ad  3bc  4bd 59. 3x2  2xy  4y2 60. 5u2  4uv  v2

65. 4(A  B)2  5(A  B)  5 66. 6(x  y)2  23(x  y)  4 67. x4  6x2  8

68. x4  x2  6

69. m4  n4

70. y4  3y2  4

71. y4  11y2  18

72. 27a2  a5b3

73. m2  2mn  n2  m  n 74. y2  2xy  x2  y  x 75. 18a3  8a(x2  8x  16) 76. 25(4x2  12xy  9y2)  9a2b2 77. x4  2x2  1  x2 78. a4  2a2b2  b4  a2b2 79. Find a factor formula for u7  v7, where the first factor is u  v and the second factor is a sum of terms of the form aunvm. Verify your formula by multiplication. 80. Show that u8  v8  (u  v)(u  v)(u2  v2)(u4  v4). 81. Use the Sieve of Eratosthenes to find all prime numbers less than 200. 82. If p and q  p  2 are both prime numbers, then p and q are called twin primes. Use the results of Problem 81 to find all twin primes less than 200. 83. To show that 12 is an irrational number, explain how the assumption that 12 is rational leads to a contradiction of Theorem 1, the fundamental theorem of arithmetic, by the following steps: (A) Suppose that 12  a b, where a and b are positive integers, b  0. Explain why a2  2b2. (B) Explain why the prime number 2 appears an even number of times (possibly 0 times) as a factor in the prime factorization of a2. (C) Explain why the prime number 2 appears an odd number of times as a factor in the prime factorization of 2b2. (D) Explain why parts (B) and (C) contradict the fundamental theorem of arithmetic.

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84. To show that 1n is an irrational number unless n is a perfect square, explain how the assumption that 1n is rational leads to a contradiction of the fundamental theorem of arithmetic by the following steps: (A) Assume that n is not a perfect square, that is, does not belong to the sequence 1, 4, 9, 16, 25, . . . . Explain why some prime number p appears an odd number of times as a factor in the prime factorization of n. (B) Suppose that 1n  a b, where a and b are positive integers, b  0. Explain why a2  nb2. (C) Explain why the prime number p appears an even number of times (possibly 0 times) as a factor in the prime factorization of a2. (D) Explain why the prime number p appears an odd number of times as a factor in the prime factorization of nb2. (E) Explain why parts (C) and (D) contradict the fundamental theorem of arithmetic.

20 inches x

x

x

x

20 inches

x

x x

x

APPLICATIONS 85. CONSTRUCTION A rectangular open-topped box is to be constructed out of 20-inch-square sheets of thin cardboard by cutting x-inch squares out of each corner and bending the sides up as indicated in the figure. Express each of the following quantities as a polynomial in both factored and expanded form. (A) The area of cardboard after the corners have been removed. (B) The volume of the box.

R-6

86. CONSTRUCTION A rectangular open-topped box is to be constructed out of 9- by 16-inch sheets of thin cardboard by cutting x-inch squares out of each corner and bending the sides up. Express each of the following quantities as a polynomial in both factored and expanded form. (A) The area of cardboard after the corners have been removed. (B) The volume of the box.

Rational Expressions: Basic Operations Z Reducing to Lowest Terms Z Multiplication and Division Z Addition and Subtraction Z Compound Fractions

A quotient of two algebraic expressions, division by 0 excluded, is called a fractional expression. If both the numerator and denominator of a fractional expression are polynomials, the fractional expression is called a rational expression. Some examples of rational expressions are the following (recall, a nonzero constant is a polynomial of degree 0): x2 2x  3x  5 2

1 x 1 4

3 x

x2  3x  5 1

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In this section we discuss basic operations on rational expressions, including multiplication, division, addition, and subtraction. Since variables represent real numbers in the rational expressions we are going to consider, the properties of real number fractions summarized in Section R-1 play a central role in much of the work that we will do. Even though not always explicitly stated, we always assume that variables are restricted so that division by 0 is excluded.

Z Reducing to Lowest Terms We start this discussion by restating the fundamental property of fractions (from Theorem 3 in Section R-1):

Z FUNDAMENTAL PROPERTY OF FRACTIONS If a, b, and k are real numbers with b, k  0, then ka a  kb b

23 3  24 4

(x  3)2

2  (x  3)x x x  0, x  3

Using this property from left to right to eliminate all common factors from the numerator and the denominator of a given fraction is referred to as reducing a fraction to lowest terms. We are actually dividing the numerator and denominator by the same nonzero common factor. Using the property from right to left—that is, multiplying the numerator and the denominator by the same nonzero factor—is referred to as raising a fraction to higher terms. We will use the property in both directions in the material that follows. We say that a rational expression is reduced to lowest terms if the numerator and denominator do not have any factors in common. Unless stated to the contrary, factors will be relative to the integers.

EXAMPLE

1

Reducing Rational Expressions Reduce each rational expression to lowest terms. (A)

(x  3)2 x2  6x  9  (x  3)(x  3) x2  9 

x3 x3

Factor numerator and denominator completely. Divide numerator and denominator by (x  3); this is a valid operation as long as x  3.

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(x  1)(x2  x  1) x3  1  (B) 2 (x  1)(x  1) x 1

Dividing numerator and denominator by (x  1) can be indicated by drawing lines through both (x  1)’s and writing the resulting quotients, 1’s.

1

x2  x  1  x1

x  1 and x  1

MATCHED PROBLEM



1

Reduce each rational expression to lowest terms. (A)

6x2  x  2 2x2  x  1

ZZZ

(B)

x4  8x 3x3  2x2  8x



CAUTION ZZZ

Remember to always factor the numerator and denominator first, then divide out any common factors. Do not indiscriminately eliminate terms that appear in both the numerator and the denominator. For example, 2x3  y2 y2

1



2x3  y2 y2 1

2x3  y2 y2

 2x3  1

Since the term y2 is not a factor of the numerator, it cannot be eliminated. In fact, (2x3  y2) y2 is already reduced to lowest terms.

Z Multiplication and Division Since we are restricting variable replacements to real numbers, multiplication and division of rational expressions follow the rules for multiplying and dividing real number fractions (Theorem 3 in Section R-1). Z MULTIPLICATION AND DIVISION If a, b, c, and d are real numbers with b, d  0, then: 1.

a c ac   b d bd

2.

a c a d    b d b c

2 x 2x   3 x1 3(x  1)

c0

2 x 2 x1    3 x1 3 x

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ZZZ EXPLORE-DISCUSS

Rational Expressions: Basic Operations

61

1

Write a verbal description of the process of multiplying two fractions. Do the same for the quotient of two fractions.

EXAMPLE

2

Multiplying and Dividing Rational Expressions Perform the indicated operations and reduce to lowest terms. 5x2

11

10x3y 10x3y (x  3)(x  3) x2  9  2   (A) 3xy  9y 4x  12x 3y(x  3) 4x(x  3) 31

21

Factor numerators and denominators; then divide any numerator and any denominator with a like common factor.

2



5x 6

1

2(2  x) 4  2x 1  (x  2)   (B) 4 4 x2

x  2 is the same as

x2 . 1

2

1

(x  2) 2x   2(x  2) 2(x  2)

b  a  (a  b), a useful change in some problems.

1



(C)

2x3  2x2y  2xy2 x y  xy 3

2

3



1 2 x3  y3

c a d a    b d b c

x  2xy  y 2

2

1

1

2x(x2  xy  y2) (x  y)2   xy(x  y)(x  y) (x  y)(x2  xy  y2) y



1

1

Divide out common factors.

1

2 y(x  y)



MATCHED PROBLEM

2

Perform the indicated operations and reduce to lowest terms. (A)

12x2y3 2xy  6xy 2

y2  6y  9 3y  9y 3

2

m n m3n  m2n2  mn3  2m2  mn  n2 2m3n2  m2n3 3

(C)



(B) (4  x) 

x2  16 5

3



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Z Addition and Subtraction Again, because we are restricting variable replacements to real numbers, addition and subtraction of rational expressions follow the rules for adding and subtracting real number fractions (Theorem 3 in Section R-1).

Z ADDITION AND SUBTRACTION For a, b, and c real numbers with b  0: 1.

a c ac   b b b

x 2 x2   x3 x3 x3

2.

a c ac   b b b

2xy2

x



x4 2xy2



x  (x  4) 2xy2

Thus, we add rational expressions with the same denominators by adding or subtracting their numerators and placing the result over the common denominator. If the denominators are not the same, we raise the fractions to higher terms, using the fundamental property of fractions to obtain common denominators, and then proceed as described. Even though any common denominator will do, our work will be simplified if the least common denominator (LCD) is used. Often, the LCD is obvious, but if it is not, the steps in the box describe how to find it.

Z THE LEAST COMMON DENOMINATOR (LCD) The LCD of two or more rational expressions is found as follows: 1. Factor each denominator completely. 2. Identify each different prime factor from all the denominators. 3. Form a product using each different factor to the highest power that occurs in any one denominator. This product is the LCD.

EXAMPLE

3

Adding and Subtracting Rational Expressions Combine into a single fraction and reduce to lowest terms. (A)

3 5 11   10 6 45

(C)

x3 x2 5  2  3x x  6x  9 x 9 2

(B)

5x 4  21 9x 6y

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63

SOLUTIONS

(A) To find the LCD, factor each denominator completely:



10  2  5 6  2  3 LCD  2  32  5  90 45  32  5 Now use the fundamental property of fractions to make each denominator 90: 3 5 11 93 15  5 2  11      10 6 45 9  10 15  6 2  45

(B)



27 75 22   90 90 90



27  75  22 80 8   90 90 9

Combine into a single fraction.



9x  32x LCD  2  32xy2  18xy2 6y2  2  3y2 2y2  4 18xy2 3x  5x 4 5x   21 2  9x 6y 2y  9x 3x  6y2 18xy2 

(C)

Multiply.

Multiply, combine.

8y2  15x2  18xy2 18xy2

x2 5 x3 x3 x2 5  2    2  3x (x  3)(x  3) x3 x  6x  9 x 9 (x  3) 2

Note: 

5 5 5   3x (x  3) x3

We have again used the fact that a  b  (b  a).

The LCD  (x  3)2(x  3). Thus, (x  3)2

(x  3)(x  2) 5(x  3)(x  3)   (x  3)2(x  3) (x  3)2(x  3) (x  3)2(x  3) (x2  6x  9)  (x2  x  6)  5(x2  9)  (x  3)2(x  3) 2 x  6x  9  x2  x  6  5x2  45  (x  3)2(x  3) 2 5x  7x  30  (x  3)2(x  3)

Expand numerators. Be careful of sign errors here.

Combine like terms.



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MATCHED PROBLEM

3

Combine into a single fraction and reduce to lowest terms. (A) (C)

5 1 6   28 10 35 y3 y 4 2

ZZZ



y2 y  4y  4 2

(B) 

1 2x  1 3   2 3 12x 4x 3x

2 2y



EXPLORE-DISCUSS 2

16 What is the value of 4 ? 2 What is the result of entering 16  4  2 on a calculator? What is the difference between 16  (4  2) and (16  4)  2? How could you use fraction bars to distinguish between these two cases when 16 writing 4 ? 2

Z Compound Fractions A fractional expression with fractions in its numerator, denominator, or both is called a compound fraction. It is often necessary to represent a compound fraction as a simple fraction—that is (in all cases we will consider), as the quotient of two polynomials. The process does not involve any new concepts. It is a matter of applying old concepts and processes in the right sequence. We will illustrate two approaches to the problem, each with its own merits, depending on the particular problem under consideration.

EXAMPLE

4

Simplifying Compound Fractions Express as a simple fraction reduced to lowest terms: 2 1 x 4 1 x2

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65

SOLUTION

Method 1. Multiply the numerator and denominator by the LCD of all fractions in the numerator and denominator—in this case, x2. (We are multiplying by 1  x2 x2.) 2 x2a  1b x 4 x2a 2  1b x

2 x2  x2 x  4 x2 2  x2 x

1

x(2  x) 2x  x2  2  (2  x)(2  x) 4x 1



x 2x

Method 2. Write the numerator and denominator as single fractions. Then treat as a quotient. 2 2x 1 x 1 x x 4  x2 2x 2x x2      x x 4 4  x2 (2  x)(2  x) x2 1 1 2  1 2 x x 

x 2x

MATCHED PROBLEM



4

Express as a simple fraction reduced to lowest terms. Use the two methods described in Example 4. 1 x 1 x x

1

ANSWERS 1. (A)

3x  2 x1

2. (A) 2x 3. (A) 4.

1 4

1 x1

TO MATCHED PROBLEMS (B)

x2  2x  4 3x  4

5 (C) mn x4 2 2y2  9y  6 3x  5x  4 (B) (C) 12x3 ( y  2)2( y  2) (B)



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Exercises

In Problems 1–8, reduce each rational expression to lowest terms.

16.

x2  9 x3  x1 x2  1

1.

42 105

17.

1 1 1   c a b

2.

360 216

18.

1 1 1   ac bc ab

3.

x1 x  3x  2

19.

2a  b 2a  3b  2 2 2 a b a  2ab  b2

4.

x2  2x  24 x6

20.

x2 x2  x2  1 (x  1)2

5.

x2  9 x2  3x  18

21. m  2 

6.

x2  9x  20 x2  16

23.

3 2  x2 2x

25.

4y 2 3   2 y2 y2 y 4

26.

2 3 4x   2 x  y x  y x y

7. 8.

2

3x2y3 x4y 2a2b4c6 6a5b3c

In Problems 9–28, perform the indicated operations and reduce answers to lowest terms. Represent any compound fractions as simple fractions reduced to lowest terms. 9.

5 11  6 15

10.

7 19  10 25

11. a

b b2 a  2b  2a 3b a

b2 b a 12. a 2 b 2a a 3b 1 1 13.  n m m n 14.  n m x1 x2  1  2 15. x2 x 4

m2 m1

22.

x1 x x1

24.

2 1  a3 3a

2

x2 1 y2 27. x 1 y

4 x x 28. 2 1 x

Problems 29–34 are calculus-related. Reduce each fraction to lowest terms. 29.

6x3(x2  2)2  2x(x2  2)3 x4 4x (x  3)  3x2(x2  3)2 4

30. 31. 32. 33. 34.

2

x6 2x(1  3x)3  9x2(1  3x)2 (1  3x)6 2x(2x  3)4  8x2(2x  3)3 (2x  3)8 2x(x  4)3  3(3  x2)(x  4)2 (x  4)6 3x2(x  1)3  3(x3  4)(x  1)2 (x  1)6

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In Problems 35–48, perform the indicated operations and reduce answers to lowest terms. Represent any compound fractions as simple fractions reduced to lowest terms. 35.

y y2  2y  8



2 1  2 y2  5y  4 y y2

Rational Expressions: Basic Operations

In Problems 53–60, imagine that the indicated “solutions” were given to you by a student whom you were tutoring in this class. (A) Is the solution correct? If the solution is incorrect, explain what is wrong and how it can be corrected. (B) Show a correct solution for each incorrect solution.

36.

x x4 x8   x6 x3 x  9x  18

53.

x2  5x  4 x2  5x x5  x x4

37.

16  m2 m1  m  3m  4 m  4

54.

x2  2x x2  2x  3  x2 x x3

38.

x1 x2  2x  1  x(1  x) x2  1

55.

(x  h)2  x2  (x  1)2  x2  2x  1 h

39.

y9 x7  ax  bx by  ay

56.

(x  h)3  x3  (x  1)3  x3  3x2  3x  1 h

40.

c2 c c2   5c  5 3c  3 1c

57.

x2  2x x2  2x  x  2 x2 1 x x2 x2  x  2

x2  16 x2  13x  36  2x2  10x  8 x3  1

58.

x3 2x  2  x  3 1 2  2   2 x1 x  1 x 1 x 1

59.

2x2 x x 2x2  x2  2x    x2 x2 x 4 x2  4

41.

2

2

42. a 43.

x3  y3 y3

x2  xy xy  y

2



x2  xy  y2 y b xy y2

a

x2  xy

x2  y2 x  2xy  y 2

2



x2  y2

44. a

xy  y2

45. a

4 1 x b  x4 x4 x2  16



x2  2xy  y2

b x2  2xy  y2

x2y  xy2

b

3 1 x4 46. a  b x2 x1 x2 2 15  2 x x 47. 5 4 1  2 x x 1

2

2

60. x 

x2  2xy  y2 x2y  xy2

y x 2 y x 48. y x  y x

Problems 49–52 are calculus-related. Perform the indicated operations and reduce answers to lowest terms. Represent any compound fractions as simple fractions reduced to lowest terms. 1 1  x xh 49. h

1 1  2 (x  h)2 x 50. h

(x  h)2 x2  xh2 x2 51. h

2x  3 2x  2h  3  x xh 52. h

67

xx2 2 x2  2  x2 x2  3x  2 x  3x  2

In Problems 61–64, perform the indicated operations and reduce answers to lowest terms. Represent any compound fractions as simple fractions reduced to lowest terms. y2 yx 61. x2 1 2 y  x2

s2 s st 62. 2 t t st

y

1

63. 2  1

2 a2

1

64. 1  1

1 1

1 x

In Problems 65 and 66, a, b, c, and d represent real numbers. 65. (A) Prove that dc is the multiplicative inverse of cd (c, d  0). (B) Use part (A) to prove that c a d a    b d b c

b, c, d  0

66. Prove that a c ac   b b b

b0

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R

Review xy  yx; identities: 0  x  x  0  x and (1)x  x(1)  x; inverses: x is the additive inverse of x and, if x  0, x1 is the multiplicative inverse of x; and distributive property: x( y  z)  xy  xz. Subtraction is defined by 1 a  b  a  (b) and division by ab  ab . Division by 0 is never allowed. Additional properties include properties of negatives: 1. (a)  a

Algebra and Real Numbers

A set is a collection of objects called elements or members of the set. Sets are usually described by listing the elements or by stating a rule that determines the elements. A set may be finite or infinite. A set with no elements is called the empty set or the null set and is denoted . A variable is a symbol that represents unspecified elements from a replacement set. A constant is a symbol for a single object. If each element of set A is also in set B, we say A is a subset of B and write A ( B. If two sets A and B have exactly the same elements, we say that A and B are equal and write A  B. The rational numbers are numbers that can be written in the form ab, where a and b are integers and b  0. A real number can be approximated to any desired precision by rational numbers. Consequently, arithmetic operations on the rational numbers can be extended to operations on real numbers. These operations satisfy basic real number properties, including associative properties: x  ( y  z)  (x  y)  z and x( yz)  (xy)z; commutative properties: x  y  y  x and

2. (a)b  (ab)  a(b)  ab 3. (a)(b)  ab 4. (1)a  a 5.

a a a   b b b

6.

a a a a    b b b b

b0 b0

Real numbers:

Real numbers (R ) NZQR Rational numbers (Q ) Irrational numbers (I )

Integers (Z )

Natural numbers (N )

Zero

Noninteger ratios of integers

Negatives of natural numbers

Real number line: Origin Negative real numbers

10

5

Coordinate

Positive real numbers

0

10

5

Positive direction

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zero properties:

69

Review

For n a natural number and a and b real numbers:

1. a  0  0

a is an nth root of b if an  b

2. ab  0 or both.

if and only if

a0

or

b0

and fraction properties (division by 0 excluded):

The principal nth root of b is denoted by b1n. If n is odd, b has one real nth root which is the principal nth root. If n is even and b 7 0, b has two real nth roots and the positive nth root is the principal nth root. If n is even and b 6 0, b has no real nth roots. Rational number exponents (even roots of negative numbers excluded):

1.

a c  b d

2.

ka a  kb b

bmn  (b1n)m  (bm)1n

3.

ac a c   b d bd

Scientific notation:

4.

a c a d    b d b c

if and only if ad  bc

a  10n

and

bmn 

1 bmn

1 a 6 10

n an integer, a in decimal form.

a c ac 5.   b b b

R-3

c ac a 6.   b b b

An nth root radical is defined by 2b  b1n, where b1n is the principal nth root of b, 1 is a radical, n is the index, and b is the radicand. Rational exponents and radicals are related by

7.

n

c ad  bc a   b d bd

R-2

bmn  (bm)1n  2bm  (b1n)m  ( 2b)m n

Properties of integer exponents (division by 0 excluded):

n

Properties of radicals (x 7 0, y 7 0):

Integer Exponents

an  a  a  p  a (n factors of a) for n a positive integer, a0  1 (a  0), and an  1 an for n a negative integer (a  0). 00 is not defined.

Radicals

n

1. 2xn  x n

n

n

2. 2xy  2x2y 3.

n x 1 x  n Ay 1y n

1. aman  amn

A radical is in simplified form if:

2. (an)m  amn 1. No radicand contains a factor to a power greater than or equal to the index of the radical.

3. (ab)m  ambm a m am 4. a b  m b b 5.

2. No power of the radicand and the index of the radical have a common factor other than 1.

1 am  amn  nm an a

3. No radical appears in a denominator. 4. No fraction appears within a radical.

Further exponent properties (division by 0 excluded): 1. (a b )  a m n p

2. a

m p

b

pm

a a b  pn bn b n

3.

pm pn

m

a b  n bm a

b n a n 4. a b  a b a b

Algebraic fractions containing radicals are rationalized by multiplying numerator and denominator by a rationalizing factor often determined by using a special product formula.

R-4

Polynomials: Basic Operations

An algebraic expression is formed by using constants and variables and the operations of addition, subtraction, multiplication, division, raising to powers, and taking roots. A polynomial is an algebraic expression formed by adding and subtracting

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constants and terms of the form axn (one variable), axnym (two variables), and so on. The degree of a term is the sum of the powers of all variables in the term, and the degree of a polynomial is the degree of the nonzero term with highest degree in the polynomial. Polynomials with one, two, or three terms are called monomials, binomials, and trinomials, respectively. Like terms have exactly the same variable factors to the same powers and can be combined by adding their coefficients. Polynomials can be added, subtracted, and multiplied by repeatedly applying the distributive property and combining like terms. Special products of binomials are:

properties. Grouping can be used to identify common factors. Second-degree polynomials can be factored by trial and error. The following special factoring formulas are useful:

1. (a  b)(a  b)  a2  b2

There is no factoring formula relative to the real numbers for u2  v2.

2. (a  b)2  a2  2ab  b2 3. (a  b)2  a2  2ab  b2

R-5

Polynomials: Factoring

A number or algebraic expression is factored if it is expressed as a product of other numbers or algebraic expressions, which are called factors. An integer greater than 1 is a prime number if its only positive integer factors are itself and 1, and a composite number otherwise. Each composite number can be factored uniquely into a product of prime numbers. A polynomial is prime relative to a given set of numbers (usually the set of integers) if (1) all its coefficients are from that set of numbers, and (2) it cannot be written as a product of two polynomials of positive degree having coefficients from that set of numbers. A nonprime polynomial is factored completely relative to a given set of numbers if it is written as a product of prime polynomials relative to that set of numbers. Common factors can be factored out by applying the distributive

CHAPTER

R

Work through all the problems in this chapter review and check answers in the back of the book. Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that type of problem is discussed. Where weaknesses show up, review appropriate sections in the text. 1. For A  {1, 2, 3, 4, 5}, B  {1, 2, 4}, and C  {4, 1, 2}, indicate true (T) or false (F): (A) 3 A (B) 5 C (C) B A (D) B ( A (E) B  C (F) A ( B

1. u2  2uv  v2  (u  v)2

Perfect Square

2. u  2uv  v  (u  v)

Perfect Square

2

2

2

3. u2  v2  (u  v)(u  v)

Difference of Squares

4. u  v  (u  v)(u  uv  v )

Difference of Cubes

5. u  v  (u  v)(u  uv  v )

Sum of Cubes

3

3

R-6

3

3

2

2

2

2

Rational Expressions: Basic Operations

A fractional expression is the ratio of two algebraic expressions, and a rational expression is the ratio of two polynomials. The rules for adding, subtracting, multiplying, and dividing real number fractions (see Section R-1 in this review) all extend to fractional expressions with the understanding that variables are always restricted to exclude division by zero. Fractions can be reduced to lowest terms or raised to higher terms by using the fundamental property of fractions: ka a  kb b

with b, k  0

A rational expression is reduced to lowest terms if the numerator and denominator do not have any factors in common relative to the integers. The least common denominator (LCD) is useful for adding and subtracting fractions with different denominators and for reducing compound fractions to simple fractions.

Review Exercises 2. Replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property: (A) Commutative (): x(y  z)  ? (B) Associative (): 2  (x  y)  ? (C) Distributive: (2  3)x  ? In Problems 3–6, perform the indicated operations and express the result in the form ab, where a and b are integers. 2 15 1 3. a  b 3 14

4.

9 5  10 12

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8 5. a  21 b 7

Review Exercises

71

Simplify Problems 30–35, and write answers using positive exponents only. All variables represent positive real numbers.

6. (41 91)1

9u8v6 3u4v8

30. 6(xy3)5

31.

32. (2  105)(3  103)

33. (x3y2)2

7. What is the degree of (a)?

34. u53u23

35. (9a4b2)12

8. What is the degree of (b)?

36. Change to radical form: 3x25

9. What is the degree of the sum of (a) and (b)?

3 37. Change to rational exponent form: 32 (xy)2

Problems 7–12 refer to the following polynomials: (a) x3  2 and (b) x5  x3  1.

10. What is the degree of the product of (a) and (b)? Simplify Problems 38–42, and express answers in simplified form. All variables represent positive real numbers.

11. Multiply (a) and (b). 12. Add (a) and (b).

3 5 4 38. 3x 2 xy

In Problems 13–18, evaluate each expression that results in a rational number. 13. 28912

14. 21613

15. 823

16. (64)53

9 12 17. a b 16

12

18. (121

12 34

 25

)

19. 5x2  3x 34  3(x  2) 4

20. (3m  5n)(3m  5n)

21. (2x  y)(3x  4y)

22. (2a  3b)2

In Problems 23–25, write each polynomial in a completely factored form relative to the integers. If the polynomial is prime relative to the integers, say so. 24. t 2  4t  6

25. 6n3  9n2  15n

2 4 1 26.  3 2 2 5b 3a 6a b y2 y2  4y  4

1 u u 29. 1 1 2 u



41.

15 3  15

8 6 42. 2 y

5x  x is an odd integer between 4 and 26 In Problems 44–49, each statement illustrates the use of one of the following real number properties or definitions. Indicate which one. Commutative () Identity () Commutative () Identity () Division Associative () Inverse () Associative () Inverse () Zero Distributive Subtraction Negatives 44. (3)  (2)  (3)  [(2)] 45. 3y  (2x  5)  (2x  5)  3y 46. (2x  3)(3x  5)  (2x  3)3x  (2x  3)5 47. 3  (5x)  (3  5)x

In Problems 26–29, perform the indicated operations and reduce to lowest terms. Represent all compound fractions as simple fractions reduced to lowest terms.

28.

6ab 13a

43. Write using the listing method:

In Problems 19–22, perform the indicated operations and simplify.

23. 9x2  12x  4

40.

39. 22x2y5 218x3y2

1 3x 27. 2  6x 3x  12x y2  2y

y2  4y  4

48.

a a  (b  c) bc

49. 3xy  0  3xy

50. Indicate true (T) or false (F): (A) An integer is a rational number and a real number. (B) An irrational number has a repeating decimal representation. 51. Give an example of an integer that is not a natural number. 52. Given the algebraic expressions: (a) 2x2  3x  5 (b) x2  1x  3 3 2 1 (c) x  x  3x (d) x2  3xy  y2 (A) Identify all second-degree polynomials. (B) Identify all third-degree polynomials.

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In Problems 53–57, perform the indicated operations and simplify. 53. (2x  y)(2x  y)  (2x  y)2

75. (x12  y12)2

77. Convert to scientific notation and simplify:

54. (m  2mn  n )(m  2mn  n ) 2

2

2

2

0.000 000 000 52 (1,300)(0.000 002)

55. 5(x  h)2  7(x  h)  (5x2  7x)

56. 2x5(x2  2)(x  3)  x[x  x(3  x)] 6 57. (x  2y)3

Evaluate Problems 78–85 to four significant digits using a calculator.

In Problems 58–64, write in a completely factored form relative to the integers. 58. (4x  y)2  9x2

59. 2x2  4xy  5y2

60. 6x3y  12x2y2  15xy3 61. (y  b)2  y  b 62. 3x3  24y3

63. y3  2y2  4y  8

64. 2x(x  4)3  3x2(x  4)2 In Problems 65–69, perform the indicated operations and reduce to lowest terms. Represent all compound fractions as simple fractions reduced to lowest terms. 65. 66.

67.

76. (3x12  y12)(2x12  3y12)

3x2(x  2)2  2x(x  2)3

78.

(20,410)(0.000 003 477) 0.000 000 022 09

79. 0.13475

80. (60.39)3

82. (0.000 000 419 9)27

5 83. 2 0.006 604

3 84. 2 3  12

85.

87.

x4 m3 2 m1  2  2m m  4m  4 m 4 y x2

a

xyxy x  3x  2 b 2x2  5x  3 2x  3x  1 2

3

1 68.

x y

1

1

1

a1  b1 69. 2 ab  ba2

70. Check the following solution. If it is wrong, explain what is wrong and how it can be corrected, and then show a correct solution. x2  3x  2 x1 x2  2x x2 2  x1 x x2 x x2 2

In Problems 71–76, perform the indicated operations, simplify and write answers using positive exponents only. All variables represent positive real numbers.

73. a

27x2y3 13 b 8x4y3

50 32 72. 2  2 3 2

2x2 24x 3

88.

3y2

B 8x2

3 90. 2 1 4x4

92.

31x 21x  1y

93.

21u  31v 21u  31v

y2 2y2  4  2

95. Rationalize the numerator:

1t  15 t5

96. Write in the form ax p  bxq, where a and b are real numbers and p and q are rational numbers: 41x  3 21x 97. Write the repeating decimal 0.545454 . . . in the form ab reduced to lowest terms, where a and b are positive integers. Is the number rational or irrational? 98. If M  54, 3, 26 and N  53, 0, 26, find: (A) 5x ƒ x M or x N6 (B) 5x ƒ x M and x N6 99. Evaluate x2  4x  1 for x  2  13. 100. Simplify: x(2x  1)(x  3)  (x  1)3 101. Factor completely with respect to the integers:

74. (a13b14)(9a13b12)32

5

91. (2 1x  51y)( 1x  1y)

94.

x y

8u1 2 u5 3 71. a 2 2 0 b a 3 b 2uv u

9 89. 2 8x6y12

2

1

1

212  312 213  313

In Problems 86–94, perform the indicated operations and express answers in simplified form. All radicands represent positive real numbers. 5 6 7 11 86. 2x2 3xy

2

81. 82.4583

4x(a2  4a  4)  9x3

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Review Exercises

73

102. Evaluate each expression on a calculator and determine which pairs have the same value. Verify these results algebraically. (A) 23  15  23  15 (B) 24  115  24  115 (C) 110

(A) Estimate the number of units produced by using 1,600 units of capital and 900 units of labor. (B) What is the effect on production if the number of units of capital and labor are doubled to 3,200 units and 1,800 units, respectively? (C) What is the effect on production of doubling the units of labor and capital at any production level?

In Problems 103–106, simplify and express answers using positive exponents only (m is an integer greater than 1).

113. ELECTRIC CIRCUIT If three electric resistors with resistances R1, R2, and R3 are connected in parallel, then the total resistance R for the circuit shown in the figure is given by

103.

8(x  2)3(x  3)2 4

104. a

2

12(x  2) (x  3)

b2 1 a2  1 b 1 b a

R

105. (x13  y13)(x23  x13y13  y23) 106. a

2

xm x

2m1

b

1(m1)

1 1 1 1   R1 R2 R3

m 7 1 Represent this compound fraction as a simple fraction.

107. Rationalize the denominator:

108. Rationalize the numerator:

1

R1

3 1 2 x

R2

3 3 2 t 2 5 t5 n1

n2

109. Write in simplified form: 2x x

2n1

R3

n 7 0

APPLICATIONS

★114.

110. CONSTRUCTION A circular fountain in a park includes a concrete wall that is 3 feet high and 2 feet thick (see the figure). If the inner radius of the wall is x feet, write an algebraic expression in terms of x that represents the volume of the concrete used to construct the wall. Simplify the expression.

CONSTRUCTION A box with a hinged lid is to be made out of a piece of cardboard that measures 16 by 30 inches. Six squares, x inches on a side, will be cut from each corner and the middle, and then the ends and sides will be folded up to form the box and its lid (see the figure). Express each of the following quantities as a polynomial in both factored and expanded form. (A) The area of cardboard after the corners have been removed. (B) The volume of the box. 30 in.

2 feet x 3 feet

111. ECONOMICS If in the United States in 2003 the total personal income was about $9,208,000,000,000 and the population was about 291,000,000, estimate to three significant digits the average personal income. Write your answer in scientific notation and in standard decimal form. 112. ECONOMICS The number of units N produced by a petroleum company from the use of x units of capital and y units of labor is approximated by N  20x12y12

16 in.

x feet

x

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R ACTIVITY Rational and Irrational Numbers

The set of real numbers can be partitioned into two disjoint subsets, the set of rational numbers and the set of irrational numbers. In this activity, we explore the connections between two different methods for representing rational numbers, and we prove that 12 is irrational. Rational numbers can be represented in two ways: as fractions, where both the numerator and denominator are integers, and as decimal expansions that are either repeating or terminating. Consider the rational number r  ab, where a and b are integers with no common factor and b  0. 1. If b  10n for a positive integer n, what kind of decimal expansion will r have? 2. If b  2m5n for positive integers m and n, what kind of decimal expansion will r have? 3. If r has a terminating decimal expansion, show that r can be expressed in the form a b, where b  2m5n. 4. Find a and b for the following repeating expansions (the overbar indicates the repeating block): (A) 0.63 (B) 0.486 (C) 0.846153 5. Find a and b for r  0.19 and then find a terminating decimal expansion for r. By the Pythagorean theorem, if the sides of a square have length 1, then the hypotenuse has length 12. The number 12 is approximately equal to 1.41421, but is not exactly equal to 1.41421. Why not? We suppose that 12  1.41421 and show that this assumption leads to a contradiction: 12  2

141,421 100,000

Square both sides.

141,4212 100,0002

Multiply both sides by 100,0002.

2(100,000)2  141,4212

Contradiction!

The last equation is a contradiction (the left-hand side is even but the right-hand side is odd), and therefore 12 is not exactly equal to 1.41421. By similar reasoning we can show that 12 is not equal to any rational number. First note that any integer is either even (belongs to {0, 2, 4, . . .}) or odd (belongs to { 1, 3, 5, . . .}), but not both. And any even integer is either twice an even (belongs to {0, 4, 8, . . .}) or twice an odd (belongs to { 2, 6, 10}), but not both. 6. In the following proof that 12 is irrational, explain case 2 and complete cases 3 and 4 by showing that each leads to a contradiction. Suppose that 12 is rational, that is, suppose that 12  a b, where a and b are integers and b  0. We will show that this assumption leads to a contradiction. Case 1. Suppose a and b are both even. Then a b can be reduced until either the numerator or denominator is odd, and one of the following cases holds. Case 2. Suppose a is odd and b is even. Then 2b2  a2. But 2b2 is even and a2 is odd (contradiction!). Case 3. Suppose a and b are both odd. Case 4. Suppose a is even and b is odd.

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Equations and Inequalities

IN this chapter we look at techniques for solving linear equations and inequalities. We introduce complex numbers and examine solution methods for several types of nonlinear equations. Throughout the chapter, we consider applications that can be solved using these techniques. Additional techniques for solving polynomial equations will be discussed in Chapter 4.

1

SECTIONS 1-1

Linear Equations and Applications

1-2

Linear Inequalities

1-3

Absolute Value in Equations and Inequalities

1-4

Complex Numbers

1-5

Quadratic Equations and Applications

1-6

Additional Equation-Solving Techniques Chapter 1 Review Chapter 1 Group Activity: Solving a Cubic Equation

CHAPTER

C

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Linear Equations and Applications Z Understanding Basic Terms Z Solving Linear Equations Z Using a Strategy to Solve Word Problems Z Solving Number and Geometric Problems Z Solving Rate–Time Problems Z Solving Mixture Problems

After discussing terminology and solution methods for linear equations, we consider a variety of word problems. The solutions of these word problems involve setting up and solving an appropriate linear equation. The solution of this linear equation then produces a solution to the word problem.

Z Understanding Basic Terms An algebraic equation is a mathematical statement that relates two algebraic expressions involving at least one variable. Some examples of equations with x as the variable are 3x  2  7 2x2  3x  5  0

1 x  1x x2 1x  4  x  1

The replacement set, or domain, for a variable is defined to be the set of numbers that are permitted to replace the variable. Z ASSUMPTION On Domains of Variables Unless stated to the contrary, we assume that the domain for a variable in an algebraic expression or equation is the set of those real numbers for which the algebraic expressions involving the variable are real numbers.

For example, the domain for the variable x in the expression 2x  4 is R, the set of all real numbers, since 2x  4 represents a real number for all replacements of x by real numbers. The domain of x in the equation 1 2  x x3

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77

is the set of all real numbers except 0 and 3. These values are excluded because the expression on the left is not defined for x  0 and the expression on the right is not defined for x  3. Both expressions represent real numbers for all other replacements of x by real numbers. The solution set for an equation is defined to be the set of elements in the domain of the variable that makes the equation true. Each element of the solution set is called a solution, or root, of the equation. To solve an equation is to find the solution set for the equation. An equation is called an identity if the equation is true for all elements from the domain of the variable. An equation is called a conditional equation if it is true for certain domain values and false for others. For example, 2x  4  2(x  2)

and

5 5  x(x  3) x2  3x

are identities, since both equations are true for all elements from the respective domains of their variables. On the other hand, the equations 3x  2  5

and

1 2  x x1

are conditional equations, since, for example, neither equation is true for the domain value 2. Knowing what we mean by the solution set of an equation is one thing; finding it is another. We introduce the idea of equivalent equations to help us find solutions. We will call two equations equivalent if they both have the same solution set for a given replacement set. To solve an equation, we perform operations on the equation to produce simpler equivalent equations. We stop when we find an equation whose solution is obvious. Then we check this obvious solution in the original equation. Any of the properties of equality given in Theorem 1 can be used to produce equivalent equations.

Z THEOREM 1 Properties of Equality For a, b, and c any real numbers: 1. If a  b, then a  c  b  c. 2. If a  b, then a  c  b  c. 3. If a  b and c  0, then ca  cb. b a 4. If a  b and c  0, then  . c c 5. If a  b, then either may replace the other in any statement without changing the truth or falsity of the statement.

Addition Property Subtraction Property Multiplication Property Division Property Substitution Property

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Z Solving Linear Equations We now turn our attention to methods of solving first-degree, or linear, equations in one variable. Z DEFINITION 1 Linear Equation in One Variable Any equation that can be written in the form ax  b  0

a0

Standard Form

where a and b are real constants and x is a variable, is called a linear, or first-degree, equation in one variable. 5x  1  2(x  3) is a linear equation, since it can be written in the standard form 3x  7  0.

EXAMPLE

1

Solving a Linear Equation Solve 5x  9  3x  7 and check. SOLUTION

We use the properties of equality to transform the given equation into an equivalent equation whose solution is obvious. 5x  9  3x  7 5x  9  9  3x  7  9 5x  3x  16 5x  3x  3x  16  3x 2x  16 2x 16  2 2 x8

Add 9 to both sides. Combine like terms. Subtract 3x from both sides. Combine like terms. Divide both sides by 2. Simplify.

The solution set for this last equation is obvious: Solution set: {8} And since the equation x  8 is equivalent to all the preceding equations in our solution, {8} is also the solution set for all these equations, including the original equation. [Note: If an equation has only one element in its solution set, we generally use the last equation (in this case, x  8) rather than set notation to represent the solution.] CHECK

5x  9  3x  7 ? 5(8)  9  3(8)  7 ? 40  9  24  7 ✓ 31  31

Substitute x  8. Simplify each side.

A true statement



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MATCHED PROBLEM

Linear Equations and Applications

1

Solve and check: 7x  10  4x  5

A  lw



We frequently encounter equations involving more than one variable. For example, if l and w are the length and width of a rectangle, respectively, the area of the rectangle is given by (see Fig. 1).

w

A  lw

l

Z Figure 1

Depending on the situation, we may want to solve this equation for l or w. To solve for w, we simply consider A and l to be constants and w to be a variable. Then the equation A  lw becomes a linear equation in w that can be solved easily by dividing both sides by l:

Area of a rectangle.

w

EXAMPLE

79

2

A l

l0

Solving an Equation with More Than One Variable Solve for P in terms of the other variables: A  P  Prt SOLUTION

A  P  Prt A  P(1  rt)

Factor to isolate P. Divide both sides by 1  rt.

A P 1  rt P

MATCHED PROBLEM

A 1  rt

Restriction: 1  rt  0



2

Solve for F in terms of C: C  59(F  32)



Z Using a Strategy to Solve Word Problems A great many practical problems can be solved using algebraic techniques—so many, in fact, that there is no one method of attack that will work for all. However, we can put together a strategy that will help you organize your approach.

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Z STRATEGY FOR SOLVING WORD PROBLEMS 1. Read the problem carefully—several times if necessary—that is, until you understand the problem, know what is to be found, and know what is given. 2. Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully. 3. If appropriate, draw figures or diagrams and label known and unknown parts. 4. Look for formulas connecting the known quantities to the unknown quantities. 5. Form an equation relating the unknown quantities to the known quantities. 6. Solve the equation and write answers to all questions asked in the problem. 7. Check and interpret all solutions in terms of the original problem—not just the equation found in step 5—because a mistake may have been made in setting up the equation in step 5.

ZZZ EXPLORE-DISCUSS

1

Translate each of the following sentences involving two numbers into an equation. (A) The first number is 10 more than the second number. (B) The first number is 15 less than the second number. (C) The first number is half the second number. (D) The first number is three times the second number. (E) Ten times the first number is 15 more than the second number.

The remaining examples in this section contain solutions to a variety of word problems illustrating both the process of setting up word problems and the techniques used to solve the resulting equations. It is suggested that you cover up a solution, try solving the problem yourself, and uncover just enough of a solution to get you going again in case you get stuck. After successfully completing an example, try the matched problem. After completing the section in this way, you will be ready to attempt a fairly large variety of applications.

Z Solving Number and Geometric Problems The first examples introduce the process of setting up and solving word problems in a simple mathematical context. Following these, the examples are of a more concrete nature.

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EXAMPLE

3

Linear Equations and Applications

81

Setting Up and Solving a Word Problem Find four consecutive even integers such that the sum of the first three exceeds the fourth by 8. SOLUTION

Let x  the first even integer, then x

x2

x4

and

x6

represent four consecutive even integers starting with the even integer x. (Remember, even integers increase by 2.) The phrase “the sum of the first three exceeds the fourth by 8” translates into an equation: Sum of the first three  Fourth  Excess x  (x  2)  (x  4)  (x  6)  8 Collect like terms. 3x  6  x  14 Subtract 6 from both sides. 2x  8 Divide both sides by 2. x4 The four consecutive integers are 4, 6, 8, and 10. CHECK

4  6  8  18 8 10

MATCHED PROBLEM

Sum of first three Excess Fourth



3

Find three consecutive odd integers such that 3 times their sum is 5 more than 8 times the middle one. 

ZZZ EXPLORE-DISCUSS

2

According to property 1 of Theorem 1, multiplying both sides of an equation by a nonzero number always produces an equivalent equation. By what number would you choose to multiply both sides of the following equation to eliminate all the fractions? x1 x 1   3 4 2 If you did not choose 12, the LCD of all the fractions in this equation, you could still solve the resulting equation, but with more effort. (For a discussion of LCDs and how to find them, see Section R-6.)

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4

Using a Diagram in the Solution of a Word Problem If one side of a triangle is one-third the perimeter, the second side is one-fifth the perimeter, and the third side is 7 meters, what is the perimeter of the triangle?

a

b

SOLUTION

Let p  the perimeter. Draw a triangle and label the sides, as shown in Figure 2. Then

c pabc p 5

p 3

pabc p p p  7 3 5

7 meters

Z Figure 2

Multiply both sides by 15, the LCD. This and the next step usually can be done mentally.

p p 15  p  15  a   7b 3 5 p p 15p  15   15   15  7 3 5 15p  5p  3p  105 15p  8p  105 7p  105 p  15

*

Collect like terms. Subtract 8p from both sides. Divide both sides by 7.

The perimeter is 15 meters. CHECK

p 15  5 3 3 p 15  3 5 5 7 15 meters

MATCHED PROBLEM

Side 1

Side 2 Side 3 Perimeter



4

If one side of a triangle is one-fourth the perimeter, the second side is 7 centimeters, and the third side is two-fifths the perimeter, what is the perimeter?  *Throughout the book, dashed boxes—called think boxes—are used to represent steps that may be performed mentally.

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Linear Equations and Applications

83

CAUTION ZZZ

A very common error occurs about now—students tend to confuse algebraic expressions involving fractions with algebraic equations involving fractions. Consider these two problems: (A) Solve:

x x   10 2 3

(B) Add:

x x   10 2 3

The problems look very much alike but are actually very different. To solve the equation in (A) we multiply both sides by 6 (the LCD) to clear the fractions. This works so well for equations that students want to do the same thing for problems like (B). The only catch is that (B) is not an equation, and the multiplication property of equality does not apply. If we multiply (B) by 6, we simply obtain an expression 6 times as large as the original! Compare the following: x x   10 2 3

(A) 6

(B)

x x   10 2 3

x x  6   6  10 2 3



3x  2x  60 5x  60 x  12

3x 2x 6  10   32 23 61

3x 2x 60   6 6 6 5x  60  6 

Z Solving Rate–Time Problems There are many types of quantity–rate–time problems and distance–rate–time problems. In general, if Q is the quantity of something produced (kilometers, words, parts, and so on) in T units of time (hours, years, minutes, seconds, and so on), then the formulas given in the box are useful. Z QUANTITY–RATE–TIME FORMULAS Q T Q  RT Q T R If Q is distance D, then

Quantity Time Quantity  (Rate)(Time) Quantity Time  Rate

R

R

Rate 

D T

D  RT

[Note: R is an average or uniform rate.]

T

D R

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ZZZ EXPLORE-DISCUSS

3

A bus leaves Milwaukee at 12:00 noon and travels due west on Interstate 94 at a constant rate of 55 miles per hour. A passenger that was left behind leaves Milwaukee in a taxicab at 1:00 P.M. in pursuit of the bus. The taxicab travels at a constant rate of 65 miles per hour. Let t represent time in hours after 12:00 noon. (A) How far has the bus traveled after t hours? (B) If t  1, how far has the taxicab traveled after t hours? (C) When will the taxicab catch up with the bus?

EXAMPLE

5

San Francisco 2,100

Honolulu

miles

A Distance–Rate–Time Problem The distance along a shipping route between San Francisco and Honolulu is 2,100 nautical miles. If one ship leaves San Francisco at the same time another leaves Honolulu, and if the former travels at 15 knots* and the latter at 20 knots, how long will it take the two ships to rendezvous? How far will they be from Honolulu and San Francisco at that time? SOLUTION

Let T  number of hours until both ships meet. Draw a diagram and label known and unknown parts. Both ships will have traveled the same amount of time when they meet.

H

D1  20T

D2  15T

20 knots

15 knots Meeting

Distance ship 1 Distance ship 2 from Honolulu from San Francisco ± ≤  ± ≤ travels to travels to meeting point meeting point D1  D2 20T  15T 35T T

SF

Total distance  ° from Honolulu ¢ to San Francisco    

2,100 2,100 2,100 60

*15 knots means 15 nautical miles per hour. There are 6,076.1 feet in 1 nautical mile, and 5,280 feet in 1 statute mile.

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Therefore, it takes 60 hours, or 2.5 days, for the ships to meet. Distance from Honolulu  20  60  1,200 nautical miles Distance from San Francisco  15  60  900 nautical miles CHECK

1,200  900  2,100 nautical miles

MATCHED PROBLEM



5

An old piece of equipment can print, stuff, and label 38 mailing pieces per minute. A newer model can handle 82 per minute. How long will it take for both pieces of equipment to prepare a mailing of 6,000 pieces? [Note: The mathematical form is the same as in Example 5.]  Some equations involving variables in a denominator can be transformed into linear equations. We can proceed in essentially the same way as in Example 5; however, we must exclude any value of the variable that will make a denominator 0. With these values excluded, we can multiply through by the LCD even though it contains a variable, and, according to Theorem 1, the new equation will be equivalent to the old.

EXAMPLE

6

A Distance–Rate–Time Problem An excursion boat takes 1.5 times as long to go 360 miles up a river as to return. If the boat cruises at 15 miles per hour in still water, what is the rate of the current? SOLUTION

Let

360 miles

x  Rate of current (in miles per hour) 15  x  Rate of boat upstream 15  x  Rate of boat downstream Time upstream  (1.5)(Time downstream) Distance upstream Distance downstream D Recall: T   (1.5) R Rate upstream Rate downstream 360 360 x  15, x  15  (1.5) 15  x 15  x 360 540 Multiply both sides by  (15  x)(15  x). 15  x 15  x Remove parentheses. 360(15  x)  540(15  x) Add 540x to both sides. 5,400  360x  8,100  540x Subtract 5,400 from both sides. 5,400  900x  8,100 Divide both sides by 900. 900x  2,700 x3 Therefore, the rate of the current is 3 miles per hour. The check is left to the reader. 

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MATCHED PROBLEM

6

A jetliner takes 1.2 times as long to fly from Paris to New York (3,600 miles) as to return. If the jet cruises at 550 miles per hour in still air, what is the average rate of the wind blowing in the direction of Paris from New York? 

ZZZ

EXPLORE-DISCUSS 4

Consider the following solution: x 2x  2 2 x2 x2 x  2x  4  2x  2 x2 Is x  2 a root of the original equation? If not, why? Discuss the importance of excluding values that make a denominator 0 when solving equations.

EXAMPLE

7

A Quantity–Rate–Time Problem An advertising firm has an old computer that can prepare a whole mailing in 6 hours. With the help of a newer model the job is complete in 2 hours. How long would it take the newer model to do the job alone? SOLUTION

Let x  time (in hours) for the newer model to do the whole job alone. a

Part of job completed b  (Rate)(Time) in a given length of time 1 Rate of old model  job per hour 6 1 Rate of new model  job per hour x Part of job completed Part of job completed ° by old model ¢  ° by new model ¢  1 whole job in 2 hours in 2 hours Rate of Time of Rate of Time of a ba ba ba b  1 Recall: Q  RT old model old model new model new model 1 1 (2)  (2)  1 x0 x 6

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1 3



2 x

1

1 3x a b 3



2 3x a b x

 3x

x



6

 3x

6

 2x

3

x

87

Multiply both sides by 3x, the LCD.

Subtract x from each side. Divide both sides by 2.

Therefore, the new computer could do the job alone in 3 hours. CHECK

Part of job completed by old model in 2 hours  2(16)  13 Part of job completed by new model in 2 hours  2(13)  23 Part of job completed by both models in 2 hours  1

MATCHED PROBLEM



7

Two pumps are used to fill a water storage tank at a resort. One pump can fill the tank by itself in 9 hours, and the other can fill it in 6 hours. How long will it take both pumps operating together to fill the tank? 

Z Solving Mixture Problems A variety of applications can be classified as mixture problems. Even though the problems come from different areas, their mathematical treatment is essentially the same.

EXAMPLE

8

A Mixture Problem How many liters of a mixture containing 80% alcohol should be added to 5 liters of a 20% solution to yield a 30% solution? SOLUTION

Let x  amount of 80% solution used. BEFORE MIXING 80% solution

AFTER MIXING

 x liters

30% solution

20% solution

 5 liters

(x  5) liters

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EQUATIONS AND INEQUALITIES

Amount of Amount of Amount of ° alcohol in ¢  ° alcohol in ¢  ° alcohol in ¢ first solution second solution mixture 0.8x  0.2(5)  0.3(x  5) 0.8x  1  0.3x  1.5 0.5x  0.5 x1 Add 1 liter of the 80% solution. CHECK

First solution Second solution Mixture

Liters of solution

Liters of alcohol

Percent alcohol

1 5 6

0.8(1)  0.8 0.2(5)  1 1.8

80 20 1.86  0.3, or 30%



MATCHED PROBLEM

8

A chemical storeroom has a 90% acid solution and a 40% acid solution. How many centiliters of the 90% solution should be added to 50 centiliters of the 40% solution to yield a 50% solution? 

ANSWERS

TO MATCHED PROBLEMS

1. x  5 2. F  95C  32 3. 3, 5, 7 4. 20 centimeters 5. 50 minutes 6. 50 miles per hour 7. 3.6 hours 8. 12.5 centiliters

1-1

Exercises

In Problems 1–16, solve each equation. 1. 4(x  5)  6(x  2)

2. 3( y  4)  2y  18

9. 0.1(t  0.5)  0.2t  0.3(t  0.4) 10. 0.1(w  0.5)  0.2w  0.2(w  0.4)

3. 5  4(w  1)  2w  2(w  4)

11. 0.35(s  0.34)  0.15s  0.2s  1.66

4. 4  3(t  2)  t  5(t  1)  7t

12. 0.35(u  0.34)  0.15u  0.2u  1.66

3a  4 7  2a  5 2

6.

2b  5 3b   4 7 2

13.

5 2 2  4 y 2 3y

14.

3w 1 4   6w 2w 3

x 2x  1 3x  4   2 3 4

8.

x 3x  1 6x  5   5 2 4

15.

z 1  2 z1 z1

16.

2 t  2 t1 t1

5. 5  7.

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S E C T I O N 1–1

In Problems 17–24, solve each equation. 17.

2m m4 4m  1   2 5 6 4

18.

3(n  2) 2n  3 4n  1   2 5 6 9

19. 1 

x3 2x  3  x2 x2

6 5 21. 1 y4 2y  8 23. 24.

38.

x2  4x  3 x2  1  x1 x1 x2  1  x2  4x  3 x1

4y 12 22. 5 y3 y3

3 3 3a  1  2  a a  4a  4 a  2a

1 x 3 39. 1 1 x x1

2

41. 1

1 1 10   2 b5 b5 b  5b  25

x

x

1 x

2 x

40.

In Problems 25–28, use a calculator to solve each equation to 3 significant digits.* 25. 3.142x  0.4835(x  4)  6.795

1 x

x1

2 x

3 y x a b 1y 1x

a

43. Solve for x in terms of y: y  1

26. 0.0512x  0.125(x  2)  0.725x 2.32x 3.76  2.32  x x2

28.

4.49x 6.08  4.49  x x3

mnp (m  n)m  (m  n)(n  p) m2  mn  mn  mp  n2  np

29. an  a1  (n  1)d for d (arithmetic progressions)

m  mn  mp  mn  n2  np 2

30. F  95C  32 for C (temperature scale) 31.

1 1 1 for f (simple lens formula)   f d1 d2

32.

1 1 1 for R1 (electric circuit)   R R1 R2

34. A  2ab  2ac  2bc for c 36. x 

3y  2 for y y3

In Problems 37 and 38, imagine that the indicated “solutions” were given to you by a student whom you were tutoring in this class. Is the solution right or wrong? If the solution is wrong, explain what is wrong and show a correct solution. 37.

m(m  n  p)  n(m  n  p) mn

APPLICATIONS

33. A  2ab  2ac  2bc for a (surface area of a rectangular solid)

2x  3 for x 3x  5

b xc

44. Let m and n be real numbers with m larger than n. Then there exists a positive real number p such that m  n  p. Find the fallacy in the following argument:

In Problems 29–36, solve for the indicated variable in terms of the other variables.

35. y 

1

x2

42. Solve for y in terms of x:

27.

89

In Problems 39–41, solve for x.

2x  3 3x  1 2 x1 x1

20.

Linear Equations and Applications

x 2x  3 4 x3 x3 x  4x  12  2x  3 x3

*Appendix A contains a brief discussion of significant digits.

These problems are grouped according to subject area. As before, the most difficult problems are marked with two stars (★★), the moderately difficult problems are marked with one star (★), and the easier problems are not marked. Numbers 45. Find a number such that 10 less than two-thirds the number is one-fourth the number. 46. Find a number such that 6 more than one-half the number is two-thirds the number. 47. Find four consecutive even integers so that the sum of the first three is 2 more than twice the fourth. 48. Find three consecutive even integers so that the first plus twice the second is twice the third.

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Geometry

(A) Let VS  Takeoff ground speed at sea level for a particular plane (in miles per hour)

49. Find the dimensions of a rectangle with a perimeter of 54 meters, if its length is 3 meters less than twice its width.

A  Altitude above sea level (in thousands of feet)

50. A rectangle 24 meters long has the same area as a square that is 12 meters on a side. What are the dimensions of the rectangle? 51. Find the perimeter of a triangle if one side is 16 feet, another side is two-sevenths the perimeter, and the third side is one-third the perimeter. 52. Find the perimeter of a triangle if one side is 11 centimeters, another side is two-fifths the perimeter, and the third side is onehalf the perimeter. Business and Economics 53. The sale price on a camera after a 20% discount is $72. What was the price before the discount? 54. A stereo store marks up each item it sells 60% above wholesale price. What is the wholesale price on a cassette player that retails at $144? 55. One employee of a computer store is paid a base salary of $2,150 a month plus an 8% commission on all sales over $7,000 during the month. How much must the employee sell in 1 month to earn a total of $3,170 for the month? 56. A second employee of the computer store in Problem 55 is paid a base salary of $1,175 a month plus a 5% commission on all sales during the month. (A) How much must this employee sell in 1 month to earn a total of $3,170 for the month? (B) Determine the sales level where both employees receive the same monthly income. If employees can select either of these payment methods, how would you advise an employee to make this selection? Earth Science ★57.

In 1970, Russian scientists began drilling a very deep borehole in the Kola Peninsula. Their goal was to reach a depth of 15 kilometers, but high temperatures in the borehole forced them to stop in 1994 after reaching a depth of 12 kilometers. They found that below 3 kilometers the temperature T increased 2.5°C for each additional 100 meters of depth. (A) If the temperature at 3 kilometers is 30°C and x is the depth of the hole in kilometers, write an equation using x that will give the temperature T in the hole at any depth beyond 3 kilometers. (B) What would the temperature be at 12 kilometers? (C) At what depth (in kilometers) would they reach a temperature of 200°C?

★58.

Because air is not as dense at high altitudes, planes require a higher ground speed to become airborne. A rule of thumb is 3% more ground speed per 1,000 feet of elevation, assuming no wind and no change in air temperature. (Compute numerical answers to 3 significant digits.)

V  Takeoff ground speed at altitude A for the same plane (in miles per hour) Write a formula relating these three quantities. (B) What takeoff ground speed would be required at Lake Tahoe airport (6,400 feet), if takeoff ground speed at San Francisco airport (sea level) is 120 miles per hour? (C) If a landing strip at a Colorado Rockies hunting lodge (8,500 feet) requires a takeoff ground speed of 125 miles per hour, what would be the takeoff ground speed in Los Angeles (sea level)? (D) If the takeoff ground speed at sea level is 135 miles per hour and the takeoff ground speed at a mountain resort is 155 miles per hour, what is the altitude of the mountain resort in thousands of feet? ★★59. An

earthquake emits a primary wave and a secondary wave. Near the surface of the Earth the primary wave travels at about 5 miles per second, and the secondary wave travels at about 3 miles per second. From the time lag between the two waves arriving at a given seismic station, it is possible to estimate the distance to the quake. Suppose a station measures a time difference of 12 seconds between the arrival of the two waves. How far is the earthquake from the station? (The epicenter can be located by obtaining distance bearings at three or more stations.)

★★60. A

ship using sound-sensing devices above and below water recorded a surface explosion 39 seconds sooner on its underwater device than on its above-water device. If sound travels in air at about 1,100 feet per second and in water at about 5,000 feet per second, how far away was the explosion? Life Science 61. A naturalist for a fish and game department estimated the total number of trout in a certain lake using the popular capture–mark–recapture technique. She netted, marked, and released 200 trout. A week later, allowing for thorough mixing, she again netted 200 trout and found 8 marked ones among them. Assuming that the ratio of marked trout to the total number in the second sample is the same as the ratio of all marked fish in the first sample to the total trout population in the lake, estimate the total number of fish in the lake. 62. Repeat Problem 61 with a first (marked) sample of 300 and a second sample of 180 with only 6 marked trout. Chemistry

★63.

How many gallons of distilled water must be mixed with 50 gallons of 30% alcohol solution to obtain a 25% solution?

★64.

How many gallons of hydrochloric acid must be added to 12 gallons of a 30% solution to obtain a 40% solution?

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S E C T I O N 1–1 ★65.

A chemist mixes distilled water with a 90% solution of sulfuric acid to produce a 50% solution. If 5 liters of distilled water is used, how much 50% solution is produced?

★ 66. A

fuel oil distributor has 120,000 gallons of fuel with 0.9% sulfur content, which exceeds pollution control standards of 0.8% sulfur content. How many gallons of fuel oil with a 0.3% sulfur content must be added to the 120,000 gallons to obtain fuel oil that will comply with the pollution control standards?

Linear Equations and Applications

91

frequency of 264 hertz (middle C on the piano), find the frequencies of the other two notes. [Hint: Set up two proportions using 4:5 and 4:6.] ★72. A

minor chord is composed of notes whose frequencies are in the ratio 10:12:15. If the first note of a minor chord is A, with a frequency of 220 hertz, what are the frequencies of the other two notes? Psychology

Rate–Time ★67. An old computer can do the weekly payroll in 5 hours. A newer

computer can do the same payroll in 3 hours. The old computer starts on the payroll, and after 1 hour the newer computer is brought on-line to work with the older computer until the job is finished. How long will it take both computers working together to finish the job? (Assume the computers operate independently.) ★68.

One pump can fill a gasoline storage tank in 8 hours. With a second pump working simultaneously, the tank can be filled in 3 hours. How long would it take the second pump to fill the tank operating alone?

★★69.

The cruising speed of an airplane is 150 miles per hour (relative to the ground). You wish to hire the plane for a 3-hour sightseeing trip. You instruct the pilot to fly north as far as she can and still return to the airport at the end of the allotted time. (A) How far north should the pilot fly if the wind is blowing from the north at 30 miles per hour? (B) How far north should the pilot fly if there is no wind?

★70.

Suppose you are at a river resort and rent a motor boat for 5 hours starting at 7 A.M. You are told that the boat will travel at 8 miles per hour upstream and 12 miles per hour returning. You decide that you would like to go as far up the river as you can and still be back at noon. At what time should you turn back, and how far from the resort will you be at that time? Music

★71.

A major chord in music is composed of notes whose frequencies are in the ratio 4:5:6. If the first note of a chord has a

73. In an experiment on motivation, Professor Brown trained a group of rats to run down a narrow passage in a cage to receive food in a goal box. He then put a harness on each rat and connected it to an overhead wire attached to a scale. In this way he could place the rat different distances from the food and measure the pull (in grams) of the rat toward the food. He found that the relationship between motivation (pull) and position was given approximately by the equation p  15d  70

30 d 170

where pull p is measured in grams and distance d in centimeters. When the pull registered was 40 grams, how far was the rat from the goal box? 74. Professor Brown performed the same kind of experiment as described in Problem 73, except that he replaced the food in the goal box with a mild electric shock. With the same kind of apparatus, he was able to measure the avoidance strength relative to the distance from the object to be avoided. He found that the avoidance strength a (measured in grams) was related to the distance d that the rat was from the shock (measured in centimeters) approximately by the equation a  43d  230

30 d 170

If the same rat were trained as described in this problem and in Problem 73, at what distance (to one decimal place) from the goal box would the approach and avoidance strengths be the same? (What do you think the rat would do at this point?) Puzzle 75. An oil-drilling rig in the Gulf of Mexico stands so that onefifth of it is in sand, 20 feet of it is in water, and two-thirds of it is in the air. What is the total height of the rig? 76. During a camping trip in the North Woods in Canada, a couple went one-third of the way by boat, 10 miles by foot, and one-sixth of the way by horse. How long was the trip? ★★77.

After exactly 12 o’clock noon, what time will the hands of a clock be together again?

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Linear Inequalities Z Understanding Inequality Relations and Interval Notation Z Solving Linear Inequalities Z Applying Linear Inequalities to Chemistry

We now turn to the problem of solving linear inequalities in one variable, such as 3(x  5)  5(x  7)  10

and

4 3  2x 6 7

Z Understanding Inequality Relations and Interval Notation The preceding mathematical forms involve the inequality, or order, relation—that is, “less than” and “greater than” relations. Just as we use  to replace the words “is equal to,” we use the inequality symbols and  to represent “is less than” and “is greater than,” respectively. While it probably seems obvious to you that 2 6 4

5 7 0

25,000 7 1

are true, it may not seem as obvious that 4 6 2

0 7 5

25,000 6 1

To make the inequality relation precise so that we can interpret it relative to all real numbers, we need a precise definition of the concept. Z DEFINITION 1 a  b and b  a For a and b real numbers, we say that a is less than b or b is greater than a and write a 6 b

or

b 7 a

if there exists a positive real number p such that a  p  b (or equivalently, b  a  p).

We certainly expect that if a positive number is added to any real number, the sum is larger than the original. That is essentially what the definition states.

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S E C T I O N 1–2

93

Linear Inequalities

When we write a b we mean a 6 b or a  b and say a is less than or equal to b. When we write ab

a

d

Z Figure 1 a b, c  d.

b

c

we mean a 7 b or a  b and say a is greater than or equal to b. The inequality symbols 6 and 7 have a very clear geometric interpretation on the real number line. If a 6 b, then a is to the left of b; if c 7 d, then c is to the right of d (Fig. 1). It is an interesting and useful fact that for any two real numbers a and b, either a 6 b, or a 7 b, or a  b. This is called the trichotomy property of real numbers. The double inequality a 6 x b means that x 7 a and x b; that is, x is between a and b, including b but not including a. The set of all real numbers x satisfying the inequality a 6 x b is called an interval and is represented by (a, b]. Thus, (a, b]  5x ƒ a 6 x b6* The number a is called the left endpoint of the interval, and the symbol “(” indicates that a is not included in the interval. The number b is called the right endpoint of the interval, and the symbol “]” indicates that b is included in the interval. An interval is closed if it contains its endpoint(s) and open if it does not contain any endpoint. Other types of intervals of real numbers are shown in Table 1. Table 1 Interval Notation Interval notation

Inequality notation

[a, b]

a x b

[a, b)

a x b

Line graph

Type

[

]

x

Closed

)

x

Half-open

]

x

Half-open

)

x

Open

[

x

Closed

(

x

Open

]

x

Closed

)

x

Open

a

b

[

a

(a, b] (a, b)

a x b a x b

[b, )

xb

(b, )

xb

( , a]

x a

( , a)

x a

b

(

a

b

(

a

b b b a a

*In general, 5x ƒ P(x)6 represents the set of all x such that statement P(x) is true. To express this set verbally, just read the vertical bar as “such that.”

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Note that the symbol “ ,” read “infinity,” used in Table 1 is not a numeral. When we write [b, ), we are simply referring to the interval starting at b and continuing indefinitely to the right. We would never write [b, ] or b x , because cannot be used as an endpoint of an interval. The interval ( , ) represents the set of real numbers R, since its graph is the entire real number line.

ZZZ

CAUTION ZZZ

It is important to note that 5 7 x3

is equivalent to [3, 5) and not to (5, 3]

In interval notation, the smaller number is always written to the left. Thus, it may be useful to rewrite the inequality as 3 x 6 5 before rewriting it in interval notation.

EXAMPLE

1

Graphing Intervals and Inequalities Write each of the following in inequality notation and graph on a real number line: (A) [2, 3)

(C) [ 2, )

(B) (4, 2)

(D) ( , 3)

SOLUTIONS

(A) 2 x 6 3 (B) 4 6 x 6 2 (C) x  2 (D) x 6 3

[

5

2

) 0

(

5

MATCHED PROBLEM

5

)

5 4 5

x

3

0

[

2

x

2

5

x 0

5

) 0

3

x



5

1

Write each of the following in interval notation and graph on a real number line: (A) 3 6 x 3

(B) 2  x  1

(C) x 7 1

(D) x 2



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S E C T I O N 1–2

ZZZ

Linear Inequalities

95

EXPLORE-DISCUSS 1

Example 1C shows the graph of the inequality x  2. What is the graph of x 6 2? What is the corresponding interval? Describe the relationship between these sets.

Since intervals are sets of real numbers, the set operations of union and intersection are often useful when working with intervals. The union of sets A and B, denoted by A  B, is the set formed by combining all the elements of A and all the elements of B. The intersection of sets A and B, denoted by A  B, is the set of elements of A that are also in B. Symbolically:

Z DEFINITION 2 Union and Intersection A ´ B  5x ƒ x is in A or x is in B6

Union:

{1, 2, 3} ´ {2, 3, 4, 5}  {1, 2, 3, 4, 5}

Intersection: A  B  5x ƒ x is in A and x is in B6 {1, 2, 3}  {2, 3, 4, 5}  {2, 3}

EXAMPLE

2

Graphing Unions and Intersections of Intervals If A  [2, 3], B  (1, 6), and C  (4, ), graph the indicated sets and write as a single interval, if possible: (A) A  B and A  B

(B) A  C and A  C

SOLUTIONS

(A) [

2 2

[

2 2

1

A  [2, 3]

]

3

6

1

3

6

1

3

(

]

(

) )

1

6

3

6

]

2

3

4

3

A  B  (1, 3]

4

(

2

]

(

3

4

2

3

4

[

A  B  [2, 6)

A  [2, 3]

(B) [

2

B  (1, 6)

C  (4, ) A  C  [2, 3]  (4, ) AC



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MATCHED PROBLEM

2

If D  [4, 1), E  (1, 3], and F  [2, ), graph the indicated sets and write as a single interval, if possible: (A) D  E

ZZZ

(B) D  E

(C) E  F

(D) E  F



EXPLORE-DISCUSS 2

Replace ? with 6 or 7 in each of the following. (A) 1 ? 3

and

2(1) ? 2(3)

(B) 1 ? 3

and

2(1) ? 2(3)

(C) 12 ? 8

and

12 8 ? 4 4

(D) 12 ? 8

and

12 8 ? 4 4

Based on these examples, describe verbally the effect of multiplying both sides of an inequality by a number.

Z Solving Linear Inequalities We now turn to the problem of solving linear inequalities in one variable, such as 2(2x  3) 6 6(x  2)  10

and

3 6 2x  3 9

The solution set for an inequality is the set of all values of the variable that make the inequality a true statement. Each element of the solution set is called a solution of the inequality. To solve an inequality is to find its solution set. Two inequalities are equivalent if they have the same solution set for a given replacement set. Just as with equations, we perform operations on inequalities that produce simpler equivalent inequalities, and continue the process until an inequality is reached whose solution is obvious. The properties of inequalities given in Theorem 1 can be used to produce equivalent inequalities.

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Linear Inequalities

97

Z THEOREM 1 Inequality Properties For a, b, and c any real numbers: 1. If a 6 b and b 6 c, then a 6 c. 2. If a 6 b, then a  c 6 b  c.

Transitive Property Addition Property

3. If a 6 b, then a  c 6 b  c.

Subtraction Property

2 6 4

2 6 4

23 6 43

2  3 6 4  3

4. If a 6 b and c is positive, then ca 6 cb. 2 6 4

3(2) 6 3(4)

5. If a 6 b and c is negative, then ca 7 cb. 2 6 4

(3)(2) 7 (3)(4)

6. If a 6 b and c is positive, then 2 6 4

2 4 6 2 2

7. If a 6 b and c is negative, then 2 6 4

a b 6 . c c a b 7 . c c

2 4 7 2 2

⎫ ⎪ ⎬ ⎪ ⎭ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Multiplication Property (Note difference between 4 and 5.)

Division Property (Note difference between 6 and 7.)

Similar properties hold if each inequality sign is reversed, or if is replaced with and  is replaced with . Thus, we find that we can perform essentially the same operations on inequalities that we perform on equations. When working with inequalities, however, we have to be particularly careful of the use of the multiplication and division properties. The order of the inequality reverses if we multiply or divide both sides of an inequality statement by a negative number.

ZZZ

EXPLORE-DISCUSS 3

Properties of equality are easily summarized: We can add, subtract, multiply, or divide both sides of an equation by any nonzero real number to produce an equivalent equation. Write a similar summary for the properties of inequalities.

Now let’s see how the inequality properties are used to solve linear inequalities. Examples 3, 4, and 5 will illustrate the process.

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3

Solving a Linear Inequality Solve and graph: 2(2x  3)  10 6 6(x  2) SOLUTION

2(2x  3)  10 6 6(x  2) 4x  6  10 6 6x  12

Remove parentheses. Combine like terms.

4x  4 6 6x  12

Add 4 to both sides.

4x  4  4 6 6x  12  4 4x 6 6x  8

Subtract 6x from both sides.

4x  6x 6 6x  8  6x 2x 6 8

Divide both sides by 2. Note that order reverses since 2 is negative

2x 8 7 2 2 x 7 4 2

3

(4, )

or

(

x

4

5

6

7

8

Solution set Graph of solution set

9

MATCHED PROBLEM

3

Solve and graph: 3(x  1)  5(x  2)  5

EXAMPLE

4





Solving a Linear Inequality Involving Fractions Solve and graph:

2x  3 4x 62 4 3

SOLUTION

2x  3 4x 62 4 3 12 

Multiply both sides by 12, the LCD.

2x  3 4x  12  6  12  2  12  4 3 3(2x  3)  72  24  4(4x) 6x  9  72  24  16x 6x  63  24  16x 6x  39  16x 10x  39 x 3.9 ]

3.9

or x

Remove parentheses. Collect like terms. Subtract 63 from both sides. Subtract 16x from both sides.

( , 3.9]

Order reverses when both sides are divided by 10, a negative number.



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S E C T I O N 1–2

MATCHED PROBLEM Solve and graph:

EXAMPLE

5

Linear Inequalities

99

4

4x  3 3x 8 6 6 3 2



Solving a Double Inequality Solve and graph: 3 4  7x 6 18 SOLUTION

We proceed as before, except we try to isolate x in the middle with a coefficient of 1. 3 4  7x 6 18

Subtract 4 from each member.

3  4 4  7x  4 6 18  4 7 7x 6 14

Divide each member by 7 and reverse each inequality.

7 7x 14  7 7 7 7 (

2

]

1  x 7 2

x

or

2 6 x 1

or

(2, 1]

1

MATCHED PROBLEM



5

Solve and graph: 3 6 7  2x 7



Z Applying Linear Inequalities to Chemistry EXAMPLE

6

Chemistry In a chemistry experiment, a solution of hydrochloric acid is to be kept between 30°C and 35°C—that is, 30 C 35. What is the range in temperature in degrees Fahrenheit if the Celsius/Fahrenheit conversion formula is C  59 (F  32)? SOLUTION

30 C 35 5 30 (F  32) 35 9

5 (F  32). 9 9 Multiply each member by . 5

Replace C with

9 9 5 9  30  (F  32)  35 5 5 9 5 54 F  32 63

Add 32 to each member.

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54  32 F  32  32 63  32 86 F 95 

The range of the temperature is from 86°F to 95°F, inclusive.

MATCHED PROBLEM

6

A film developer is to be kept between 68°F and 77°F—that is, 68 F 77. What is the range in temperature in degrees Celsius if the Celsius/Fahrenheit conversion formula is F  95C  32? 

TO MATCHED PROBLEMS )

1. (A) (3, 3] 5

3

0

5

4

1

(B) (C)

x

0

]

x

2

5

D  E  [ 4, 3]

]

1

(

5

3

D  E  (1, 1)

) 1

3

E  F  (1, )

(

1

(D)

5

0 1

5

1

x

2

(

5

4

1

5

]

1 0

(D) ( , 2] 2. (A) [

x

3

[

(B) [1, 2] (C) (1, )

[

ANSWERS

2

3

[

]

2

6

E  F  [2, 3]

3

6

3. x 4 or ( , 4]

7

4. x 7 6 or (6, )

]

x

4

0

( 5

x

6

5. 5 7 x  0 or 0 x 6 5 or [0, 5)

12 1

)

[ 0

5

x 6

6. 20 C 25: the range in temperature is from 20°C to 25°C

1-2

Exercises

In Problems 1–6, rewrite in inequality notation and graph on a real number line.

In Problems 7–12, rewrite in interval notation and graph on a real number line.

1. [ 8, 7 ]

2. (4, 8)

3. [6, 6)

7. 2 6 x 6

4. (3, 3 ]

5. [6, )

6. ( , 7)

10. 4 x 6 5

8. 5 x 5 11. x 2

9. 7 6 x 6 8 12. x 7 3

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In Problems 13–16, write in interval and inequality notation. 13. 10

[

[

14. 10

15.

10

)

5

5

0 0

5 5

5

]

]

10

5

x 5

10

In Problems 17–24, replace each ? with  or  to make the resulting statement true. and

12  5 ? 6  5

18. 4 ? 2

and

4  7 ? 2  7

19. 6 ? 8

and

6  3 ? 8  3

20.

4?9

and

42?92

21.

2 ? 1

and

2(2) ? 2(1)

22. 3 ? 2

and

4(3) ? 4(2)

23.

and

2 6 ? 2 2

and

10 15 ? 5 5

17.

12 ? 6

2?6

24. 10 ? 15

51.

q q4 3 7 1 7 3

53.

2x 1 3 2x  (x  3)  (x  2) 5 2 3 10

52.

p p2 p  4 3 2 4

10

( 0

101

In Problems 51–66, solve and graph.

x 10

x

0

16. 10

x 5

Linear Inequalities

In Problems 25–38, solve and graph. 25. 7x  8 6 4x  7

26. 4x  8  x  1

27. 3  x  5(3  x)

28. 2(x  3)  5 6 5  x

N 29. 7 4 2

M 30. 2 3

31. 5t 6 10

32. 7n  21

33. 3  m 6 4(m  3)

34. 2(1  u)  5u

B 1B 35. 2  4 3

y3 y 36. 1 7 4 2

37. 4 6 5t  6 21

38. 2 3m  7 6 14

In Problems 39–50, graph the indicated set and write as a single interval, if possible.

2 1 x x 54. (x  7)  7 (3  x)  3 4 2 6 55. 4 95x  32 68

56. 1 23A  5 11

57. 12 6 34 (2  x) 24

58. 24 23 (x  5) 6 36

59. 16 6 7  3x 31

60. 1 9  2x 6 5

61. 6 6 25 (1  x) 4

62. 15 7  25x 21

63. 0.1(x  7) 6 0.8  0.05x 64. 0.4(x  5) 7 0.3x  17 65. 0.3x  2.04  0.04(x  1) 66. 0.02x  5.32 0.5(x  2) Problems 67–72 are calculus-related. For what real number(s) x does each expression represent a real number? 67. 11  x

68. 1x  5

69. 13x  5

70. 17  2x

71.

1 22x  3 4

72.

1 25  6x 4

73. What can be said about the signs of the numbers a and b in each case? (A) ab 7 0 (B) ab 6 0 a a (C) 7 0 (D) 6 0 b b 74. What can be said about the signs of the numbers a, b, and c in each case? ab (A) abc 7 0 (B) 6 0 c 2 a a (C) (D) 7 0 6 0 bc bc

39. (5, 5)  [ 4, 7 ]

40. (5, 5)  [4, 7]

75. Replace each question mark with or , as appropriate: (A) If a  b  1, then a ? b. (B) If u  v  2, then u ? v.

41. [1, 4)  (2, 6]

42. [ 1, 4)  (2, 6]

76. For what p and q is p  q 6 p  q?

43. ( , 1)  (2, )

44. ( , 1)  (2, )

45. ( , 1)  [ 3, 7)

46. (1, 6] ´ [ 9, )

77. If both a and b are negative numbers and ba is greater than 1, then is a  b positive or negative?

47. [ 2, 3 ]  (1, 5)

48. [ 2, 3 ]  (1, 5)

49. ( , 4)  (1, 6 ]

50. (3, 2)  [0, )

78. If both a and b are positive numbers and b a is greater than 1, then is a  b positive or negative?

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79. Indicate true (T) or false (F): (A) If p 7 q and m 7 0, then mp 6 mq. (B) If p 6 q and m 6 0, then mp 7 mq. (C) If p 7 0 and q 6 0, then p  q 7 q. 80. Assume that m 7 n 7 0; then mn 7 n2 mn  m2 7 n2  m2 m(n  m) 7 (n  m)(n  m) m 7 nm 0 7 n But it was assumed that n 7 0. Find the error. Prove each inequality property in Problems 81–84, given a, b, and c are arbitrary real numbers. 81. If a 6 b, then a  c 6 b  c. 82. If a 6 b, then a  c 6 b  c. 83. (A) If a 6 b and c is positive, then ca 6 cb. (B) If a 6 b and c is negative, then ca 7 cb. b a 6 . c c b a (B) If a 6 b and c is negative, then 7 . c c

84. (A) If a 6 b and c is positive, then

APPLICATIONS Write all your answers using inequality notation. 85. EARTH SCIENCE In 1970, Russian scientists began drilling a very deep borehole in the Kola Peninsula. Their goal was to reach a depth of 15 kilometers, but high temperatures in the borehole forced them to stop in 1994 after reaching a depth of 12 kilometers. They found that the approximate temperature x kilometers below the surface of the Earth is given by T  30  25(x  3)

3 x 12

where T is temperature in degrees Celsius. At what depth is the temperature between 150°C and 250°C, inclusive? 86. EARTH SCIENCE As dry air moves upward it expands, and in so doing it cools at a rate of about 5.5°F for each 1,000-foot rise up to about 40,000 feet. If the ground temperature is 70°F, then the temperature T at height h is given approximately by T  70  0.0055h. For what range in altitude will the temperature be between 26°F and 40°F, inclusive? 87. BUSINESS AND ECONOMICS An electronics firm is planning to market a new graphing calculator. The fixed costs are $650,000 and the variable costs are $47 per calculator. The

wholesale price of the calculator will be $63. For the company to make a profit, it is clear that revenues must be greater than costs. (A) How many calculators must be sold for the company to make a profit? (B) How many calculators must be sold for the company to break even? (C) Discuss the relationship between the results in parts A and B. 88. BUSINESS AND ECONOMICS A video game manufacturer is planning to market a new version of its game machine. The fixed costs are $550,000 and the variable costs are $120 per machine. The wholesale price of the machine will be $140. (A) How many game machines must be sold for the company to make a profit? (B) How many game machines must be sold for the company to break even? (C) Discuss the relationship between the results in parts A and B. 89. BUSINESS AND ECONOMICS The electronics firm in Problem 87 finds that rising prices for parts increases the variable costs to $50.50 per calculator. (A) Discuss possible strategies the company might use to deal with this increase in costs. (B) If the company continues to sell the calculators for $63, how many must they sell now to make a profit? (C) If the company wants to start making a profit at the same production level as before the cost increase, how much should they increase the wholesale price? 90. BUSINESS AND ECONOMICS The video game manufacturer in Problem 88 finds that unexpected programming problems increases the fixed costs to $660,000. (A) Discuss possible strategies the company might use to deal with this increase in costs. (B) If the company continues to sell the game machines for $140, how many must they sell now to make a profit? (C) If the company wants to start making a profit at the same production level as before the cost increase, how much should they increase the wholesale price? 91. ENERGY If the power demands in a 110-volt electric circuit in a home vary between 220 and 2,750 watts, what is the range of current flowing through the circuit? (W  EI, where W  Power in watts, E  Pressure in volts, and I  Current in amperes.) 92. PSYCHOLOGY A person’s IQ is given by the formula IQ 

MA 100 CA

where MA is mental age and CA is chronological age. If 80 IQ 140 for a group of 12-year-old children, find the range of their mental ages.

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1-3

Absolute Value in Equations and Inequalities

103

Absolute Value in Equations and Inequalities Z Relating Absolute Value and Distance Z Solving Absolute Value Equations and Inequalities Z Using Absolute Value to Solve Radical Inequalities

We can express the distance between two points on a number line using the concept of absolute value. As a result, absolute values often appear in equations and inequalities that are associated with distance. In this section we define absolute value and we show how to solve equations and inequalities that involve absolute value.

Z Relating Absolute Value and Distance We start with a geometric definition of absolute value. If a is the coordinate of a point on a real number line, then the distance from the origin to a is represented by |a| and is referred to as the absolute value of a. Thus, |5|  5, since the point with coordinate 5 is five units from the origin, and  6   6, since the point with coordinate 6 is six units from the origin (Fig. 1). 6  6

5  5 x

6

0

5

Z Figure 1 Absolute value.

Symbolically, and more formally, we define absolute value as follows:

Z DEFINITION 1 Absolute Value

x 



x x

if x 6 0 if x  0

3

 (3)

3

4  4

[Note: x is positive if x is negative.]

Both the geometric and nongeometric definitions of absolute value are useful, as will be seen in the material that follows. Remember: The absolute value of a number is never negative.

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EXAMPLE

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EQUATIONS AND INEQUALITIES

1

Absolute Value of a Real Number (A)    3     3

Since   3.14,   3 is positive.

(B)  3     (3  )    3

Since 3   is negative

MATCHED PROBLEM



1

Write without the absolute value sign: (A)  8 

3 (B)  2 9  2

(C)  12 

3 (D)  2  2 9



Following the same reasoning used in Example 1, the Theorem 1 can be proved (see Problem 77 in Exercises 1-3).

Z THEOREM 1 For all real numbers a and b, b  a  a  b

We use this result in defining the distance between two points on a real number line.

Z DEFINITION 2 Distance Between Points A and B Let A and B be two points on a real number line with coordinates a and b, respectively. The distance between A and B is given by d(A, B)   b  a  This distance is also called the length of the line segment joining A and B.

EXAMPLE

2

Distance Between Points on a Number Line Find the distance between points A and B with coordinates a and b, respectively, as given. (A) a  4, b  9

(B) a  9, b  4

(C) a  0, b  6

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Absolute Value in Equations and Inequalities

105

SOLUTIONS d (A, B)  9  4  5  5

(A)

A

0

B

5

x 10

d(A, B)  4  9  5  5

(B)

B

0

A

5

x 10

d(A, B)  6  0  6  6

(C) 0

A

x

B

5

10

It should be clear, since |b  a|  |a  b|, that d(A, B)  d(B, A) Hence, in computing the distance between two points on a real number line, it does not matter how the two points are labeled—point A can be to the left or to the right of point B. Note also that if A is at the origin O, then d(O, B)   b  0    b 

MATCHED PROBLEM



2

Use the number line shown here to find the indicated distances. A 10

B 5

O 0

C

D 5

x 10

(A) d(C, D)

(B) d(D, C)

(C) d(A, B)

(D) d(A, C)

(E) d(O, A)

(F) d(D, A)



Z Solving Absolute Value Equations and Inequalities The interplay between algebra and geometry is an important tool when working with equations and inequalities involving absolute value. For example, the algebraic statement x  1  2 can be interpreted geometrically as stating that the distance from x to 1 is 2.

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ZZZ

EXPLORE-DISCUSS 1

Write geometric interpretations of the following algebraic statements: (A)  x  1  6 2

EXAMPLE

3

(B) 0 6  x  1  6 2

(C)  x  1  7 2

Solving Absolute Value Problems Geometrically Interpret geometrically, solve, and graph. Write solutions in both inequality and interval notation, where appropriate. (A)  x  3   5

(B)  x  3  6 5

(C) 0 6  x  3  6 5

(D)  x  3  7 5

SOLUTIONS

(A) Geometrically, |x  3| represents the distance between x and 3. Thus, in |x  3|  5, x is a number whose distance from 3 is 5. That is, x  3  5  2 The solution set is {2, 8}. 5

or

8

This is not interval notation.

5 x

2

3

8

(B) Geometrically, in |x  3|  5, x is a number whose distance from 3 is less than 5; that is, 2 6 x 6 8 The solution set is (2, 8).

This is interval notation.

)

(

2

3

x

8

(C) The form 0  |x  3|  5 is frequently encountered in calculus and more advanced mathematics. Geometrically, x is a number whose distance from 3 is less than 5, but x cannot equal 3. Thus, 2 6 x 6 8

x3

or

Hole

)

(

2

3

8

x

(2, 3)  (3, 8)

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107

Absolute Value in Equations and Inequalities

(D) Geometrically, in  x  3  7 5, x is a number whose distance from 3 is greater than 5; that is, x 6 2

or

)

(

2

ZZZ

( , 2)  (8, )

x 7 8

3

8

CAUTION ZZZ

Do not confuse solutions like 2 6 x

and

x 6 8

which can also be written as 2 6 x 6 8

or

(2, 8)

with solutions like x 6 2

or

x 7 8

which cannot be written as a double inequality or as a single interval.

We summarize the preceding results in Table 1. Table 1 Geometric Interpretation of Absolute Value Equations and Inequalities Form (d 0) Geometric interpretation

Solution

Graph d

x  c  d x  c 6 d 0 6 x  c 6 d x  c 7 d

Distance between x and c is equal to d.

5c  d, c  d6

Distance between x and c is less than d.

(c  d, c  d )

Distance between x and c is less than d, but x  c.

(c  d, c)  (c, c  d )

Distance between x and c is greater than d.

( , c  d )  (c  d, )

cd

(

cd

(

cd

)

cd

d c

cd

c

cd

c

cd

c

cd

x

)

x

)

x

(

x



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EQUATIONS AND INEQUALITIES

MATCHED PROBLEM

3

Interpret geometrically, solve, and graph. Write solutions in both inequality and interval notation, where appropriate. (A)  x  2   6

(B)  x  2  6 6

(C) 0 6  x  2  6 6

(D)  x  2  7 6

[Hint: |x  2|  |x  (2)|.]

EXAMPLE

4



Interpreting Verbal Statements Algebraically Express each verbal statement as an absolute value equation or inequality. (A) x is 4 units from 2. (B) y is less than 3 units form 5. (C) t is no more than 5 units from 7. (D) w is no less than 2 units form 1. SOLUTIONS



|x  2|  4

(B)

d( y, 5) 

| y  5| 6 3

(C)

d(t, 7)



|t  7| 5

(D) d(w, 1) 

|w  1|  2

(A) d(x, 2)

MATCHED PROBLEM



4

Express each verbal statement as an absolute value equation or inequality. (A) x is 6 units from 5. (B) y is less than 7 units from 6. (C) w is no less than 3 units from 2. (D) t is no more than 4 units from 3.



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S E C T I O N 1–3

ZZZ

109

Absolute Value in Equations and Inequalities

EXPLORE-DISCUSS 2

Describe the set of numbers that satisfies each of the following: (A) 2 7 x 7 1

(B) 2 7 x 6 1

(C) 2 6 x 7 1

(D) 2 6 x 6 1

Explain why it is never necessary to use double inequalities with inequality symbols pointing in different directions. Standard mathematical notation requires that all inequality symbols in an expression must point in the same direction.

Reasoning geometrically as before (noting that |x|  |x  0|) leads to Theorem 2. Z THEOREM 2 Properties of Equations and Inequalities Involving x For p 0: 1.  x   p is equivalent to

x  p or

x  p.

p

2.  x  6 p is equivalent to p 6 x 6 p. 3.  x  7 p is equivalent to

x 6 p or

0

p

0

p

0

p

(

p

)

x 7 p.

p

x

)

x

(

x

If we replace x in Theorem 2 with ax  b, we obtain the more general Theorem 3.

Z THEOREM 3 Properties of Equations and Inequalities Involving |ax  b| For p 0: 1.  ax  b   p

is equivalent to

ax  b  p

2.  ax  b  6 p

is equivalent to

p 6 ax  b 6 p.

3.  ax  b  7 p

is equivalent to

ax  b 6 p

ax  b  p.

or

or

ax  b 7 p.

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EXAMPLE

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EQUATIONS AND INEQUALITIES

5

Solving Absolute Value Problems Solve, and write solutions in both inequality and interval notation, where appropriate. (A)  3x  5   4

(B)  x  6 5

(C)  2x  1  6 3

(D)  7  3x  2

SOLUTIONS

(A)  3x  5   4 3x  5  4 3x  5  4 5  4 x 3 x  3, 13 or 53, 13 6

(B)  x  6 5 5 6 x 6 5 or (5, 5)

(C)  2x  1  6 3 3 6 2x  1 6 3 2 6 2x 6 4 1 6 x 6 2 or (1, 2)

(D)  7  3x  2 2 7  3x 2 9 3x 5 3  x  53 5 3 x 3 or [ 53, 3]

MATCHED PROBLEM



5

Solve, and write solutions in both inequality and interval notation, where appropriate. (A)  2x  1   8

EXAMPLE

6

(B)  x  7

(C)  3x  3  9

(D)  5  2x  6 9

Solving Absolute Value Inequalities Solve, and write solutions in both inequality and interval notation. (A)  x  7 3

(B)  2x  1   3

(C)  7  3x  7 2

SOLUTIONS

(A)  x  7 3 x 6 3 or x 7 3 ( , 3)  (3, )

Use Theorem 2 to remove absolute value. Inequality notation Interval notation



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Absolute Value in Equations and Inequalities

(B)  2x  1   3 2x  1 3 or 2x  1  3 2x 2 or 2x  4 x 1 or x2 ( , 1]  [2, ) (C)  7  3x  7 2 7  3x 6 2 3x 6 9

111

Use Theorem 3 to remove absolute value. Add 1 to both sides. Divide both sides by 2. Inequality notation Interval notation Use Theorem 3 to remove absolute value.

or or

7  3x 7 2 3x 7 5

x 7 3 or ( , 53)  (3, )

x 6

5 3

Subtract 7 from both sides. Divide both sides by 3 and reverse the order of the inequality. Inequality notation Interval notation

MATCHED PROBLEM



6

Solve, and write solutions in both inequality and interval notation. (A)  x   5

EXAMPLE

7

(B)  4x  3  7 5

(C)  6  5x  7 16



An Absolute Value Problem with Two Cases Solve:  x  4   3x  8 SOLUTION

Theorem 3 does not apply directly, since we do not know that 3x  8 is positive. However, we can use the definition of absolute value and two cases: x  4  0 and x  4  0. Case 1. x  4  0 (that is, x  4) For this case, the possible values of x are in the set 5x ƒ x  46.  x  4   3x  8 x  4  3x  8 2x  12 x6

|a|  a for a  0

A solution, since 6 is among the possible values of x

The check is left to the reader. Case 2. x  4  0 (that is, x  4) In this case, the possible values of x are in the set 5x ƒ x 6 46.  x  4   3x  8 (x  4)  3x  8 x  4  3x  8 4x  4 x1

|a|  a for a  0

Not a solution, since 1 is not among the possible values of x

Combining both cases, we see that the only solution is x  6.

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EQUATIONS AND INEQUALITIES CHECK

As a final check, we substitute x  6 and x  1 in the original equation.  x  4   3x  8 ?  6  4   3(6)  8 ✓ 10  10

MATCHED PROBLEM

 x  4   3x  8 ?  1  4   3(1)  8 5  5



7

Solve:  3x  4   x  5



Z Using Absolute Value to Solve Radical Inequalities In Section R-3, we found that if x is positive or 0, then 2x2  x If x is negative, however, we must write 2x2  x

2(2)2  (2)  2

Thus, for x any real number, 2x2 

x x



if x 6 0 if x  0

But this is exactly how we defined  x  at the beginning of this section (see Definition 1). Thus, for x any real number, 2x2   x 

EXAMPLE

8

(1)

Solving a Radical Inequality Use equation (1) to solve: 2(x  2)2  5 Write your answers in both inequality and interval notation. SOLUTION

2(x  2)2  5 x  2  5 5  x  2  5 3  x  7 or [3, 7]

Use equation (1). Use Theorem 3. Add 2 to each member. Inequality notation Interval notation



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S E C T I O N 1–3

113

Absolute Value in Equations and Inequalities

MATCHED PROBLEM

8

Use equation (1) to solve: 2(x  2)2 6 3 

Write your answers in both inequality and interval notation.

ANSWERS

TO MATCHED PROBLEMS

3 3 1. (A) 8 (B) 2 92 (C) 12 (D) 2 92 2. (A) 4 (B) 4 (C) 6 (D) 11 (E) 8 (F) 15 3. (A) x is a number whose distance from 2 is 6. x  8, 4 or 58, 46

8

2

x

4

(B) x is a number whose distance from 2 is less than 6. 8 6 x 6 4 or (8, 4) ( ) 8

2

x

4

(C) x is a number whose distance from 2 is less than 6, but x cannot equal 2. 8 6 x 6 4, x  2, or (8, 2)  (2, 4) ( ) 8

2

(D) x is a number whose distance from 2 is greater than 6. x 6 8 or x 7 4, or ( , 8)  (4, ) ) 8

x

4

(

2

x

4

4. (A)  x  5   6 (B)  y  6  6 7 (C)  w  2   3 (D)  t  3  4 5. (A) x  72, 92 or 572, 92 6 (B) 7 x 7 or [7, 7] (C) 4 x 2 or (D) 2 6 x 6 7 or (2, 7) [4, 2] 6. (A) x 5 or x  5, or ( , 5]  [5, ) (B) x 6 12 or x 7 2, or 1 22 (C) x 6 2 or x 7 5 , or ( , 2)  (225, ) ( , 2)  (2, ) 7. x  14, 92 or 514, 92 6 8. 5 6 x 6 1 or (5, 1)

1-3

Exercises

In Problems 1–8, simplify, and write without absolute value signs. Do not replace radicals with decimal approximations.

In Problems 13–18, use the number line shown to find the indicated distances.

1.  15 

2.  34 

3.  (6)  (2) 

4.  (2)  (6) 

10

5.  5  15 

6.  17  2 

13. d(B, O)

14. d(A, B)

15. d(O, B)

7.  15  5 

8.  2  17 

16. d(B, A)

17. d(B, C)

18. d(D, C)

In Problems 9–12, find the distance between points A and B with coordinates a and b respectively, as given. 9. a  7, b  5 11. a  5, b  7

A

B 5

O

C

0

5

D

x 10

Write each of the statements in Problems 19–28 as an absolute value equation or inequality.

10. a  3, b  12

19. x is 4 units from 3.

12. a  9, b  17

20. y is 3 units from 1.

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EQUATIONS AND INEQUALITIES

21. m is 5 units from 2.

Problems 63–66 are calculus-related. Solve and graph. Write each solution using interval notation.

22. n is 7 units from 5. 23. x is less than 5 units from 3. 24. z is less than 8 units from 2. 25. p is more than 6 units from 2.

63. 0 6  x  3  6 0.1

64. 0 6  x  5  6 0.01

65. 0 6  x  c  6 d

66. 0 6  x  4  6 d

In Problems 67–74, for what values of x does each hold?

26. c is no greater than 7 units from 3.

67.  x  2   2x  7

68.  x  4   3x  8

27. q is no less than 2 units from 1.

69.  3x  5   2x  6

70.  7  2x   5  x

28. d is no more than 4 units from 5.

71.  x    x  3   3

72.  x    x  5   5

In Problems 29–40, solve, interpret geometrically, and graph. When applicable, write answers using both inequality notation and interval notation. 29.  y  5   3

30.  t  3   4

31.  y  5  6 3

32.  t  3  6 4

33.  y  5  7 3

34.  t  3  7 4

35.  u  8   3

36.  x  1   5

37.  u  8  3

38.  x  1  5

39.  u  8   3

40.  x  1   5

73.  2x  7    6  3x   8 74.  3x  1    3  2x   11 75. What are the possible values of

x ? x

76. What are the possible values of

x  1 ? x1

77. Prove that  b  a    a  b  for all real numbers a and b. 78. Prove that  x 2  x2 for all real numbers x.

In Problems 41–58, solve each equation or inequality. When applicable, write answers using both inequality notation and interval notation. 41.  3x  7  4

42.  5y  2   8

43.  4  2t  7 6

44.  10  4s  6 6

45.  7m  11   3

46.  4  5n  8

47.  12w  34  6 2

48.  13z  56   1

49.  0.2u  1.7   0.5

50.  0.5v  2.5  7 1.6

51.

 95 C

 32  6 31

52.

 59 (F

 32)  6 40

53. 2x 6 2

54. 2m 7 3

55. 2(1  3t)2 2

56. 2(3  2x)2 6 5

57. 2(2t  3)2 7 3

58. 2(3m  5)2  4

2

2

79. Prove that the average of two numbers is between the two numbers; that is, if m 6 n, then m 6

mn 6 n 2

80. Prove that for m 6 n, d am,

mn mn b  da , nb 2 2

81. Prove that  m    m . 82. Prove that  m    n  if and only if m  n or m  n. 83. Prove that for n  0, `

m m ` n n

84. Prove that  mn    m  n . In Problems 59–62, solve and write answers in inequality notation. Round decimals to three significant digits. 59.  2.25  1.02x  1.64 60.  0.962  0.292x  2.52 61.  21.7  11.3x   15.2 62.  195  55.5x   315

85. Prove that  m  m  m . 86. Prove the triangle inequality: m  n m  n Hint: Use Problem 85 to show that  m    n  m  n  m    n 

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87. If a and b are real numbers, prove that the maximum of a and b is given by max(a, b)  12 [a  b   a  b  ] 88. Prove that the minimum of a and b is given by min(a, b)  12 [a  b   a  b  ]

APPLICATIONS 89. STATISTICS Inequalities of the form `

xm ` 6 n s

occur frequently in statistics. If m  45.4, s  3.2, and n  1, solve for x. 90. STATISTICS Repeat Problem 89 for m  28.6, s  6.5, and n  2. ★91.

BUSINESS The daily production P in an automobile assembly plant is within 20 units of 500 units. Express the daily production as an absolute value inequality.

1-4

Complex Numbers

115

★92.

CHEMISTRY In a chemical process, the temperature T is to be kept within 10°C of 200°C. Express this restriction as an absolute value inequality.

93. APPROXIMATION The area A of a region is approximately equal to 12.436. The error in this approximation is less than 0.001. Describe the possible values of this area both with an absolute value inequality and with interval notation. 94. APPROXIMATION The volume V of a solid is approximately equal to 6.94. The error in this approximation is less than 0.02. Describe the possible values of this volume both with an absolute value inequality and with interval notation. SIGNIFICANT DIGITS If N  2.37 represents a measurement, then we assume an accuracy of 2.37  0.005. Express the accuracy assumption using an absolute value inequality.

★95.

★96.

SIGNIFICANT DIGITS If N  3.65 103 is a number from a

measurement, then we assume an accuracy of 3.65 103  5 106. Express the accuracy assumption using an absolute value inequality.

Complex Numbers Z Understanding Complex Number Terminology Z Performing Operations with Complex Numbers Z Relating Complex Numbers and Radicals Z Solving Equations Involving Complex Numbers

The Pythagoreans (500–275 B.C.) found that the simple equation x2  2

(1)

had no rational number solutions. If equation (1) were to have a solution, then a new kind of number had to be invented—an irrational number. The irrational numbers 12 and 12 are both solutions to equation (1). Irrational numbers were not put on a firm mathematical foundation until the nineteenth century. The rational and irrational numbers together constitute the real number system. Is there any need to consider another number system? Yes, if we want the simple equation x2  1

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to have a solution. If x is any real number, then x2  0. Thus, x2  1 cannot have any real number solutions. Once again a new type of number must be invented, a number whose square can be negative. These new numbers are among the numbers called complex numbers. The complex numbers evolved over a long period of time, but, like the real numbers, it was not until the nineteenth century that they were placed on a firm mathematical foundation (see Table 1). Table 1 Brief History of Complex Numbers Approximate date

Person

Event

Heron of Alexandria

First recorded encounter of a square root of a negative number

Mahavira of India

Said that a negative has no square root, since it is not a square

1545

Cardano of Italy

Solutions to cubic equations involved square roots of negative numbers.

1637

Descartes of France

Introduced the terms real and imaginary

1748

Euler of Switzerland

Used i for 11

1832

Gauss of Germany

Introduced the term complex number

50 850

Z Understanding Complex Number Terminology We start the development of the complex number system by defining a complex number and listing some special terms related to complex numbers.

Z DEFINITION 1 Complex Number A complex number is a number of the form a  bi

Standard Form

where a and b are real numbers and i is called the imaginary unit.

The imaginary unit i introduced in Definition 1 is not a real number. It is a special symbol used in the representation of the elements in this new complex number system. Some examples of complex numbers are 3  2i 0  3i

 5i 5  0i 1 2

The notation 3  2i is shorthand for 3  (2)i.

2  13i 0  0i

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117

Particular kinds of complex numbers are given special names as follows:

Z DEFINITION 2 Special Terms i a  bi a and b real numbers a  bi b0 0  bi  bi b0 bi a  0i  a a 0  0  0i a  bi

EXAMPLE

1

Imaginary Unit Complex Number Imaginary Number Pure Imaginary Number Imaginary Part of a  bi Real Number Real Part of a  bi Zero Conjugate of a  bi

Complex Numbers Identify the real part, the imaginary part, and the conjugate of each of the following numbers: (A) 3  2i

(B) 2  5i

(C) 7i

(D) 6

SOLUTIONS

(A) Real part: 3; imaginary part: 2i; conjugate: 3  2i (B) Real part: 2; imaginary part: 5i; conjugate: 2  5i (C) Real part: 0; imaginary part: 7i; conjugate: 7i 

(D) Real part: 6; imaginary part: 0; conjugate: 6

MATCHED PROBLEM

1

Identify the real part, the imaginary part, and the conjugate of each of the following numbers: (A) 6  7i

(B) 3  8i

(C) 4i

(D) 9



In Definition 2, notice that we identify a complex number of the form a  0i with the real number a, a complex number of the form 0  bi, b  0, with the pure imaginary number bi, and the complex number 0  0i with the real number 0. Thus, a real number is also a complex number, just as a rational number is also a real number. Any complex number that is not a real number is called an imaginary number. If we combine the set of all real numbers with the set of all imaginary numbers, we obtain C, the set of complex numbers. The relationship of the complex number system to the other number systems we have studied is shown in Figure 1.

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Z Figure 1 Complex numbers and important subsets

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Complex numbers (C ) NZQRC Real numbers (R ) Rational numbers (Q) Integers (Z ) Natural numbers (N )

Zero

Imaginary numbers

Irrational numbers (I )

Noninteger ratios of integers Negatives of natural numbers

Z Performing Operations with Complex Numbers To use complex numbers, we must know how to add, subtract, multiply, and divide them. We start by defining equality, addition, and multiplication.

Z DEFINITION 3 Equality and Basic Operations 1. Equality: a  bi  c  di if and only if a  c and b  d 2. Addition: (a  bi)  (c  di)  (a  c)  (b  d )i 3. Multiplication: (a  bi)(c  di)  (ac  bd)  (ad  bc)i

In Section R-1 we listed the basic properties of the real number system. Using Definition 3, it can be shown that the complex number system possesses the same properties. That is, 1. Addition and multiplication of complex numbers are commutative and associative operations. 2. There is an additive identity and a multiplicative identity for complex numbers. 3. Every complex number has an additive inverse or negative. 4. Every nonzero complex number has a multiplicative inverse or reciprocal. 5. Multiplication distributes over addition. As a consequence of these properties, we can manipulate complex number symbols of the form a  bi just like we manipulate binomials of the form a  bx, as long as we remember that i is a special symbol for the imaginary unit, not for a real number. Thus, you will not have to memorize the definitions of addition and multiplication of complex numbers. We now discuss these operations and some of their properties. Others will be considered in Exercises 1-4.

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EXAMPLE

2

Complex Numbers

119

Addition of Complex Numbers Carry out each operation and express the answer in standard form: (A) (2  3i)  (6  2i)

(B) (5  4i)  (0  0i)

SOLUTIONS

(A) We could apply the definition of addition directly, but it is easier to use complex number properties. (2  3i)  (6  2i)  2  3i  6  2i

Remove parentheses.

 (2  6)  (3  2)i

Combine like terms.

8i (B) (5  4i)  (0  0i)  5  4i  0  0i  5  4i

MATCHED PROBLEM



2

Carry out each operation and express the answer in standard form: (A) (3  2i)  (6  4i)

(B) (0  0i)  (7  5i)



Example 2B and Matched Problem 2B illustrate the following general result: For any complex number a  bi, (a  bi)  (0  0i)  (0  0i)  (a  bi)  a  bi Thus, 0  0i is the additive identity or zero for the complex numbers. We anticipated this result in Definition 2 when we identified the complex number 0  0i with the real number 0. The additive inverse or negative of a  bi is a  bi because (a  bi)  (a  bi)  (a  bi)  (a  bi)  0

EXAMPLE

3

Negation and Subtraction Carry out each operation and express the answer in standard form: (A) (4  5i)

(B) (7  3i)  (6  2i)

(C) (2  7i)  (2  7i)

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EQUATIONS AND INEQUALITIES SOLUTIONS

(A) (4  5i)

 (1)(4  5i)

 4  5i

(B) (7  3i)  (6  2i)  7  3i  6  2i  1  5i

Remove parentheses. Combine like terms.

(C) (2  7i)  (2  7i)  2  7i  2  7i  0

MATCHED PROBLEM



3

Carry out each operation and express the answer in standard form: (A) (3  2i)

(B) (3  5i)  (1  3i)

(C) (4  9i)  (4  9i)



Now we turn our attention to multiplication. First, we use the definition of multiplication to see what happens to the complex unit i when it is squared: a

b

c

d

i2  (0  1i)(0  1i) a

c

b

d

a

d

b

c

 (0  0  1  1)  (0  1  1  0)i  1  0i  1 Thus, we have proved that i 2  1 Just as was the case with addition and subtraction, multiplication of complex numbers can be carried out by using the properties of complex numbers rather than the definition of multiplication. We just replace i2 with 1 each time it occurs.

EXAMPLE

4

Multiplying Complex Numbers Carry out each operation and express the answer in standard form: (A) (2  3i)(6  2i)

(B) 1(3  5i)

(C) i(1  i)

(D) (3  4i)(3  4i)

SOLUTIONS

(A) (2  3i)(6  2i)

 2(6  2i)  3i(6  2i)  12  4i  18i  6i2  12  14i  6(1)  18  14i

Replace i 2 with 1.

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(B) 1(3  5i)

 1  3  1  5i

Complex Numbers

121

 3  5i

(C) i(1  i)  i  i2  i  1  1  i (D) (3  4i)(3  4i)  9  12i  12i  16i2  9  16  25

MATCHED PROBLEM



4

Carry out each operation and express the answer in standard form: (A) (5  2i)(4  3i)

(B) 3(2  6i)

(C) i(2  3i)

(D) (2  3i)(2  3i)



For any complex number a  bi, 1(a  bi)  (a  bi)1  a  bi (see Example 4B). Thus, 1 is the multiplicative identity for complex numbers, just as it is for real numbers. Earlier we stated that every nonzero complex number has a multiplicative inverse or reciprocal. We will denote this as a fraction, just as we do with real numbers. Thus, 1 a  bi

is the reciprocal of

a  bi

a  bi  0

The following important property of the conjugate of a complex number is used to express reciprocals and quotients in standard form.

Z THEOREM 1 Product of a Complex Number and Its Conjugate (a  bi)(a  bi)  a2  b2

EXAMPLE

5

Reciprocals and Quotients Write each expression in standard form: (A)

1 2  3i

(B)

7  3i 1i

A real number

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EQUATIONS AND INEQUALITIES SOLUTIONS

(A) Multiply numerator and denominator by the conjugate of the denominator: 1 1 2  3i   2  3i 2  3i 2  3i 



2  3i 2  3i 2  49 4  9i

2  3i 2 3   i 13 13 13

This answer can be checked by multiplication: (2  3i) a

CHECK

(B)

2 3 4 6 6 9 2  ib   i i i 13 13 13 13 13 13 4 9   1 13 13

7  3i 7  3i 1  i   1i 1i 1i 

CHECK



7  7i  3i  3i2 1  i2

4  10i  2  5i 2 (1  i)(2  5i)  2  5i  2i  5i2  7  3i

MATCHED PROBLEM



5

Carry out each operation and express the answer in standard form: (A)

EXAMPLE

6

1 4  2i

(B)

6  7i 2i



Combined Operations Carry out the indicated operations and write each answer in standard form: (A) (3  2i)2  6(3  2i)  13

(B)

2  3i 2i

SOLUTIONS

(A) (3  2i)2  6(3  2i)  13  9  12i  4i2  18  12i  13  9  12i  4  18  12i  13 0

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Complex Numbers

123

(B) If a complex number is divided by a pure imaginary number, we can make the denominator real by multiplying numerator and denominator by i. 2  3i i 2i  3i2 2i  3 3     i 2 2i i 2 2 2i

MATCHED PROBLEM



6

Carry out the indicated operations and write each answer in standard form: (A) (3  2i)2  6(3  2i)  13

ZZZ EXPLORE-DISCUSS

(B)

4i 3i



1

Natural number powers of i take on particularly simple forms: i i2  1 i3  i2  i  (1)i  i i4  i2  i2  (1)(1)  1

i5 i6 i7 i8

   

i4 i4 i4 i4

   

i  (1)i  i i2  1(1)  1 i3  1(i)  i i4  1  1  1

In general, what are the possible values for i n, n a natural number? Explain how you could easily evaluate i n for any natural number n. Then evaluate each of the following: (A) i17

(B) i24

(C) i38

(D) i 47

Z Relating Complex Numbers and Radicals Recall that we say that a is a square root of b if a2  b. If x is a positive real number, then x has two square roots, the principal square root, denoted by 1x, and its negative,  1x (Section R-3). If x is a negative real number, then x still has two square roots, but now these square roots are imaginary numbers. Z DEFINITION 4 Principal Square Root of a Negative Real Number The principal square root of a negative real number, denoted by 1a, where a is positive, is defined by 1a  i 1a

13  i 13

19  i 19  3i

The other square root of a, a 7 0, is  1a  i1a.

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Note in Definition 4 that we wrote i1a and i 13 in place of the standard forms 1a i and 13 i. We follow this convention whenever it appears that i might accidentally slip under a radical sign ( 1a i  1a i, but 1a i  i1a). Definition 4 is motivated by the fact that (i1a)2  i2a  a

EXAMPLE

7

Complex Numbers and Radicals Write in standard form: (A) 14

(B) 4  15

(C)

3  15 2

(D)

1 1  19

SOLUTIONS

(A) 14  i 14  2i

(B) 4  15  4  i 15

(C)

3  15 3  i15 3 15    i 2 2 2 2

(D)

1  (1  3i ) 1 1   1  3i (1  3i)  (1  3i ) 1  19 

1  3i 1  3i 1 3    i 2 10 10 10 1  9i

MATCHED PROBLEM



7

Write in standard form: (A) 116

ZZZ

(B) 5  17

(C)

5  12 2

(D)

1 3  14

EXPLORE-DISCUSS 2

From Theorem 1 in Section R-3, we know that if a and b are positive real numbers, then 1a 1b  1ab (2) Thus, we can evaluate expressions like 19 14 two ways: 19 14  1(9)(4)  136  6

and

19 14  (3)(2)  6

Evaluate each of the following two ways. Is equation (2) a valid property to use in all cases? (A) 19 14

(B) 19 14

(C) 19 14



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S E C T I O N 1–4

ZZZ

Complex Numbers

125

CAUTION ZZZ

Note that in Example 7D, we wrote 1  19  1  3i before proceeding with the simplification. Writing a complex number in standard form is a necessary step because some of the properties of radicals that are true for real numbers turn out not to be true for complex numbers. In particular, for positive real numbers a and b, 1a 1b  1ab

but

1a 1b  1(a)(b)

(See Explore-Discuss 2.)

Z Solving Equations Involving Complex Numbers EXAMPLE

8

Equations Involving Complex Numbers (A) Solve for real numbers x and y: (3x  2)  (2y  4)i  4  6i (B) Solve for complex number z: (3  2i)z  3  6i  8  4i SOLUTIONS

(A) Equate the real and imaginary parts of each side of the equation to form two equations: Real Parts

Imaginary Parts

3x  2  4 3x  6 x  2

2y  4  6 2y  10 y5

(B) (3  2i)z  3  6i  8  4i (3  2i)z  11  10i 11  10i 3  2i (11  10i)(3  2i)  (3  2i)(3  2i)

z

Add 3  6i to both sides. Divide both sides by 3  2i. Multiply numerator and denominator by 3  2i.

Simplify.

13  52i 13  1  4i 

A check is left to the reader.



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MATCHED PROBLEM

8

(A) Solve for real numbers x and y: (2y  7)  (3x  4)i  1  i (B) Solve for complex number z: (1  3i)z  4  5i  3  2i



Early resistance to these new numbers is suggested by the words used to name them: complex and imaginary. In spite of this early resistance, complex numbers have come into widespread use in both pure and applied mathematics. They are used extensively, for example, in electrical engineering, physics, chemistry, statistics, and aeronautical engineering. Our first use of them will be in connection with solutions of second-degree equations in Section 1-5.

ANSWERS

TO MATCHED PROBLEMS

1. (A) Real part: 6; imaginary part: 7i; conjugate: 6  7i (B) Real part: 3; imaginary part: 8i; conjugate: 3  8i (C) Real part: 0; imaginary part: 4i; conjugate: 4i (D) Real part: 9; imaginary part: 0; conjugate: 9 2. (A) 9  2i (B) 7  5i 3. (A) 3  2i (B) 2  2i (C) 0 4. (A) 26  7i (B) 6  18i (C) 3  2i (D) 13 5. (A) 15  101 i (B) 1  4i 6. (A) 0 (B) 13  43 i 7. (A) 4i (B) 5  i17 (C) 52  (122)i (D) 133  132 i 8. (A) x  1, y  4 (B) z  2  i

1-4

Exercises

For each number in Problems 1–12, find (A) real part, (B) imaginary part, and (C) conjugate.

In Problems 13–38, perform the indicated operations and write each answer in standard form.

2. 6i  4

3 5 3.   i 2 6

13. (3  5i)  (2  4i)

14. (4  i)  (5  3i)

15. (8  3i)  (5  6i)

16. (1  2i)  (4  7i)

4. 4.2  9.7i

5. 6.5  2.1i

3 4 6.  i 5 5

17. (9  5i)  (6  2i)

18. (3  7i)  (2  5i)

7. i

8. 6

9. 4

19. (3  4i)  (5  6i)

20. (4  2i)  (1  i)

21. 2  (3i  5)

22. (2i  7)  4i

1. 2  9i

10. 2i

11. 5  i12

12. 4  i17

23. (2i)(4i)

24. (3i)(5i)

26. (4i)(2  3i)

25. 2i(4  6i) 27. (1  2i)(3  4i)

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28. (2  i)(5  6i)

29. (3  i)(4  i)

30. (5  2i)(4  3i)

31. (2  9i)(2  9i)

32. (3  8i)(3  8i)

1 33. 2  4i

34.

i 3i

37.

7i 2i

35.

4  3i 1  2i

36. 38.

3  5i 2i

5  10i 3  4i

In Problems 39–46, evaluate and express results in standard form. 39. 12 18

40. 13 112

41. 12 18

42. 13 112

43. 12 18

44. 13 112

45. 12 18

46. 13 112

In Problems 47–56, convert imaginary numbers to standard form, perform the indicated operations, and express answers in standard form. 47. (2  14)  (5  19)

Complex Numbers

127

In Problems 65–68, for what real values of x does each expression represent an imaginary number? 65. 13  x

66. 15  x

67. 12  3x

68. 13  2x

Use a calculator to compute Problems 69–72. Write in standard form a  bi, where a and b are computed to three significant digits. 69. (3.17  4.08i)(7.14  2.76i) 70. (6.12  4.92i)(1.82  5.05i) 71.

8.14  2.63i 3.04  6.27i

72.

7.66  3.33i 4.72  2.68i

In Problems 73–76, solve for x and y. 73. (2x  1)  (3y  2)i  5  4i 74. 3x  ( y  2)i  (5  2x)  (3y  8)i 75.

(1  x)  ( y  2)i 2i 1i

76.

(2  x)  ( y  3)i  3  i 1i

48. (3  14)  (8  125) In Problems 77–80, solve for z. Express answers in standard form.

49. (9  19)  (12  125) 50. (2  136)  (4  149)

77. (2  i)z  i  4i

51. (3  14)(2  149)

78. (3  i)z  2  i

52. (2  11)(5  19) 5  14 53. 7 55.

1 2  19

79. 3iz  (2  4i)  (1  2i)z  3i 6  164 54. 2 56.

1 3  116

Write Problems 57–62 in standard form. 2 57. 5i

1 58. 3i

1  3i 59. 2i

2i 60. 3i

61. (2  3i)2  2(2  3i)  9 62. (2  i)2  3(2  i)  5 63. Evaluate x  2x  2 for x  1  i.

80. (2  i)z  (1  4i)  (1  3i)z  (4  2i) 81. Explain what is wrong with the following “proof ” that 1  1: 1  i2  11 11  1(1)(1)  11  1 82. Explain what is wrong with the following “proof ” that 1 i  i. What is the correct value of 1 i? 1 11 1 1    11  i  i 11 11 A 1 In Problems 83–86, perform the indicated operations, and write each answer in standard form. 83. (a  bi)(a  bi)

84. (u  vi)(u  vi)

85. (a  bi)(c  di)

86.

2

64. Evaluate x  2x  2 for x  1  i. 2

a  bi c  di

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87. Show that i4k  1, k a natural number 88. Show that i4k1  i, k a natural number Supply the reasons in the proofs for the theorems stated in Problems 89 and 90. 89. Theorem: The complex numbers are commutative under addition. Proof: Let a  bi and c  di be two arbitrary complex numbers; then: Statement 1. (a  bi)  (c  di)  (a  c)  (b  d )i 2.  (c  a)  (d  b)i 3.  (c  di)  (a  bi) Reason 1. 2. 3.

Reason 1. 2. 3. Letters z and w are often used as complex variables, where z  x  yi, w  u  vi, and x, y, u, v are real numbers. The conjugates of z and w, denoted by z and w, respectively, are given by z  x  yi and w  u  vi. In Problems 91–98, express each property of conjugates verbally and then prove the property. 91. zz is a real number. 92. z  z is a real number. 93. z  z if and only if z is real. 94. z  z 95. z  w  z  w

90. Theorem: The complex numbers are commutative under multiplication. Proof: Let a  bi and c  di be two arbitrary complex numbers; then:

96. z  w  z  w 97. zw  z  w 98. zw  z  w

Statement 1. (a  bi)  (c  di)  (ac  bd )  (ad  bc)i 2.  (ca  db)  (da  cb)i 3.  (c  di)(a  bi)

1-5

Quadratic Equations and Applications Z Using Factoring to Solve Quadratic Equations Z Using the Square Root Property to Solve Quadratic Equations Z Using Completing the Square to Solve Quadratic Equations Z Using the Quadratic Formula to Solve Quadratic Equations Z Solving Applications Involving Quadratic Equations

The next class of equations we consider are the second-degree polynomial equations in one variable, called quadratic equations.

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Z DEFINITION 1 Quadratic Equation A quadratic equation in one variable is any equation that can be written in the form ax2  bx  c  0

a0

Standard Form

where x is a variable and a, b, and c are constants.

Now that we have discussed the complex number system, we will use complex numbers when solving equations. Recall that a solution of an equation is also called a root of the equation. A real number solution of an equation is called a real root, and an imaginary number solution is called an imaginary root. In this section we develop methods for finding all real and imaginary roots of a quadratic equation.

Z Using Factoring to Solve Quadratic Equations If ax2  bx  c can be written as the product of two first-degree factors, then the quadratic equation can be quickly and easily solved. The method of solution by factoring rests on the zero property of complex numbers, which is a generalization of the zero property of real numbers introduced in Section R-1.

Z ZERO PROPERTY If m and n are complex numbers, then mn0

EXAMPLE

1

if and only if

m  0 or n  0 (or both)

Solving Quadratic Equations by Factoring Solve by factoring: (A) 6x2  19x  7  0

(B) x2  6x  5  4

SOLUTIONS

(A)

6x2  19x  7  0 Factor left side. (2x  7)(3x  1)  0 Apply the zero property. 2x  7  0 or 3x  1  0 7 x2 x  13 The solution set is 513, 72 6.

(C) 2x2  3x

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(B) x2  6x  5  4 x2  6x  9  0 (x  3)2  0 x3

Write in standard form. Factor left side. Apply the zero property.

The solution set is {3}. The equation has one root, 3. But since it came from two factors, we call 3 a double root. (C)

2x2  3x Write in standard form. 2x2  3x  0 Factor left side. x(2x  3)  0 Apply the zero property. x0 or 2x  3  0 x  32 Solution set: 50, 32 6



MATCHED PROBLEM

1

Solve by factoring: (A) 3x2  7x  20  0

ZZZ

(B) 4x2  12x  9  0

(C) 4x2  5x

CAUTION ZZZ

1. One side of an equation must be 0 before the zero property can be applied. Thus, x2  6x  5  4 (x  1)(x  5)  4 does not imply that x  1  4 or x  5  4. See Example 1B for the correct solution of this equation. 2. The equations 2x2  3x

and

2x  3

are not equivalent. The first has solution set 50, 32 6, while the second has solution set 5 32 6. The root x  0 is lost when each member of the first equation is divided by the variable x. See Example 1C for the correct solution of this equation.

Do not divide both sides of an equation by an expression containing the variable for which you are solving. You may be dividing by 0.



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Z Using the Square Root Property to Solve Quadratic Equations We now turn our attention to quadratic equations that do not have the first-degree term—that is, equations of the special form ax2  c  0

a0

The method of solution of this special form makes direct use of the square root property:

Z SQUARE ROOT PROPERTY If A2  C, then A  1C.

ZZZ

EXPLORE-DISCUSS 1

Determine if each of the following pairs of equations is equivalent or not. Explain your answer. (A) x2  4

and

x  |2|

(B) x  4

and

x  2

(C) x  14

and

x2

(D) x  14

and

x  2

2

The use of the square root property is illustrated in Example 2.

EXAMPLE

2

Using the Square Root Property Solve using the square root property: (A) 2x2  3  0

(B) 3x2  27  0

(C) (x  12)2  54

SOLUTIONS

(A) 2x2  3  0 x2  32 16 x   232   2

Solve for x2. Apply the square root property. Solution set: e

16 16 , f 2 2

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(B) 3x2  27  0 x2  9 x   19 3i (C) (x  12)2  54 x  12   254 1 15 x  2 2 1  15  2

Solve for x2. Apply the square root property. Solution set: 53i, 3i6

Apply the square root property. Subtract

1 2

from both sides, and simplify 254 .



MATCHED PROBLEM

2

Solve using the square root property: (A) 3x2  5  0

(B) 2x2  8  0

(C) (x  13)2  29



Note: It is common practice to represent solutions of quadratic equations informally by the last equation (Example 2C) rather than by writing a solution set using set notation (Examples 2A and 2B). From now on, we will follow this practice unless a particular emphasis is desired.

ZZZ

EXPLORE-DISCUSS 2

Replace ? in each of the following with a number that makes the equation valid. (A) (x  1)2  x2  2x  ?

(B) (x  2)2  x2  4x  ?

(C) (x  3)2  x2  6x  ?

(D) (x  4)2  x2  8x  ?

Replace ? in each of the following with a number that makes the trinomial a perfect square. (E) x2  10x  ?

(F) x2  12x  ?

(G) x2  bx  ?

Z Using Completing the Square to Solve Quadratic Equations The methods of square root and factoring are generally fast when they apply; however, there are equations, such as x2  6x  2  0 (see Example 4A), that cannot be solved directly by these methods. A more general procedure must be developed to take care of this type of equation—for example, the method of completing the square.* This method is based on the process of transforming the standard quadratic equation *We will find many other uses for this important method.

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ax2  bx  c  0 into the form (x  A)2  B where A and B are constants. The last equation can easily be solved by using the square root property. But how do we transform the first equation into the second? The following brief discussion provides the key to the process. What number must be added to x2  bx so that the result is the square of a firstdegree polynomial? There is a simple mechanical rule for finding this number, based on the square of the following binomials: (x  m)2  x2  2mx  m2 (x  m)2  x2  2mx  m2 In either case, we see that the third term on the right is the square of one-half of the coefficient of x in the second term on the right. This observation leads directly to the rule for completing the square.

Z COMPLETING THE SQUARE To complete the square of a quadratic expression of the form x2  bx, add the square of one-half the coefficient of x; that is, add (b2)2. Thus, x2  bx

x2  5x

b 2 b 2 x2  bx  a b  ax  b 2 2

5 2 5 2 x2  5x  a b  ax  b 2 2

To complete the square in a quadratic equation, add (b2)2 to both sides of the equation. See Example 4.

EXAMPLE

3

Completing the Square Complete the square for each of the following: (A) x2  3x

(B) x2  bx

SOLUTIONS

(A) x2  3x x2  3x  94  (x  32)2

Add a

3 2 9 b ; that is, and factor. 2 4

(B) x2  bx x2  bx 

b2 b 2  ax  b 4 2

Add a

b2 b 2 and factor. b ; that is, 2 4



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MATCHED PROBLEM

3

Complete the square for each of the following: (A) x2  5x

(B) x2  mx



You should note that the rule for completing the square applies only if the coefficient of the second-degree term is 1. This causes little trouble, however, as you will see. We now solve two equations by the method of completing the square.

EXAMPLE

4

Solution by Completing the Square Solve by completing the square: (A) x2  6x  2  0

(B) 2x2  4x  3  0

SOLUTIONS

(A) x2  6x  2  0 x2  6x  2

Add 2 to both sides to obtain the form x2  bx on the left side. Complete the square on the left side and add (2b )2  (62 )2  9 to both sides.

x2  6x  9  2  9 (x  3)2  11 x  3   111 x  3  111 (B) 2x2  4x  3  0 x2  2x  32  0

Factor the left side. Use the square root property. Add 3 to both sides.

Make the leading coefficient 1 by dividing by 2. Subtract 32 from both sides.

x2  2x  32 2 x  2x  1  32  1 (x  1)2  12 x  1   212 x  1  i212 12 1 i 2

MATCHED PROBLEM

Complete the square on the left side and add 2 (2b )2  (2 2 )  1 to both sides. Factor the left side. Use the square root property. Add 1 to both sides and simplify 212 .

Answer in a  bi form.



4

Solve by completing the square: (A) x2  8x  3  0

(B) 3x2  12x  13  0



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Z Using the Quadratic Formula to Solve Quadratic Equations Now consider the general quadratic equation with unspecified coefficients: ax2  bx  c  0 ax2  bx  c  0 b c x2  x   0 a a b c x2  x   a a x2 

a0 Make the leading coefficient 1 by dividing by a. Subtract

c from both sides. a

Complete the square on the left side and add b2 b 2 to both sides. a b  2a 4a2

b b2 b2 c x 2 2 a a 4a 4a

Factor the left side and combine terms on the right side.

b 2 b2  4ac b  2a 4a2 b b2  4ac x  2a B 4a2 b 2b2  4ac x  2a 2a b  2b2  4ac  2a

ax 

Use the square root property. b2  4ac b to both sides and simplify 2a B 4a2 (see Problem 73 in Exercises 1-5). Add 

Combine terms on the right side.

We have thus derived the well-known and widely used quadratic formula: Z THEOREM 1 Quadratic Formula If ax2  bx  c  0, a  0, then x

b  2b2  4ac 2a

The quadratic formula should be memorized and used to solve quadratic equations when other methods fail or are more difficult to apply.

EXAMPLE

5

Using the Quadratic Formula Solve 2x  32  x2 by use of the quadratic formula. Leave the answer in simplest radical form.

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2x  32  x2 4x  3  2x2

Multiply both sides by 2. Write in standard form.

Identify a, b, and c and use the quadratic 2x2  4x  3  0 formula: a  2, b  4, c  3 b  2b2  4ac x 2a (4)  2(4)2  4(2)(3)  2(2) 4  140 4  2110 2  110    4 4 2

ZZZ

CAUTION ZZZ

1. 42  (4)2 2. 2  3.



110 2  110  2 2

4  2110  2110 4

42  16 and (4)2  16 2

110 4  110  2 2

2(2  110) 4  2 110 2  110   4 4 2

MATCHED PROBLEM

5

Solve x2  52  3x by use of the quadratic formula. Leave the answer in simplest radical form.  We conclude this part of the discussion by noting that b2  4ac in the quadratic formula is called the discriminant and gives us useful information about the corresponding roots as shown in Table 1. Table 1 Discriminant and Roots Discriminant b2  4ac

Roots of ax2  bx  c  0 a, b, and c real numbers, a  0

Positive

Two distinct real roots

0

One real root (a double root)

Negative

Two imaginary roots, one the conjugate of the other

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EXAMPLE

6

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137

Using the Discriminant Find the number of real roots of each quadratic equation. (A) 2x2  4x  1  0

(B) 2x2  4x  2  0

(C) 2x2  4x  3  0

SOLUTIONS

(A) b2  4ac  (4)2  4(2)(1)  8 7 0; two real roots (B) b2  4ac  (4)2  4(2)(2)  0; one real (double) root (C) b2  4ac  (4)2  4(2)(3)  8 6 0; no real roots (two imaginary roots) 

MATCHED PROBLEM

6

Find the number of real roots of each quadratic equation. (A) 3x2  6x  5  0

(B) 3x2  6x  1  0

(C) 3x2  6x  3  0



Z Solving Applications Involving Quadratic Equations We now consider several applications that involve finding solutions to quadratic equations. At this point, you should review the strategy for solving word problems presented in Section 1-1.

Z STRATEGY FOR SOLVING WORD PROBLEMS 1. Read the problem carefully—several times if necessary—that is, until you understand the problem, know what is to be found, and know what is given. 2. Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully. 3. If appropriate, draw figures or diagrams and label known and unknown parts. 4. Look for formulas connecting the known quantities to the unknown quantities. 5. Form an equation relating the unknown quantities to the known quantities. 6. Solve the equation and write answers to all questions asked in the problem. 7. Check and interpret all solutions in terms of the original problem—not just the equation found in step 5—since a mistake may have been made in setting up the equation in step 5.

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7

Setting Up and Solving a Word Problem The sum of a number and its reciprocal is

13 6.

Find all such numbers.

SOLUTION

Let x  the number; then: x

1 13  x 6

Multiply both sides by 6x. [Note: x  0.]

1 13 (6x)x  (6x)  (6x) x 6 6x2  6  13x 6x  13x  6  0 (2x  3)(3x  2)  0 2x  3  0 or 3 x2

Subtract 13x from both sides.

2

Thus, two such numbers are CHECK

3 2 3 2

Factor the left side. Use the zero property.

3x  2  0 x  23

Solve each equation for x.

and 32.  23 

MATCHED PROBLEM

13 6

2 3

 32  136



7

The sum of two numbers is 23 and their product is 132. Find the two numbers. [Hint: If one number is x, then the other number is 23  x.] 

EXAMPLE

8

A Distance–Rate–Time Problem An excursion boat takes 1.6 hours longer to go 36 miles up a river than to return. If the rate of the current is 4 miles per hour, what is the rate of the boat in still water? SOLUTION

Let x  Rate of boat in still water x  4  Rate downstream x  4  Rate upstream

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a

Time Time ba b  1.6 upstream downstream 36 36   1.6 x4 x4 36(x  4)  36(x  4)  1.6(x  4)(x  4) 36x  144  36x  144  1.6x2  25.6 1.6x2  313.6 x2  196 x  1196  14

T

139

D , x  4, x  4 R

Multiply both sides by (x  4)(x  4), the LCD. Remove parentheses. Combine like terms and isolate 1.6x 2 on one side of the equation. Divide both sides by 1.6. Use the square root property.

The rate in still water is 14 miles per hour. [Note: 1196  14 must be discarded, since it doesn’t make sense in the problem to have a negative rate.] CHECK

Time upstream 

D 36   3.6 R 14  4

Time downstream 

D 36  2 R 14  4 1.6

MATCHED PROBLEM

Difference of times



8

Two boats travel at right angles to each other after leaving a dock at the same time. One hour later they are 25 miles apart. If one boat travels 5 miles per hour faster than the other, what is the rate of each? [Hint: Use the Pythagorean theorem,* remembering that distance equals rate times time.] 

EXAMPLE

9

A Quantity–Rate–Time Problem A payroll can be completed in 4 hours by two computers working simultaneously. How many hours are required for each computer to complete the payroll alone if the older model requires 3 hours longer than the newer model? Compute answers to two decimal places. SOLUTION

Let x  Time for new model to complete the payroll alone x  3  Time for old model to complete the payroll alone 4  Time for both computers to complete the payroll together a

c b

*Pythagorean theorem: A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the two shorter sides: c2  a2  b2.

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Then, 1  Rate for new model x 1  Rate for old model x3

Completes

1 of the payroll per hour x

Completes

1 of the payroll per hour x3

Part of job Part of job completed by completed by ± ≤  ± ≤  1 whole job new model in old model in 4 hours 4 hours 1 1  1 x  0, x  3 (4)  (4) x x3 4 4   1 Multiply both sides by x x3 4(x  3)  4x  x(x  3) 4x  12  4x  x2  3x x2  5x  12  0

x(x  3).

Remove parentheses. Write in standard form. Use the quadratic formula.

5  173 x 2 x

5  173  6.77 2

5  173  1.77 is 2 discarded since x cannot be negative.

x  3  9.77 The new model would complete the payroll in 6.77 hours working alone, and the old model would complete the payroll in 9.77 hours working alone. CHECK

1 1 ? (4)  (4)  1 6.77 9.77 ✓ 1.000 259  1

Note: We do not expect the check to be exact, since we rounded the answers to two decimal places. An exact check would be produced by using x  (5  173) 2. The latter is left to the reader. 

MATCHED PROBLEM

9

Two technicians can complete a mailing in 3 hours when working together. Alone, one can complete the mailing 2 hours faster than the other. How long will it take each person to complete the mailing alone? Compute the answers to two decimal places.  In Example 10, we introduce some concepts from economics that will be used throughout this book. The quantity of a product that people are willing to buy during some period of time is called the demand for that product. The price p of a product

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and the demand q for that product are often related by a price–demand equation of the following form: q  a  bp

q is the number of items that can be sold at $p per item.

The constants a and b in a price–demand equation are usually determined by using historical data and statistical analysis. The amount of money received from the sale of q items at $p per item is called the revenue and is given by R  (Number of items sold)  (Price per item)  qp  q(a  bp)

EXAMPLE

10

Price and Demand The daily price–demand equation for whole milk in a chain of supermarkets is q  5,600  800p where p is the price per gallon and q is the number of gallons sold per day. Find the price(s) that will produce a revenue of $9,500. Round answer(s) to two decimal places. SOLUTION

The revenue equation is R  qp  (5,600  800p)p  5,600p  800p2 To produce a revenue of $9,500, p must satisfy 5,600p  800p2  9,500 9,500  5,600p  800p2  0 p2  7p  11.875  0

Subtract 9,500 from both sides. Divide both sides by 800. Use the quadratic formula with a  1, b  7, and c  11.875.

7  11.5 2  2.89, 4.11

p

Selling whole milk for either $2.89 per gallon or $4.11 per gallon will produce a revenue of $9,500. 

MATCHED PROBLEM

10

Refer to Example 10. Find the prices that will produce a revenue of (A) $9,300

(B) $9,800



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ANSWERS 1. (A) x  4, 53

TO MATCHED PROBLEMS (B) x  32 (a double root)

(C) x  0, 54

2. (A) x  or 115 3 (B) x  2i (C) x  (1  12)3 3. (A) x2  5x  254  (x  52)2 (B) x2  mx  (m2 4)  [x  (m2)] 2 4. (A) x  4  119 (B) x  (6  i 13)3 or 2  ( 133)i 5. x  (3  119) 2 6. (A) No real roots (two imaginary roots) (B) Two real roots (C) One real (double) root 7. 11 and 12 8. 15 and 20 miles per hour 9. 5.16 and 7.16 hours 10. (A) $2.71, $4.29 (B) $3.50  253

1-5

Exercises

Leave all answers involving radicals in simplified radical form unless otherwise stated.

In Problems 1–6, solve by factoring. 1. 2x  8x

2. 2y  5y  3

3. 4t  9  12t

4. 3s  6s

5. 3w  13w  10

6. 16x2  9  24x

2

2

2

2

8. n2  16  0

9. c2  9  0

10. d 2  36  0

11. 4y2  9  0

12. 9x2  25  0

13. 25z2  32  0

14. 16w2  27  0

15. (s  1)2  5

16. (t  2)2  3

17. (n  3)2  4

18. (m  4)2  1

In Problems 19–26, use the discriminant to determine the number of real roots of each equation and then solve each equation using the quadratic formula. 19. x  2x  1  0

20. y  4y  7  0

21. x  2x  3  0

22. y2  4y  1  0

23. 2t 2  8  6t

24. 9s2  2  12s

25. 2t 2  1  6t

26. 9s2  7  12s

2 2

27. x2  4x  1  0

28. y2  4y  3  0

29. 2r2  10r  11  0

30. 2s2  6s  7  0

31. 4u2  8u  15  0

32. 4v2  16v  23  0

33. 3w2  4w  3  0

34. 3z2  8z  1  0

2

In Problems 7–18, solve by using the square root property. 7. m2  25  0

In Problems 27–34, solve by completing the square.

2

In Problems 35–52, solve by any method. 35. 12x2  7x  10

36. 9x2  9x  4

37. (2y  3)2  5

38. (3m  2)2  4

39. x2  3x  1

40. x2  2x  2

41. 7n2  4n

42. 8u2  3u  0

43. 1 

8 4  x x2

44.

2 3  21 u u

47.

4 1 2   x2 x3 x1

45.

24 24 1 10  m 10  m

46.

1.2 1.2  1 y y1

48.

3 2 4   x1 x3 x2

49.

x2 x2 x1  2 1 x3 3x x 9

50.

11 2x  3 x3   2x x2 x2  4

51.  3u  2   u2

52.  12  7x   x2

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In Problems 53–56, solve for the indicated variable in terms of the other variables. Use positive square roots only. 53. s  12gt 2

54. a2  b2  c2 for a

for t

55. P  EI  RI 2

for I

56. A  P(1  r)2 for r

Solve Problems 57–60 to three significant digits. 57. 2.07x2  3.79x  1.34  0

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143

74. Find the error in the following “proof ” that two arbitrary numbers are equal to each other: Let a and b be arbitrary numbers such that a  b. Then (a  b)2  a2  2ab  b2  b2  2ab  a2 (a  b)2  (b  a)2 abba 2a  2b ab

58. 0.610x2  4.28x  2.93  0 59. 4.83x2  2.04x  3.18  0

APPLICATIONS

60. 5.13x  7.27x  4.32  0

75. NUMBERS Find two numbers such that their sum is 21 and their product is 104.

61. Consider the quadratic equation

76. NUMBERS Find all numbers with the property that when the number is added to itself the sum is the same as when the number is multiplied by itself.

2

x  4x  c  0 2

where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table 1. 62. Consider the quadratic equation x2  2x  c  0 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table 1. Solve Problems 63–66 and leave answers in simplified radical form (i is the imaginary unit). 63. 13x2  8 12x  4 13

64. 2 12x  13  13x2

65. x2  2ix  3

66. x2  2ix  3

In Problems 67 and 68, find all solutions. 67. x3  1  0

68. x4  1  0

69. Can a quadratic equation with rational coefficients have one rational root and one irrational root? Explain. 70. Can a quadratic equation with real coefficients have one real root and one imaginary root? Explain. 71. Show that if r1 and r2 are the two roots of ax2  bx  c  0, then r1r2  ca. 72. For r1 and r2 in Problem 71, show that r1  r2  ba. 73. In one stage of the derivation of the quadratic formula, we replaced the expression  2(b2  4ac)4a2 with 2b2  4ac2a What justifies using 2a in place of  2a ?

77. NUMBERS Find two consecutive positive even integers whose product is 168. 78. NUMBERS The sum of a number and its reciprocal is Find the number.

10 3.

79. GEOMETRY If the length and width of a 4- by 2-inch rectangle are each increased by the same amount, the area of the new rectangle will be twice that of the original. What are the dimensions of the new rectangle (to two decimal places)? 80. GEOMETRY Find the base b and height h of a triangle with an area of 2 square feet if its base is 3 feet longer than its height and the formula for area is A  12bh. 81. PRICE AND DEMAND The daily price–demand equation for hamburgers at a fast-food restaurant is q  1,600  200p where q is the number of hamburgers sold daily and p is the price of one hamburger (in dollars). Find the demand and the revenue when the price of a hamburger is $3. 82. PRICE AND DEMAND The weekly price–demand equation for medium pepperoni pizzas at a fast-food restaurant is q  8,000  400p where q is the number of pizzas sold weekly and p is the price of one medium pepperoni pizza (in dollars). Find the demand and the revenue when the price is $8. 83. PRICE AND DEMAND Refer to Problem 81. Find the price p that will produce each of the following revenues. Round answers to two decimal places. (A) $2,800 (B) $3,200 (C) $3,400

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84. PRICE AND DEMAND Refer to Problem 82. Find the price p that will produce each of the following revenues. Round answers to two decimal places. (A) $38,000 (B) $40,000 (C) $42,000 85. PUZZLE Two planes travel at right angles to each other after leaving the same airport at the same time. One hour later they are 260 miles apart. If one travels 140 miles per hour faster than the other, what is the rate of each? 86. NAVIGATION A speedboat takes 1 hour longer to go 24 miles up a river than to return. If the boat cruises at 10 miles per hour in still water, what is the rate of the current?

[Hint: Use Euclid’s theorem* to find a relationship between the length and width of the building.] ★92.

ARCHITECTURE An architect is designing a small A-frame cottage for a resort area. A cross section of the cottage is an isosceles triangle with an area of 98 square feet. The front wall of the cottage must accommodate a sliding door that is 6 feet wide and 8 feet high (see the figure). Find the width and height of the cross section of the cottage. [Recall: The area of a triangle with base b and altitude h is bh2.]

★87.

ENGINEERING One pipe can fill a tank in 5 hours less than another. Together they can fill the tank in 5 hours. How long would it take each alone to fill the tank? Compute the answer to two decimal places.

★★88.

ENGINEERING Two gears rotate so that one completes 1 more revolution per minute than the other. If it takes the smaller gear 1 second less than the larger gear to complete 1 5 revolution, how many revolutions does each gear make in 1 minute?

★89.

6 feet

PHYSICS—ENGINEERING For a car traveling at a speed of v

miles per hour, under the best possible conditions the shortest distance d necessary to stop it (including reaction time) is given by the empirical formula d  0.044v2  1.1v, where d is measured in feet. Estimate the speed of a car that requires 165 feet to stop in an emergency. ★90.

PHYSICS—ENGINEERING If a projectile is shot vertically into the air (from the ground) with an initial velocity of 176 feet per second, its distance y (in feet) above the ground t seconds after it is shot is given by y  176t  16t2 (neglecting air resistance). (A) Find the times when y is 0, and interpret the results physically. (B) Find the times when the projectile is 16 feet off the ground. Compute answers to two decimal places.

★91.

8 feet

93. TRANSPORTATION A delivery truck leaves a warehouse and travels north to factory A. From factory A the truck travels east to factory B and then returns directly to the warehouse (see the figure). The driver recorded the truck’s odometer reading at the warehouse at both the beginning and the end of the trip and also at factory B, but forgot to record it at factory A (see the table). The driver does recall that it was farther from the warehouse to factory A than it was from factory A to factory B. Since delivery charges are based on distance from the warehouse, the driver needs to know how far factory A is from the warehouse. Find this distance.

Factory A

CONSTRUCTION A developer wants to erect a rectangular

Factory B

building on a triangular-shaped piece of property that is 200 feet wide and 400 feet long (see the figure). Find the dimensions of the building if its cross-sectional area is 15,000 square feet.

Warehouse 200 feet 15,000 square feet

*Euclid’s theorem: If two triangles are similar, their corresponding sides are proportional: c

a 400 feet

b

a

c b

a b c   a¿ b¿ c¿

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Odometer readings

★★94.

Warehouse

5 2 8 4 6

Factory A

5 2 ? ? ?

Factory B

5 2 9 3 7

Warehouse

5 3 0 0 2

Additional Equation-Solving Techniques

145

enclose an area of 100,000 square feet. Find the length of the straightaways and the diameter of the semicircles to the nearest foot. [Recall: The area A and circumference C of a circle of di2 ameter d are given by A  d 4 and c  d.]

1

CONSTRUCTION A 4 -mile track for racing stock cars con-

100,000 square feet

sists of two semicircles connected by parallel straightaways (see the figure). In order to provide sufficient room for pit crews, emergency vehicles, and spectator parking, the track must

1-6

Additional Equation-Solving Techniques Z Solving Equations Involving Radicals Z Revisiting Equations Involving Absolute Value Z Solving Equations Involving Rational Exponents

In this section we show that raising each side of an equation to the same power can lead to the solutions of the equation. We also show that if changing the variable in an equation transforms the equation into a quadratic equation, then any of the solution techniques discussed in Section 1-5 can be applied to the transformed equation.

Z Solving Equations Involving Radicals In solving an equation involving a radical like x  1x  2 it appears that we can remove the radical by squaring each side and then proceed to solve the resulting quadratic equation. Thus, x  1x  2 Square both sides. 2 2 x  ( 1x  2) Recall that ( 1a)2  a if a  0. 2 x x2 Subtract x  2 from both sides. 2 x x20 Factor the left side. (x  2)(x  1)  0 Use the zero property. x20 or x10 x2 or x  1

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Now we check these results in the original equation. Check: x  2 x  1x  2 ? 2  12  2 ? 2  14 ✓ 22

Check: x  1 x  1x  2 ? 1  11  2 ? 1  11 1  1

Thus, 2 is a solution, but 1 is not. These results are a special case of Theorem 1.

Z THEOREM 1 Squaring Operation on Equations If both sides of an equation are squared, then the solution set of the original equation is a subset of the solution set of the new equation. Equation x3 x2  9

Solution Set {3} {3, 3}

This theorem provides us with a method of solving some equations involving radicals. It is important to remember that any new equation obtained by raising both members of an equation to the same power may have solutions, called extraneous solutions, that are not solutions of the original equation. On the other hand, any solution of the original equation must be among those of the new equation. Every solution of the new equation must be checked in the original equation to eliminate extraneous solutions.

ZZZ

EXPLORE-DISCUSS 1

Squaring both sides of the equations x  1x and x   1x produces the new equation x2  x. Find the solutions to the new equation and then check for extraneous solutions in each of the original equations.

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ZZZ

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147

CAUTION ZZZ

Remember that 19 represents the positive square root of 9 and 19 represents the negative square root of 9. It is correct to use the symbol  to combine these two roots when solving an equation: x2  9

implies

x   19  3

But it is incorrect to use  when evaluating the positive square root of a number: 19  3

EXAMPLE

1

19  3

Solving Equations Involving Radicals Solve: (A) x  1x  4  4

(B) 12x  3  1x  2  2

SOLUTIONS

x  1x  4  4 1x  4  4  x

(A)

Isolate radical on one side. Square both sides.

(1x  4)2  (4  x)2 x  4  16  8x  x2 x  9x  20  0 (x  5)(x  4)  0 2

CHECK

Write in standard form. Factor left side. Use the zero property.

x50

or

x40

x5

or

x4

x5 x  1x  4  4 ? 5  15  4  4 64

x4 x  1x  4  4 ? 4  14  4  4 ✓ 44

This shows that 4 is a solution to the original equation and 5 is extraneous. Thus, x4

Only one solution

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(B) To solve an equation that contains more than one radical, isolate one radical at a time and square both sides to eliminate the isolated radical. Repeat this process until all the radicals are eliminated. 12x  3  1x  2  2 12x  3  1x  2  2

Isolate one of the radicals. Square both sides.

(12x  3)2  ( 1x  2  2)2 2x  3  x  2  41x  2  4 x  1  41x  2

Isolate the remaining radical. Square both sides.

(x  1)2  (4 1x  2)2 x2  2x  1  16(x  2) x2  14x  33  0 (x  3)(x  11)  0

CHECK

x30

or

x3

or

x3 12x  3  1x  2  2 ? 12(3)  3  13  2  2 ✓ 22

Write in standard form. Factor left side. Use the zero property.

x  11  0 x  11 x  11 12x  3  1x  2  2 ? 12(11)  3  111  2  2 ✓ 22

Both solutions check. Thus, x  3, 11

MATCHED PROBLEM

Two solutions



1

Solve: (A) x  5  1x  3 (B) 12x  5  1x  2  5



Z Revisiting Equations Involving Absolute Value Squaring both sides of an equation can be a useful operation even if the equation does not involve any radicals.

EXAMPLE

2

Absolute Value Equations Revisited Solve the following equation by squaring both sides: |x  4|  3x  8

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149

SOLUTION

|x  4|  3x  8

Square both sides.

|x  4|2  (3x  8)2

Use |u|2  u2 and expand each side.

x2  8x  16  9x2  48x  64 8x2  56x  48  0 x2  7x  6  0 (x  1)(x  6)  0

CHECK

Write in standard form. Divide both sides by 8. Factor the left side. Use the zero property.

x10

or

x60

x1

or

x6

x1 |x  4|  3x  8 ? |1  4|  3(1)  8 ? |5|  5 5  5

x6  x  4   3x  8 ? |6  4|  3(6)  8 ?  10   10 ✓

10  10

Thus, x  6 is the only solution. Compare this solution with the solution of Example 7, Section 1-3. Squaring both sides eliminates the need to consider two separate cases. 

MATCHED PROBLEM

2

Solve the following equation by squaring both sides: 3x  4  x  4

ZZZ



CAUTION ZZZ

When squaring an expression like 1x  2  2, be certain to correctly apply the formula for squaring the sum of two terms (see Section R-4): (u  v)2 

u2



2uv



v2

(1x  2  2)2  ( 1x  2)2  2( 1x  2)(2)  (2)2  x  2  41x  2  4 Do not omit the middle term in this product: ( 1x  2  2)2  x  2  4

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Z Solving Equations Involving Rational Exponents To solve the equation x23  x13  6  0 write it in the form (x13)2  x13  6  0 You can now recognize that the equation is quadratic in x13. So, we solve for x13 first, and then solve for x. We can solve the equation directly or make the substitution u  x13, solve for u, and then solve for x. Both methods of solution are shown below. Method I. Direct solution: (x13)2  x13  6  0 (x  3)(x13  2)  0 x1  3  3 or x13  2

Factor left side.

13

(x13)3  33

Use the zero property. Cube both sides.

(x13)3  (2)3

x  27

x  8

Solution set: {8, 27} Method II. Using substitution: Let u  x13, solve for u, and then solve for x. u2  u  6  0 (u  3)(u  2)  0 u  3, 2

Factor left side. Use the zero property.

Replacing u with x13, we obtain x13  3 x  27

or

x13  2 x  8

Cube both sides.

Solution set: {8, 27} In general, if an equation that is not quadratic can be transformed to the form au2  bu  c  0 where u is an expression in some other variable, then the equation is said to be quadratic in u and is called an equation of quadratic type. Once recognized as an equation of quadratic type, an equation often can be solved using quadratic methods.

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ZZZ

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151

EXPLORE-DISCUSS 2

Which of the following is an equation of quadratic type? (A) 3x4  2x2  7

(B) 7x5  3x2  3

(C) 2x5  4x2 1x  6

(D) 8x2 1x  5x1 1x  2

In general, if a, b, c, m, and n are nonzero real numbers, when is axm  bxn  c  0 an equation of quadratic type?

EXAMPLE

3

Solving Equations of Quadratic Type Solve: (A) x4  3x2  4  0

(B) 3x25  6x15  2  0

SOLUTIONS

(A) The equation is quadratic in x2. We solve for x2 and then for x: (x2)2  3x2  4  0 (x2  4)(x2  1)  0 x2  4 or x  2 or

Factor the left side. Use the zero property.

x2  1 x  i

Use the square root property.

Solution set: {2, 2, i, i} Since we did not raise each side of the equation to a natural number power, we do not have to check for extraneous solutions. (You should still check the accuracy of the solutions.) (B) The equation 3x25  6x15  2  0 is quadratic in x15. We substitute u  x15 and solve for u: 3u2  6u  2  0 6  112 3  13 u  6 3 5 xu 3  13 5 a b 3

Use the quadratic formula.

Substitute u 

3  13 . 3

Write with positive exponent.

Thus, the two solutions are xa

5 3 b 3  13

Two real solutions



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MATCHED PROBLEM

3

Solve: (A) x4  3x2  4  0

ZZZ

(B) 3x25  x15  2  0



EXPLORE-DISCUSS 3

Solve the equation m  71m  12  0 two ways: (A) By squaring

(B) By making a substitution

Which method do you prefer? Why?

EXAMPLE

4

Setting Up and Solving a Word Problem The diagonal of a rectangle is 10 inches, and the area is 45 square inches. Find the dimensions of the rectangle correct to one decimal place. SOLUTION

h

10

inc

x

Draw a rectangle and label the dimensions as shown in Figure 1. From the Pythagorean theorem,

es

y

x2  y2  102 Thus, y  2100  x2

Z Figure 1

Since the area of the rectangle is given by xy, we have x2100  x2  45 x2(100  x2)  2,025 100x2  x4  2,025 (x2)2  100x2  2,025  0 100  21002  4(1)(2,025) x2  2 x2  50  5119 x  250  5119

Square both sides. Remove parentheses. Write as a quadratic in x2. Use the quadratic formula. Simplify. Use the square root property. Discard the negative solutions since x  0.

If x  250  5119  8.5, then y  2100  x2  2100  (50  5119)  250  5119  5.3

Substitute x  250  5119 Simplify.

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153

Thus, the dimensions of the rectangle to one decimal place are 8.5 inches by 5.3 inches. Notice that if x  250  5119, then y  250  5119, and the dimensions are still 8.5 inches by 5.3 inches. CHECK

(8.5)(5.3)  45.05  45 28.52  5.32  1100.34  10

Area: Diagonal:

Note: An exact check can be obtained by using 250  5119 and 250  5119 in place of these decimal approximations. This is left to the reader. 

MATCHED PROBLEM

4

If the area of a right triangle is 24 square inches and the hypotenuse is 12 inches, find the lengths of the legs of the triangle correct to one decimal place. 

ANSWERS 1. (A) x  7 2. x  0, 4

TO MATCHED PROBLEMS (B) x  2

3. (A) x  1, 2i

(B) x  1, 

243 32

4. 11.2 inches by 4.3 inches

1-6

Exercises

In Problems 1–6, determine the validity of each statement. If a statement is false, explain why.

In Problems 19–24, determine if the equation is an equation of quadratic type. Do not solve.

1. If x2  5, then x   15.

2. 125  5

19. 3x5  4x2  9  0

20. 4y6  7y3  17  0

3. ( 1x  1  1)2  x

4. ( 1x  1)2  1  x

21. 6t45  11t 25  8  0

22. 5w2  2w1  4  0

5. If x3  2, then x  8.

6. If x13  8, then x  2.

23. 2y  1y  5  0

3 24. 3x  1 x  13  0

Solve: 7. 1x  2  4

In Problems 25–48, solve the equation 8. 1x  4  2

25. 13t  2  1  21t

26. 15t  4  21t  1

9. 13y  2  y  2

10. 14y  1  5  y

27. m4  2m2  15  0

28. m4  4m2  12  0

11. 15w  6  w  2

12. 12w  3  w  1

29. 3x  2x2  2

30. x  25x2  9

13.  2x  1   x  2

14.  2x  2   5  x

31. 2y23  5y13  12  0

32. 3y23  2y13  8  0

15.  x  5   7  2x

16.  x  7   1  2x

33. (m2  2m)2  2(m2  2m)  15

17.  3x  4   2x  5

18.  3x  1   x  1

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34. (m2  2m)2  6(m2  2m)  16 35. 12t  3  2  1t  2 36. 12x  1  1x  5  3 37. 1w  3  12  w  3

62. DESIGN A food-processing company packages an assortment of their products in circular metal tins 12 inches in diameter. Four identically sized rectangular boxes are used to divide the tin into eight compartments (see the figure). If the crosssectional area of each box is 15 square inches, find the dimensions of the boxes correct to one decimal place.

38. 1w  7  2  13  w 39. 18  z  1  1z  5 40. 13z  1  2  1z  1 41. 24x2  12x  1  6x  9 42. 6x  24x2  20x  17  15 43. y2  2y1  3  0

44. y2  3y1  4  0

45. 2t4  5t2  2  0

46. 15t4  23t2  4  0

47. 3z1  3z1/2  1  0

48. 2z1  3z1/2  2  0

★63.

CONSTRUCTION A water trough is constructed by bending a 4- by 6-foot rectangular sheet of metal down the middle and attaching triangular ends (see the figure). If the volume of the trough is 9 cubic feet, find the width correct to two decimal places.

Solve Problems 49–52 two ways: by squaring and by substitution. 49. 4m  81m  5  0

50. 4m  8 1m  21  0

51. 2w  31w  14

52. 3w  5 1w  12

6 feet

2 feet

In Problems 53–60, solve the equation. 53. 17  2x  1x  2  1x  5 54. 11  3x  12x  1  1x  2 55. 3  x4  5x2

56. 2  4x4  7x2

57. 2 1x  5  0.01x  2.04

★64.

DESIGN A paper drinking cup in the shape of a right circular cone is constructed from 125 square centimeters of paper (see the figure). If the height of the cone is 10 centimeters, find the radius correct to two decimal places.

58. 3 1x  1  0.05x  2.9 59. 2x25  5x15  1  0

r

60. x25  3x15  1  0

APPLICATIONS 61. MANUFACTURING A lumber mill cuts rectangular beams from circular logs (see the figure). If the diameter of the log is 16 inches and the cross-sectional area of the beam is 120 square inches, find the dimensions of the cross section of the beam correct to one decimal place.

h

Lateral surface area: S  ␲r r 2  h 2

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1

Review

155

Review

Linear Equations and Applications

A solution or root of an equation is a number in the domain or replacement set of the variable that when substituted for the variable makes the equation a true statement. To solve an equation is to find its solution set. An equation is an identity if it is true for all values from the domain of the variable and a conditional equation if it is true for some domain values and false for others. Two equations are equivalent if they have the same solution set. The properties of equality are used to solve equations: 1. If a  b, then a  c  b  c.

Addition Property

2. If a  b, then a  c  b  c.

Subtraction Property

4. If a  b and c  0, then

An equation that can be written in the standard form ax  b  0, a  0, is a linear or first-degree equation. If Q is quantity produced or distance traveled at an average or uniform rate R in T units of time, then the quantity–rate–time formulas are R

1-2

1. Read the problem carefully—several times if necessary—that is, until you understand the problem, know what is to be found, and know what is given. 2. Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully. 3. If appropriate, draw figures or diagrams and label known and unknown parts. 4. Look for formulas connecting the known quantities to the unknown quantities. 5. Form an equation relating the unknown quantities to the known quantities. 6. Solve the equation and write answers to all questions asked in the problem. 7. Check and interpret all solutions in terms of the original problem—not just the equation found in step 5—since a mistake may have been made in setting up the equation in step 5.

Division Property

5. If a  b, then either may replace Substitution Property the other in any statement without changing the truth or falsity of statement.

3. If a  b and c  0, then ca  cb. Multiplication Property

Z STRATEGY FOR SOLVING WORD PROBLEMS

a b  . c c

Q T

T

Q  RT

Q R

Linear Inequalities

The inequality symbols , , ,  are used to express inequality relations. Line graphs, interval notation, and the set operations of union and intersection are used to describe inequality relations. A solution of a linear inequality in one variable is a value of the variable that makes the inequality a true statement. Two inequalities are equivalent if they have the same solution set. Inequality properties are used to solve inequalities: 1. If a 6 b and b 6 c, then a 6 c.

Transitive Property

2. If a 6 b, then a  c 6 b  c.

Addition Property

3. If a 6 b, then a  c 6 b  c.

Subtraction Property

4. If a 6 b and c 7 0, then ca 6 cb. ⎫ Multiplication ⎬ 5. If a 6 b and c 6 0, then ca 7 cb. ⎭ Property 6. If a 6 b and c 7 0, then

a b 6 . ⎫ c c ⎪ ⎪

⎬ Division Property

a b ⎪ 7. If a 6 b and c 6 0, then 7 . ⎪ c c ⎭

The order of an inequality reverses if we multiply or divide both sides of an inequality statement by a negative number.

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Absolute Value in Equations and Inequalities

The absolute value of a number x is the distance on a real number line from the origin to the point with coordinate x and is given by x 

x x



can be used to find reciprocals and quotients. If a 7 0, then the principal square root of the negative real number a is 1a  i1a.

1-5

if x 6 0 if x  0

Quadratic Equations and Applications

A quadratic equation in standard form is an equation that can be written in the form

The distance between points A and B with coordinates a and b, respectively, is d(A, B)   b  a , which has the following geometric interpretations:

ax2  bx  c  0

a0

x  c  d

Distance between x and c is equal to d.

where x is a variable and a, b, and c are constants. Methods of solution include:

x  c 6 d

Distance between x and c is less than d.

1. Factoring and using the zero property:

0 6 x  c 6 d

Distance between x and c is less than d, but x  c.

x  c 7 d

mn0

if and only if

m  0 or n  0 (or both)

2. Using the square root property: If A2  C, then A  1C

Distance between x and c is greater than d.

3. Completing the square: Equations and inequalities involving absolute values are solved using the following relationships for p 0:

b 2 b 2 x2  bx  a b  ax  b 2 2

1.  x   p is equivalent to x  p or x  p.

4. Using the quadratic formula:

2.  x  6 p is equivalent to p 6 x 6 p.

x

3.  x  7 p is equivalent to x 6 p or x 7 p. These relationships also hold if x is replaced with ax  b. For x any real number, 2x2   x .

1-4

Complex Numbers

A complex number in standard form is a number in the form a  bi, where a and b are real numbers and i is the imaginary unit. If b  0, then a  bi is also called an imaginary number. If a  0, then 0  bi  bi is also called a pure imaginary number. If b  0, then a  0i  a is a real number. The real part of a  bi is a and the imaginary part is bi. The complex zero is 0  0i  0. The conjugate of a  bi is a  bi. Equality, addition, and multiplication are defined as follows: 1. a  bi  c  di

if and only if

a  c and b  d

2. (a  bi)  (c  di)  (a  c)  (b  d )i 3. (a  bi)(c  di)  (ac  bd )  (ad  bc)i Since complex numbers obey the same commutative, associative, and distributive properties as real numbers, most operations with complex numbers are performed by using these properties and the fact that i2  1. The property of conjugates, (a  bi)(a  bi)  a2  b2

b  2b2  4ac 2a

If the discriminant b2  4ac is positive, the equation has two distinct real roots; if the discriminant is 0, the equation has one real double root; and if the discriminant is negative, the equation has two imaginary roots, each the conjugate of the other.

1-6

Additional Equation-Solving Techniques

A square root radical can be eliminated from an equation by isolating the radical on one side of the equation and squaring both sides of the equation. The new equation formed by squaring both sides may have extraneous solutions. Consequently, every solution of the new equation must be checked in the original equation to eliminate extraneous solutions. If an equation contains more than one radical, then the process of isolating a radical and squaring both sides can be repeated until all radicals are eliminated. If a substitution transforms an equation into the form au2  bu  c  0, where u is an expression in some other variable, then the equation is an equation of quadratic type that can be solved by quadratic methods.

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1

1. 0.05x  0.25(30  x)  3.3

Solve and graph Problems 3–5. 4.  y  9  6 5

5.  3  2x   5 6. Find the real part, the imaginary part, and the conjugate: (A) 9  4i (B) 5i (C) 10 7. Perform the indicated operations and write the answers in standard form: (A) (3  2i)  (6  8i) (B) (3  3i)(2  3i) 13  i (C) 5  3i

9. 2x  4x 2

10. 2x2  7x  3

11. m2  m  1  0

12. y 

13. 15x  6  x  0

2

3 2(y

 1)

14. For what values of x does 13  5x represent a real number? Solve Problems 15 and 16. 15.

7 10  4x  2 2x x  3x  10

Solve Problems 23–28. 5 2 5 23. au  b  2 4 25.

x 2 3  x  3 x x6 2

3 2  u u2

26. 2x23  5x13  12  0

27. m4  5m2  36  0 28. 1y  2  15y  1  3 Solve Problems 29–33, and round answers to three significant digits.

16.

1 1u u3   2u  2 6 3u  3

30. 1.52  0.770  2.04x  5.33 31.  9.71  3.62x  7 5.48 32.

3.77  8.47i 6.82  7.06i

33. 6.09x2  4.57x  8.86  0 Solve Problems 34–36 for the indicated variable in terms of the other variables. 34. P  M  Mdt

for M (mathematics of finance)

35. P  EI  RI

for I (electrical engineering)

Solve and graph Problems 17–19. x3 2x 17. 5 8 3

24. 1 

29. 2.15x  3.73(x  0.930)  6.11x

Solve Problems 8–13. 8. 2x  7  0

20. If the coordinates of A and B on a real number line are 8 and 2, respectively, find: (A) d(A, B) (B) d(B, A)

22. Convert to a  bi forms, perform the indicated operations, and write the final answers in standard form: (A) (2  14)  (3  19) 2  11 4  125 (B) (C) 3  14 14

5x 4x x2   1 3 2 4

2

19. 2(1  2m)2  3

21. Perform the indicated operations and write the final answers in standard form: (A) (3  i)2  2(3  i)  3 (B) i27

Solve Problems 1 and 2.

3. 3(2  x)  2  2x  1

157

Review Exercises

Work through all the problems in this chapter review and check answers in the back of the book. Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that type of problem is discussed. Where weaknesses show up, review appropriate sections in the text.

2.

Review Exercises

36. x  18.  3x  8  7 2

4y  5 2y  1

2

for y

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37. Find the error in the following “solution” and then find the correct solution. 3 4  2 x  4x  3 x  3x  2 4x2  12x  8  3x2  12x  9 2

x 1 2

x  1

x1

or

38. Consider the quadratic equation x2  6x  c  0 where c is a real number. Discuss the relationship between the values of c and the three types of roots listed in Table 1 in Section 1-5. 39. For what values of a and b is the inequality a  b 6 b  a true? 40. If a and b are negative numbers and a 7 b, then is ab greater than 1 or less than 1? 1

41. Solve for x in terms of y: y  1

1 1x

42. Solve and graph: 0 6  x  6  6 d

Solve Problems 43 and 44. 43. 2x2  13x  12 44. 4  8x2  x4 45. Evaluate: (a  bi) a

a b  2 ib, a, b  0 2 a b a  b2 2

APPLICATIONS

(B) What is the maximum safe sustained heart rate for a 20-year-old? (C) If the maximum safe sustained heart rate for a person is 126 beats per minute, how old is the person? ★48.

CHEMISTRY A chemical storeroom has an 80% alcohol solution and a 30% alcohol solution. How many milliliters of each should be used to obtain 50 milliliters of a 60% solution?

★49.

RATE–TIME An excursion boat takes 2 hours longer to go 45 miles up a river than to return. If the boat’s speed in still water is 12 miles per hour, what is the rate of the current?

★50. RATE–TIME A crew of four practices by rowing up a river for

a fixed distance and then returning to their starting point. The river has a current of 3 km/h. (A) Currently the crew can row 15 km/h in still water. If it takes them 25 minutes to make the round-trip, how far upstream did they row? (B) After some additional practice the crew cuts the round-trip time to 23 minutes. What is their still-water speed now? Round answers to one decimal place. (C) If the crew wants to increase their still-water speed to 18 km/h, how fast must they make the round-trip? Express answer in minutes rounded to one decimal place. 51. COST ANALYSIS Cost equations for manufacturing companies are often quadratic in nature. If the cost equation for manufacturing inexpensive calculators is C  x2  10x  31, where C is the cost of manufacturing x units per week (both in thousands), find: (A) The output for a $15 thousand weekly cost (B) The output for a $6 thousand weekly cost 52. BREAK-EVEN ANALYSIS The manufacturing company in Problem 51 sells its calculators to wholesalers for $3 each. Thus, its revenue equation is R  3x, where R is revenue and x is the number of units sold per week (both in thousands). Find the break-even point(s) for the company—that is, the output at which revenue equals cost. ★53.

46. NUMBERS Find a number such that subtracting its reciprocal from the number gives 16 15 . 47. SPORTS MEDICINE The following quotation was found in a sports medicine handout: “The idea is to raise and sustain your heart rate to 70% of its maximum safe rate for your age. One way to determine this is to subtract your age from 220 and multiply by 0.7.” (A) If H is the maximum safe sustained heart rate (in beats per minute) for a person of age A (in years), write a formula relating H and A.

CHEMISTRY If the temperature T of a solution must be kept within 5°C of 110°C, express this restriction as an absolute value inequality.

★54.

DESIGN The pages of a textbook have uniform margins of 2 centimeters on all four sides (see the figure). If the area of the entire page is 480 square centimeters and the area of the printed portion is 320 square centimeters, find the dimensions of the page.

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2

2

2

159

the figure). If the area of each triangle is 24 square feet, find the base correct to two decimal places.

2

2

Group Activity

2 2

2

8 feet

Figure for 54. ★55.

DESIGN A landscape designer uses 8-foot timbers to form a pattern of isosceles triangles along the wall of a building (see

CHAPTER

ZZZ GROUP

1 ACTIVITY Solving a Cubic Equation

If a, b, and c are real numbers with a  0, then the quadratic equation ax2  bx  c  0 can be solved by a variety of methods, including the quadratic formula. How can we solve the cubic equation ax3  bx2  cx  d  0,

a0

(1)

and is there a formula for the roots of this equation? The first published solution of equation (1) is generally attributed to the Italian mathematician Girolamo Cardano (1501–1576) in 1545. His work led to a complicated formula for the roots of equation (1) that involves topics that are discussed later in this text. For now, we will use Cardano’s method to find a real solution in special cases of equation (1). Note that because a is nonzero, we can always multiply both sides of (1) by 1a to make the coefficient of x3 equal to 1. To solve the cubic equation x3  6x2  6x  5  0 first substitute xy

b 6 y y2 3 3

(2)

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in equation (2) and simplify to obtain the reduced cubic y3  6y = 9

(3)

Next introduce two new variables u and v defined by 3uv  6

Note that 6 is the coefficient of y in equation (3).

and u3  v3  9

Note that 9 is the constant term in equation (3).

Solve the first equation for v  2u and substitute for v in the second, to obtain 2 3 b 9 u 8 u3  3  9 u u6  8  9u3 u6  9u3  8  0 3 (u  8)(u3  1)  0 u3  a

u3  8  0 u3  8 3 u 2 82 2 2 v   1 u 2

Simplify.

Multiply both sides by u3. Subtract 9u3 from both sides. Factor this quadratic in u3. Use the zero property.

u3  1  0

or

u3  1 3 u 2 11 2 2 v   2 u 1

or or or

Take the cube root of each side.

Use v  2u to determine v.

If we choose either u  2 and v  1 or u  1 and v  2, then yuv3 is a solution of equation (3) (verify this) and xy2325 is a solution of equation (2) (verify this also). The steps we followed in solving equation (2) are summarized next.

Cardano’s Method for Solving a Cubic Equation Let x3  bx2  cx  d  0 Step 1. Substitute x  y  b3 to obtain the reduced cubic y3  my  n. Step 2. Define u and v by m  3uv and n  u3  v3. Use v  m/3u to write n  u3  a

m 3 b 3u

and solve for u3 by factoring this quadratic in u3 or by using the quadratic formula. Then solve for u.

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161

Step 3. Using either of the solutions found in step 2, xy

b b uv 3 3

is a solution to x3  bx2  cx  d  0 (A) The key to Cardano’s method is to recognize that if u and v are defined as in step 2, then y  u  v is a solution of the reduced cubic. Verify this by substituting y  u  v, m  3uv, and n  u3  y3 in y3  my  n and show that the result is an identity. (B) Use Cardano’s method to solve x3  6x2  3x  8  0

(4)

Use a calculator to find a decimal approximation of your solution and check your answer by substituting this approximate value in equation (4). (C) Use Cardano’s method to solve x3  6x2  9x  6  0

(5)

Use a calculator to find a decimal approximation of your solution and check your answer by substituting this approximate value in equation (5). n 2 m 3 (D) In step 2 of Cardano’s method, show that u3 is real if a b  a b. 2 3

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Graphs

C

2

SECTIONS 2-1

Cartesian Coordinate Systems

2-2

Distance in the Plane

2-3

Equations of a Line

2-4

Linear Equations and Models Chapter 2 Review Chapter 2 Group Activity: Rates of Change

CHAPTER

ANALYTIC geometry is the study of the relationship between geometric forms, such as circles and lines, and algebraic forms, such as equations and inequalities. The key to this relationship is the Cartesian coordinate system, named after the French mathematician and philosopher René Descartes (1596–1650) who was the first to combine the study of algebra and geometry into a single discipline. We begin by discussing some of the basic tools used to graph equations. Next we cover the relationship between equations of straight lines and their graphs. We conclude by considering models that involve linear equations.

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2-1

Cartesian Coordinate Systems Z Reviewing Cartesian Coordinate Systems Z Graphing: Point by Point Z Using Symmetry as an Aid in Graphing

In Chapter 1 we discussed algebraic methods for solving equations. In this section we show how to find a geometric representation ( graph) of an equation. Examining the graph of an equation often results in additional insight into the nature of the equation’s solutions.

Z Reviewing Cartesian Coordinate Systems y 10

II

I x

10

10

III

IV

10

Z Figure 1 Cartesian coordinate system.

y 10

R(5, 10)

Q(10, 5) a 10

Origin b (0, 0)

x 10

P (a, b)

10

Z Figure 2 Coordinates in a plane.

Just as a real number line is formed by establishing a one-to-one correspondence between the points on a line and the elements in the set of real numbers, we can form a real plane by establishing a one-to-one correspondence between the points in a plane and elements in the set of all ordered pairs of real numbers. This can be done by means of a Cartesian coordinate system. Recall that to form a Cartesian or rectangular coordinate system, we select two real number lines, one horizontal and one vertical, and let them cross through their origins, as indicated in Figure 1. Up and to the right are the usual choices for the positive directions. These two number lines are called the horizontal axis and the vertical axis, or, together, the coordinate axes. The horizontal axis is usually referred to as the x axis and the vertical axis as the y axis, and each is labeled accordingly. Other labels may be used in certain situations. The coordinate axes divide the plane into four parts called quadrants, which are numbered counterclockwise from I to IV (see Fig. 1). Now we want to assign coordinates to each point in the plane. Given an arbitrary point P in the plane, pass horizontal and vertical lines through the point (Fig. 2). The vertical line will intersect the horizontal axis at a point with coordinate a, and the horizontal line will intersect the vertical axis at a point with coordinate b. These two numbers written as the ordered pair* (a, b) form the coordinates of the point P. The first coordinate a is called the abscissa of P; the second coordinate b is called the ordinate of P. The abscissa of Q in Figure 2 is 10, and the ordinate of Q is 5. The coordinates of a point can also be referenced in terms of the axis labels. The x coordinate of R in Figure 2 is 5, and the y coordinate of R is 10. The point with coordinates (0, 0) is called the origin. *An ordered pair of real numbers is a pair of numbers in which the order is specified. We now use (a, b) as the coordinates of a point in a plane. In Chapter 1 we used (a, b) to represent an interval on a real number line. These concepts are not the same. You must always interpret the symbol (a, b) in terms of the context in which it is used.

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Cartesian Coordinate Systems

165

The procedure we have just described assigns to each point P in the plane a unique pair of real numbers (a, b). Conversely, if we are given an ordered pair of real numbers (a, b), then, reversing this procedure, we can determine a unique point P in the plane. Thus, There is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs of real numbers. This correspondence is often referred to as the fundamental theorem of analytic geometry. Because of this correspondence, we regularly speak of the point (a, b) when we are referring to the point with coordinates (a, b). We also write P  (a, b) to identify the coordinates of the point P. Thus, in Figure 2, referring to Q as the point (10, 5) and writing R  (5, 10) are both acceptable statements.

Z Graphing: Point by Point Given any set of ordered pairs of real numbers S, the graph of S is the set of points in the plane corresponding to the ordered pairs in S. The fundamental theorem of analytic geometry enables us to look at algebraic forms (a set of ordered pairs) geometrically and to look at geometric forms (a graph) algebraically. We begin by considering an algebraic form, an equation in two variables: y  x2  4

(1)

A solution to equation (1) is an ordered pair of real numbers (a, b) such that b  a2  4 The solution set of equation (1) is the set of all its solutions. More formally, Solution set of equation (1): 5(x, y) ƒ y  x2  46 To find a solution to equation (1) we simply replace one of the variables with a number and solve for the other variable. For example, if x  2, then y  22  4  0, and the ordered pair (2, 0) is a solution. Similarly, if y  5, then 5  x2  4, x2  9, x  3, and the ordered pairs (3, 5) and (3, 5) are solutions. Sometimes replacing one variable with a number and solving for the other variable will introduce imaginary numbers. For example, if y  5 in equation (1), then 5  x2  4 x2  1 x   11  i Thus, (i, 5) and (i, 5) are solutions to y  x2  4. However, the coordinates of a point in a rectangular coordinate system must be real numbers.

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For that reason, when graphing an equation, we only consider those values of the variables that produce real solutions to the equation. The graph of an equation in two variables is the graph of its solution set. In equation (1), we find that its solution set will have infinitely many elements and its graph will extend off any paper we might choose, no matter how large. Thus, to sketch the graph of an equation, we include enough points from its solution set so that the total graph is apparent. This process is called point-by-point plotting.

EXAMPLE

Graphing an Equation Using Point-by-Point Plotting

1

Sketch a graph of y  x2  4. SOLUTION

We make up a table of solutions—ordered pairs of real numbers that satisfy the given equation.

y

y  x2  4

x

4

3

2

1

0

1

2

3

4

y

12

5

0

3

4

3

0

5

12

15

(4, 12)

(4, 12) 10

(3, 5)

5

(2, 0)

(3, 5)

(2, 0)

5

5

(1, 3) 5

Z Figure 3

(1, 3) (0, 4)

x

After plotting these solutions, if there are any portions of the graph that are unclear, we plot additional points until the shape of the graph is apparent. Then we join all these plotted points with a smooth curve, as shown in Figure 3. Arrowheads are used to indicate that the graph continues beyond the portion shown here with no significant changes in shape. The resulting figure is called a parabola. Notice that if we fold the paper along the y axis, the right side will match the left side. We say that the graph is symmetric with respect to the y axis and call the y axis the axis of the parabola. More will be said about parabolas later in the text. 

MATCHED PROBLEM Sketch a graph of y2  x.

1 

This book contains a number of activities that use an electronic device capable of displaying graphs to emphasize the connection between graphic, numeric, and algebraic viewpoints. The two most common devices are handheld graphing calculators and computers with appropriate software. All these activities are clearly marked and easily omitted if no such device is available.

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Technology Connections To graph the equation in Example 1 on a graphing calculator, we first enter the equation in the calculator’s equation editor* [Fig. 4(a)]. Using Figure 3 as a guide, we next enter values for the window variables [Fig. 4(b)], and then we graph the equation

167

Cartesian Coordinate Systems

[Fig. 4(c)]. The values of the window variables, shown in red in Figure 4(c), are not displayed on the calculator screen. We add them to give you additional insight into the graph. Compare the graphs in Figure 3 and Figure 4(c). They are similar in shape, but they are not identical. The discrepancy is due to the difference in the axes scales. In Figure 3, one unit on the x axis is equal to one unit on the y axes. In Figure 4(c), one unit on the x axis is equal to about three units on the y axis. We will have more to say about axes scales later in this section. 15

5

5

5

Enter the equation. (a)

Enter the window variables. (b)

Graph the equation. (c)

Z Figure 4

*See the Technology Index for a list of graphing calculator terms used in this book.

ZZZ EXPLORE-DISCUSS

1

To graph the equation y  x3  2x, we use point-by-point plotting to obtain the graph in Figure 5. (A) Do you think this is the correct graph of the equation? If so, why? If not, why?

y 5

(B) Add points on the graph for x  2, 0.5, 0.5, and 2. (C) Now, what do you think the graph looks like? Sketch your version of the graph, adding more points as necessary. (D) Write a short statement explaining any conclusions you might draw from parts A, B, and C.

x

y

1 1 0 0 1 1

5

5

5

Z Figure 5

x

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Graphs illustrate the relationship between two quantities, one represented by the x coordinates and the other by the y coordinates. If no equation for the graph is available, we can find specific examples of this relationship by estimating coordinates of points on the graph. Example 2 illustrates this process.

2

Ozone Levels The ozone level is measured in parts per billion (ppb). The ozone level during a 12-hour period in a suburb of Milwaukee, Wisconsin, on a particular summer day is given in Figure 6 (source: Wisconsin Department of Natural Resources). Use this graph to estimate the following ozone levels to the nearest integer and times to the nearest quarter hour. (A) The ozone level at 6 P.M. (B) The highest ozone level and the time when it occurs. (C) The time(s) when the ozone level is 90 ppb.

120

100

Parts per billion (ppb)

EXAMPLE

80

60

40

20

0 Noon 1

2

3

4

5

6

7

8

9

10

11

12

Z Figure 6 Ozone level.

SOLUTIONS

(A) The ordinate of the point on the graph with abscissa 6 is 97 ppb (see Fig. 7). (B) The highest ozone level is 109 ppb at 3 P.M. (C) The ozone level is 90 ppb at about 12:30 P.M. and again at 10 P.M.

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120

100 97 90

Cartesian Coordinate Systems

169

B

109

Parts per billion (ppb)

bar67388_ch02_163-191

A C

C

80

60

40

20

0

12:30

2

3

4

5

6

7

8

9

10

11

12



Z Figure 7

MATCHED PROBLEM

2

Use Figure 6 to estimate the following ozone levels to the nearest integer and times to the nearest quarter hour. (A) The ozone level at 7 P.M. (B) The time(s) when the ozone level is 100 ppb.



An important aspect of this course, and later in calculus, is the development of tools that can be used to analyze graphs. A particularly useful tool is symmetry, which we now discuss.

Z Using Symmetry as an Aid in Graphing We noticed that the graph of y  x2  4 in Example 1 is symmetric with respect to the y axis; that is, the two parts of the graph coincide if the paper is folded along the y axis. Similarly, we say that a graph is symmetric with respect to the x axis if the parts above and below the x axis coincide when the paper is folded along the x axis. To make the intuitive idea of folding a graph along a line more concrete, we introduce two related concepts—reflection and symmetry. Z DEFINITION 1 Reflection 1. The reflection through the y axis of the point (a, b) is the point (a, b). 2. The reflection through the x axis of the point (a, b) is the point (a, b). 3. The reflection through the origin of the point (a, b) is the point (a, b). 4. To reflect a graph just reflect each point on the graph.

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EXAMPLE

GRAPHS

3

Reflections In a Cartesian coordinate system, plot the point P(4, 2) along with its reflection through (A) the y axis, (B) the x axis, (C) and the origin. SOLUTION y 5

C(4, 2)

B(4, 2)

5

5

A(4, 2)

x

P(4, 2)



5

MATCHED PROBLEM

3

In a Cartesian coordinate system, plot the point P(3, 5) along with its reflection through (A) the x axis, (B) the y axis, and (C) the origin.  Z DEFINITION 2 Symmetry A graph is symmetric with respect to 1. The x axis if (a, b) is on the graph whenever (a, b) is on the graph— reflecting the graph through the x axis does not change the graph. 2. The y axis if (a, b) is on the graph whenever (a, b) is on the graph— reflecting the graph through the y axis does not change the graph. 3. The origin if (a, b) is on the graph whenever (a, b) is on the graph— reflecting the graph through the origin does not change the graph.

Figure 8 illustrates these three types of symmetry. y (a, b)

(a, b)

x

Z Figure 8 Symmetry.

Symmetry with respect to x axis (b)

(a, b) x

x

x (a, b)

(a, b)

(a, b)

(a, b)

Symmetry with respect to y axis (a)

y

y

y

(a, b)

Symmetry with respect to origin (c)

(a, b)

(a, b)

Symmetry with respect to y axis, x axis, and origin (d)

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ZZZ EXPLORE-DISCUSS

Cartesian Coordinate Systems

171

2

If a graph possesses two of the three types of symmetry in Definition 1, must it also possess the third? Explain.

Given an equation, if we can determine the symmetry properties of its graph ahead of time, we can save a lot of time and energy in sketching the graph. For example, we know that the graph of y  x2  4 in Example 1 is symmetric with respect to the y axis, so we can carefully sketch only the right side of the graph; then reflect the result through the y axis to obtain the whole sketch—the point-by-point plotting is cut in half! The tests for symmetry are given in Theorem 1. These tests are easily applied and are very helpful aids to graphing. Recall, two equations are equivalent if they have the same solution set.

Z THEOREM 1 Tests for Symmetry

EXAMPLE

4

Symmetry with respect to the:

Equation is equivalent when:

y axis

x is replaced with x

x axis

y is replaced with y

Origin

x and y are replaced with x and y

Using Symmetry as an Aid to Graphing Test the equation y  x3 for symmetry and sketch its graph. SOLUTION

Test y Axis Replace x with x:

Test x Axis Replace y with y:

Test Origin Replace x with x and y with y:

y  (x)3 y  x3

y  x3 y  x3

y  (x)3 y  x3 y  x3

The only test that produces an equivalent equation is replacing x with x and y with y. Thus, the only symmetry property for the graph of y  x3 is symmetry with respect to the origin.

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Note that positive values of x produce positive values for y and negative values of x produce negative values for y. Thus, the graph occurs in the first and third quadrants. First we make a careful sketch in the first quadrant [Fig. 9(a)]. It is easier to perform a reflection through the origin if you first reflect through one axis [Fig. 9(b)] and then through the other axis [Fig. 9(c)]. (See Explore-Discuss 2.)

GRAPH

x

0

1

2

y

0

1

8

y

y

y  x3

10

y

10

10

(2, 8) (1, 1) 5

5

(2, 8)

x

(2, 8)

(1, 1) 5

(1, 1)

5

(1, 1)

x

5

10

(a)

(1, 1)

(b)

(c)



MATCHED PROBLEM

4

Test the equation y  x for symmetry and sketch its graph.

5

x

10

Z Figure 9

EXAMPLE

5

(2, 8)

(2, 8) 10

y  x3



Using Symmetry as an Aid to Graphing Test the equation y  x for symmetry and sketch its graph. SOLUTION

Test y Axis Replace x with x:

Test x Axis Replace y with y:

Test Origin Replace x with x and y with y:

y   x  y  x

y   x  y   x 

y   x  y   x  y   x 

Thus, the only symmetry property for the graph of y  x is symmetry with respect to the y axis. Since x is never negative, this graph occurs in the first and second quadrants. We make a careful sketch in the first quadrant; then reflect this graph through the y axis to obtain the complete sketch shown in Figure 10.

GRAPH

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Cartesian Coordinate Systems

173

y 5

y  x 5

x

0

2

4

y

0

2

4

5

x

5



Z Figure 10

MATCHED PROBLEM

5

Test the equation y   x  for symmetry and sketch its graph.

EXAMPLE

6



Using Symmetry as an Aid to Graphing Test the equation y2  x2  4 for symmetry and sketch its graph. SOLUTION

Since (x)2  x2 and (y)2  y2, the equation y2  x2  4 will be unchanged if x is replaced with x or if y is replaced with y. Thus, the graph is symmetric with respect to the y axis, the x axis, and the origin. We need to make a careful sketch in only the first quadrant, reflect this graph through the y axis, and then reflect everything through the x axis. To find quadrant I solutions, we solve the equation for either y in terms of x or x in terms of y. We choose to solve for y. y2  x2  4 y2  x2  4 y  2x2  4 To obtain the quadrant I portion of the graph, we sketch y  2x2  4 for x  0, 1, 2, . . . . The final graph is shown in Figure 11. x

0

1

2

3

4

y

2

15  2.2

18  2.8

113  3.6

120  4.5

y 5

y 2  x2  4

(3, √13)

(4, √20)

(2, √8) (0, 2) (1, √5) 5

5

x

5

Z Figure 11



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MATCHED PROBLEM

6

Test the equation 2y2  x2  2 for symmetry and sketch its graph.



To graph y2  x2  4 on a graphing calculator, we enter both 2x2  4 and 2x2  4 in the equation editor [Fig. 12(a)] and graph.

Technology Connections

5

5

5

5

(a)

(b)

Z Figure 12

ANSWERS

TO MATCHED PROBLEMS

1.

2. (A) 96 ppb 3.

y

P(3, 5)

5

5

5

10

(B) 1:45 P.M. and 5 P.M. y 5

B(3, 5)

x 5

5

5

A(3, 5)

5

C(3, 5)

x

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4. Symmetric with respect to the origin

Cartesian Coordinate Systems

5. Symmetric with respect to the y axis

y

y

5

5

5

5

x

5

5

5

x

5

6. Symmetric with respect to the x axis, the y axis, and the origin y 5

5

5

x

5

2-1

Exercises

In Problems 1–10, give a verbal description of the indicated subset of the plane in terms of quadrants and axes. 1. 5(x, y) ƒ x  06

2. 5(x, y) ƒ x 7 0, y 7 06

3. 5(x, y) ƒ x 6 0, y 6 06

4. 5(x, y) ƒ y  06

5. 5(x, y) ƒ x 7 0, y 6 06 7. 5(x, y) ƒ x 7 0, y  06 9. 5(x, y) ƒ xy 6 06

6. 5(x, y) ƒ y 6 0, x  06

8. 5(x, y) ƒ x 6 0, y 7 06

13. (0, 2), (1, 3), (4, 5), (2, 1) 14. (2, 0), (3, 2), (1, 4), (3, 5) In Problems 15–18, find the coordinates of points A, B, C, and D and the coordinates of the indicated reflections. 15. Reflect A, B, C, and D through the y axis.

10. 5(x, y) ƒ xy 7 06

y 5

[Hint: In Problems 9 and 10, consider two cases.]

A

D

In Problems 11–14, plot the given points in a rectangular coordinate system.

5

C

B

11. (5, 0), (3, 2), (4, 2), (4, 4) 12. (0, 4), (3, 2), (5, 1), (2, 4)

5

5

x

175

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GRAPHS

16. Reflect A, B, C, and D through the x axis. y

In Problems 27–30, use the graph to estimate to the nearest integer the missing coordinates of the indicated points. (Be sure you find all possible answers.) 27. (A) (8, ?) (D) (?, 6)

5

A

(B) (5, ?) (E) (?, 5)

C

5

5

(C) (0, ?) (F) (?, 0)

y

x 10

D

B

5 10

10

x

17. Reflect A, B, C, and D through the origin. y

10

5

B C 5

D

5

x

28. (A) (3, ?) (D) (?, 3)

(B) (5, ?) (E) (?, 4)

(C) (0, ?) (F) (?, 0)

y

A 5

10

18. Reflect A, B, C, and D through the x axis and then through the y axis.

10

10

x

y 10

5

C

A D

5

5

x

29. (A) (1, ?) (D) (?, 6)

(B) (8, ?) (E) (?, 4)

B

(C) (0, ?) (F) (?, 0)

y

5 10

Test each equation in Problems 19–26 for symmetry with respect to the x axis, y axis, and the origin. Sketch the graph of the equation. 19. y  2x  4

20. y  12x  1

21. y  12x

22. y  2x

23.  y   x

24.  y   x

25.  x    y 

26. y  x

10

10

10

x

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30. (A) (6, ?) (D) (?, 2)

(B) (6, ?) (E) (?, 1)

(C) Complete the following table for y  3x2  x3 and sketch a graph of the equation.

(C) (0, ?) (F) (?, 0)

y

x

1

0

1

2

10

10

(D) Write a short statement explaining any conclusions you might draw from parts (A), (B), and (C).

x

The figures in Problems 33 and 34 show a portion of a graph. Extend the given graph to one that exhibits the indicated type of symmetry.

10

31. (A) Sketch a graph based on the solutions in the following table. x

2

0

2

y

2

0

2

y

33. (A) x axis only (B) y axis only (C) origin only (D) x axis, y axis, and origin

5

5

(B) Sketch a graph based on the solutions in the following table. x

1

0

1

y

2

0

2

1

1

5

5

2

5

5

y (D) Write a short statement explaining any conclusions you might draw from parts (A), (B), and (C). 32. (A) Sketch a graph based on the solutions in the following table. x

1

1

3

y

4

2

0

(B) Sketch a graph based on the solutions in the following table.

x

y

34. (A) x axis only (B) y axis only (C) origin only (D) x axis, y axis, and origin

0

5

5

(C) Complete the following table for y  x3  3x and sketch a graph of the equation. 2

3

y

10

x

177

Cartesian Coordinate Systems

Test each equation in Problems 35–40 for symmetry with respect to the x axis, the y axis, and the origin. Do not sketch the graph. 35. x2y2  4

36. x3y3  8

37. x2  xy2  3

38. x  4xy  2y2  7

39. x2  4y  y2  9

40. x2  2xy  3y2  6

Test each equation in Problems 41–52 for symmetry with respect to the x axis, the y axis, and the origin. Sketch the graph of the equation. 41. y2  x  2

42. y2  x  2

x

0

1

2

43. y  x2  1

44. y  2  x2

y

0

2

4

45. 4y2  x2  1

46. 4x2  y2  1

x

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47. y3  x

48. y  x4

49. y  0.6x2  4.5

50. x  0.8y2  3.5

51. y  x23

52. y23  x

53. (A) Graph the triangle with vertices A  (1, 1), B  (7, 2), and C  (4, 6). (B) Now graph the triangle with vertices A  (1, 1), B  (7, 2), and C  (4, 6) in the same coordinate system. (C) How are these two triangles related? How would you describe the effect of changing the sign of the y coordinate of all the points on a graph? 54. (A) Graph the triangle with vertices A  (1, 1), B  (7, 2), and C  (4, 6). (B) Now graph the triangle with vertices A  (1, 1), B  (7, 2), and C  (4, 6) in the same coordinate system. (C) How are these two triangles related? How would you describe the effect of changing the sign of the x coordinate of all the points on a graph? 55. (A) Graph the triangle with vertices A  (1, 1), B  (7, 2), and C  (4, 6). (B) Now graph the triangle with vertices A  (1, 1), B  (7, 2), and C  (4, 6) in the same coordinate system. (C) How are these two triangles related? How would you describe the effect of changing the signs of the x and y coordinates of all the points on a graph? 56. (A) Graph the triangle with vertices A  (1, 2), B  (1, 4), and C  (3, 4). (B) Now graph y  x and the triangle obtained by reversing the coordinates for each vertex of the original triangle: A  (2, 1), B  (4, 1), B  (4, 3). (C) How are these two triangles related? How would you describe the effect of reversing the coordinates of each point on a graph?

71. If a graph is symmetric with respect to the x axis and to the origin, must it be symmetric with respect to the y axis? Explain your answer. 72. If a graph is symmetric with respect to the y axis and to the origin, must it be symmetric with respect to the x axis? Explain your answer.

APPLICATIONS 73. BUSINESS After extensive surveys, the marketing research department of a producer of popular cassette tapes developed the demand equation

where n is the number of units (in thousands) retailers are willing to buy per day at $p per tape. The company’s daily revenue R (in thousands of dollars) is given by R  np  (10  p)p

74. BUSINESS Repeat Problem 73 for the demand equation n8p

59. x 2  ( y  1)2  4

60. ( y  2)2  x 2  9

Test each equation in Problems 61–70 for symmetry with respect to the x axis, the y axis, and the origin. Sketch the graph of the equation. 61. y3   x 

62.  y   x3

64. xy  1

65. y  6x  x

63. xy  1 2

66. y  x  6x

67. y   x   1

68. y  4 x   1

69.  xy   2 y   6

70.  xy    y   4

2

2

4p8

75. PRICE AND DEMAND The quantity of a product that consumers are willing to buy during some period of time depends on its price. The price p and corresponding weekly demand q for a particular brand of diet soda in a city are shown in the figure. Use this graph to estimate the following demands to the nearest 100 cases. (A) What is the demand when the price is $6.00 per case? (B) Does the demand increase or decrease if the price is increased from $6.00 to $6.30 per case? By how much? (C) Does the demand increase or decrease if the price is decreased from $6.00 to $5.70? By how much? (D) Write a brief description of the relationship between price and demand illustrated by this graph. p $7

Price per case

58. x3  y2  8

5  p  10

Graph the revenue equation for the indicated values of p.

In Problems 57–60, solve for y, producing two equations, and then graph both of these equations in the same viewing window. 57. 2x  y2  3

5  p  10

n  10  p

$6

$5 2,000

q 3,000

4,000

Number of cases

2

76. PRICE AND SUPPLY The quantity of a product that suppliers are willing to sell during some period of time depends on

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its price. The price p and corresponding weekly supply q for a particular brand of diet soda in a city are shown in the figure. Use this graph to estimate the following supplies to the nearest 100 cases. (A) What is the supply when the price is $5.60 per case? (B) Does the supply increase or decrease if the price is increased from $5.60 to $5.80 per case? By how much? (C) Does the supply increase or decrease if the price is decreased from $5.60 to $5.40 per case? By how much? (D) Write a brief description of the relationship between price and supply illustrated by this graph.

Cartesian Coordinate Systems

179

(A) The temperature at 7:00 P.M. (B) The lowest temperature and the time when it occurs. (C) The time(s) when the temperature is 52°F. 79. PHYSICS The speed (in meters per second) of a ball swinging at the end of a pendulum is given by v  0.5 12  x where x is the vertical displacement (in centimeters) of the ball from its position at rest (see the figure).

p

Price per case

$7

$6

$5 2,000

q 3,000

x

4,000

Number of cases

77. TEMPERATURE The temperature during a spring day in the Midwest is given in the figure. Use this graph to estimate the following temperatures to the nearest degree and times to the nearest hour. (A) The temperature at 9:00 A.M. (B) The highest temperature and the time when it occurs. (C) The time(s) when the temperature is 49°F.

(A) Graph v for 0  x  2. (B) Describe the relationship between this graph and the physical behavior of the ball as it swings back and forth. 80. PHYSICS The speed (in meters per second) of a ball oscillating at the end of a spring is given by v  4 225  x2 where x is the vertical displacement (in centimeters) of the ball from its position at rest (positive displacement measured downwards—see the figure).

70

60

50

x0 x 0

40 Midnight

6 AM

Noon

6 PM

Midnight

78. TEMPERATURE Use the graph in Problem 77 to estimate the following temperatures to the nearest degree and times to the nearest half hour.

(A) Graph v for 5  x  5. (B) Describe the relationship between this graph and the physical behavior of the ball as it oscillates up and down.

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GRAPHS

2-2

Distance in the Plane Z Distance Between Two Points Z Midpoint of a Line Segment Z Circles

Two basic problems studied in analytic geometry are 1. Given an equation, find its graph. 2. Given a figure (line, circle, parabola, ellipse, etc.) in a coordinate system, find its equation. The first problem was discussed in Section 2-1. In this section, we introduce some tools that are useful when studying the second problem.

Z Distance Between Two Points Given two points P1 and P2 in a rectangular coordinate system, we denote the distance between P1 and P2 by d(P1, P2). We begin with an example.

Distance Between Two Points Find the distance between the points P1  (1, 2) and P2  (4, 6). SOLUTION

Connecting the points P1, P2, and P3  (4, 2) with straight line segments forms a right triangle (Fig. 1). y

P

5

2)

P2  (4, 6) 1,

1

P1  (1, 2)

d( P

EXAMPLE

6  2  4 P3  (4, 2)

4  1  3 5

10

x

Z Figure 1

From the figure, we see that lengths of the legs of the triangle are d(P1, P3)   4  1   3

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181

and d(P3, P2)   6  2   4 The length of the hypotenuse is d(P1, P2), the distance we are seeking. Applying the Pythagorean theorem (see Appendix B), we have [d(P1, P2)] 2    

[ d(P1, P3)] 2  [ d(P3, P2)] 2 32  42 9  16 25

Thus, d(P1, P2)  125  5

MATCHED PROBLEM



1

Find the distance between the points P1  (1, 2) and P2  (13, 7).



The ideas used in Example 1 can be generalized to any two distinct points in the plane. If P1  (x1, y1) and P2  (x2, y2) are two points in a rectangular coordinate system (Fig. 2), then [d(P1, P2)] 2   x2  x1 2   y2  y1 2  (x2  x1)2  ( y2  y1)2

Because N2  N2.

y

Z Figure 2 Distance between two points.

P1  (x1, y1)

) , P2

y2  y1

d(P

P2  (x2, y2) y2

1

x2  x1 x1

x

y1 (x2, y1) x2

Thus, Z THEOREM 1 Distance Formula The distance between P1  (x1, y1) and P2  (x2, y2) is d(P1, P2)  2(x2  x1)2  ( y2  y1)2

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GRAPHS

2

Using the Distance Formula Find the distance between the points (3, 5) and (2, 8).* SOLUTION

Let (x1, y1)  (3, 5) and (x2, y2)  (2, 8). Then, d  2[(2)  (3)] 2  [(8)  5] 2  2(2  3)2  (8  5)2  212  (13)2  21  169  2170 Notice that if we choose (x1, y1)  (2, 8) and (x2, y2)  (3, 5), then d  2 [(3)  (2)] 2  [5  (8)] 2  21  169  2170 so it doesn’t matter which point we designate as P1 or P2.

MATCHED PROBLEM



2

Find the distance between the points (6, 3) and (7, 5).



Z Midpoint of a Line Segment

ZZZ EXPLORE-DISCUSS

1

(A) Graph the line segment L joining the points (1, 2) and (7, 10). (B) Find the average a of the x coordinates of these two points. (C) Find the average b of the y coordinates of these two points. (D) Plot the point (a, b). Is it on the line segment L? (E) Find the distance between (1, 2) and (a, b) and the distance between (a, b) and (7, 10). How would you describe the point (a, b)?

The midpoint of the line segment between two points is the point on the line segment that is equidistant from each of the points. A formula for finding the midpoint is given in Theorem 2. The proof is discussed in the exercises. *We often speak of the point (a, b) when we are referring to the point with coordinates (a, b). This shorthand, though not accurate, causes little trouble, and we will continue the practice.

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183

Z THEOREM 2 Midpoint Formula The midpoint of the line segment joining P1  (x1, y1) and P2  (x2, y2) is Ma

x1  x2 y1  y2 , b 2 2

The point M is the unique point satisfying 1 d(P1, M )  d(M, P2)  d(P1, P2) 2

Note that the coordinates of the midpoint are simply the averages of the respective coordinates of the two given points.

EXAMPLE

3

Using the Midpoint Formula Find the midpoint M of the line segment joining A  (3, 2) and B  (4, 5). Plot A, B, and M and verify that d(A, M )  d(M, B)  12d(A, B). SOLUTION

We use the midpoint formula with (x1, y1)  (3, 2) and (x2, y2)  (4, 5) to obtain the coordinates of the midpoint M. Ma

x1  x2 y1  y2 , b 2 2

3  4 2  (5) , b 2 2 1 3 a , b 2 2  (0.5, 1.5)

a

Substitute x1  3, y1  2, x2  4, and y2  5.

Simplify.

The fraction form of M is probably more convenient for plotting the point. The decimal form is more convenient for computing distances. We plot the three points (Fig. 3) and compute the distances d(A, M ), D(M, B), and d(A, B). y 5

A  (3, 2) 5

1

5

Z Figure 3

5

x

M  2 ,  2 

B  (4, 5)

3

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Find the distances d(A, M ), d(M, B), and 12 d(A, B): d(A, M )  2(3  0.5)2  [2  (1.5)] 2  212.25  12.25  224.5 d(M, B)  2(0.5  4)2  [1.5  (5)] 2  212.25  12.25  224.5 d(A, B)  2(3  4)2  [2  (5)] 2  249  49  298 1 1 98 d(A, B)  198   124.5  d(A, M )  d(M, B) 2 2 A 4 This verifies that M is the midpoint of the line segment joining A and B.

MATCHED PROBLEM



3

Find the midpoint M of the line segment joining A  (4, 1) and B  (3, 5). Plot A, B, and M and verify that d(A, M )  d(M, B)  12 d(A, B). 

EXAMPLE

4

Using the Midpoint Formula If M  (1, 1) is the midpoint of the line segment joining A  (3, 1) and B  (x, y), find the coordinates of B. SOLUTION

From the midpoint formula, we have M  (1, 1)  a

3  x 1  y , b 2 2

We equate the corresponding coordinates and solve the resulting equations for x and y: 1  y 2 2  1  y

3  x 2 2  3  x

1

1

2  3  3  x  3 5x Thus, B  (5, 3)

MATCHED PROBLEM

*

2  1  1  y  1 3y 

4

If M  (1, 1) is the midpoint of the line segment joining A  (1, 5) and B  (x, y), find the coordinates of B.  *Throughout the book, dashed boxes—called think boxes—are used to represent steps that may be performed mentally.

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185

Z Circles The distance-between-two-points formula would be helpful if its only use were to find actual distances between points, such as in Example 2. However, its more important use is in finding equations of figures in a rectangular coordinate system. We start with an example.

EXAMPLE

5

Equations and Graphs of Circles Write an equation for the set of all points that are 5 units from the origin. Graph your equation. SOLUTION

The distance between a point (x, y) and the origin is d  2(x  0)2  ( y  0)2  2x2  y2 So, an equation for the set of points that are 5 units from the origin is 2x2  y2  5 We square both sides of this equation to obtain an equation that does not contain any radicals. x2  y2  25 Since (x)2  x2 and (y)2  y2, the graph will be symmetric with respect to the x axis, y axis, and origin. We make up a table of solutions, sketch the curve in first quadrant, and use symmetry properties to produce a familiar geometric object—a circle (Fig. 4). x

y

0

5

3

4

4

3

5

0

y (0, 5) (3, 4) (4, 3)

(3, 4) (4, 3) (5, 0) x

(5, 0)

(4, 3) (3, 4)

(4, 3) (3, 4) (0, 5)



Z Figure 4

MATCHED PROBLEM

5

Write an equation for the set of all points that are three units from the origin. Graph your equation. 

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Technology Connections Refer to Example 5. To graph this circle on a graphing calculator, first we solve x 2  y 2  25 for y:

x2  y2  25

The graph in Figure 5(a) doesn’t look like a circle. (A circle is as wide as it is tall.) This distortion is caused by the difference between axes scales. One unit on the x axis appears to be longer than one unit on the y axis. Most graphing calculators have an option called ZSquare under the zoom menu [Fig. 6(a)] that automatically adjusts the x axis scale [Fig. 6(b)] to produce a squared viewing window. The graph of a circle in a squared viewing window is not distorted [Fig. 6(c)].

y2  25  x2 y   225  x2 Next we enter y  225  x2 and y  225  x2 in the equation editor of a graphing calculator [Fig. 5(a)], enter appropriate window variables [Fig. 5(b)], and graph [Fig. 5(c)]. 5

5

5

5

(a)

(b)

(c)

Z Figure 5 5

7.6

7.6

5

(a)

(b)

(c)

Z Figure 6

In Example 5, we began with a verbal description of a set of points, produced an algebraic equation that these points must satisfy, constructed a numerical table listing some of these points, and then drew a graphical representation of this set of points. The interplay between verbal, algebraic, numerical, and graphical concepts is one of the central themes of this book.

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Now we generalize the ideas introduced in Example 5.

Z DEFINITION 1 Circle A circle is the set of all points in a plane equidistant from a fixed point. The fixed distance is called the radius, and the fixed point is called the center.

Let’s find the equation of a circle with radius r (r 0) and center C at (h, k) in a rectangular coordinate system (Fig. 7). The circle consists of all points P  (x, y) satisfying d(P, C )  r; that is, all points satisfying

y

r

P  (x, y)

2(x  h)2  ( y  k)2  r

C  (h, k)

r 7 0

x

or, equivalently, (x  h)2  ( y  k)2  r 2

Z Figure 7

r 7 0

Circle.

Z THEOREM 3 Standard Form of the Equation of a Circle Circle with radius r and center at (h, k): (x  h)2  ( y  k)2  r2

EXAMPLE

6

Equations and Graphs of Circles Find the equation of a circle with radius 4 and center at C  (3, 6). Graph the equation.

y (3, 10)

r 7 0

10

SOLUTION C  (3, 6)

(7, 6)

r4

(1, 6) 5

(3, 2) 5

(x  3)2  (y  6)2  16

Z Figure 8

x

C  (h, k)  (3, 6) and r  4 (x  h)2  ( y  k)2  r2 [x  (3)] 2  ( y  6)2  42 (x  3)2  ( y  6)2  16 To graph the equation, plot the center and a few points on the circle (the easiest points to plot are those located 4 units from the center in either in the horizontal or vertical direction), then draw a circle of radius 4 (Fig. 8). 

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MATCHED PROBLEM

6

Find the equation of a circle with radius 3 and center at C  (3, 2). Graph the equation. 

EXAMPLE

7

Finding the Center and Radius of a Circle Find the center and radius of the circle with equation x2  y2  6x  4y  23. SOLUTION

We transform the equation into the form (x  h)2  (y  k)2  r2 by completing the square relative to x and relative to y (see Section 1-5). From this standard form we can determine the center and radius. x2  y2  6x  4y  23 (x2  6x )  ( y2  4y )  23 2 2 (x  6x  9)  ( y  4y  4)  23  9  4 (x  3)2  ( y  2)2  36 [x  (3)] 2  ( y  2)2  62 Center: C(h, k)  C(3, 2) Radius: r  136  6

MATCHED PROBLEM

Group together the terms involving x and those involving y. Complete the squares. Factor each trinomial. Write 3 as (3) to identify h.



7

Find the center and radius of the circle with equation x2  y2  8x  10y  25. 

ANSWERS

TO MATCHED PROBLEMS

1. 13 2. 1173 3. M  (12, 2)  (0.5, 2); d(A, B)  185; d(A, M )  121.25  d(M, B)  12 d(A, B) y 5

A  (4, 1) 5

5

5

B  (3, 5)

4. B  (3, 3)

x

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5. x2 + y2  9

Distance in the Plane

189

6. (x  3)2  ( y  2)2  9 y

y 5

(0, 3) (3, 0)

(3, 1) (3, 0)

5

5

x

(0, 2)

(0, 3)

5

5

C (3, 2)

x

5

(6, 2)

(3, 5)

7. (x  4)2  ( y  5)2  16; radius: 4, center: (4, 5)

2-2

Exercises

In Problems 1–8, find the distance between each pair of points and the midpoint of the line segment joining the points. Leave distance in radical form. 1. (1, 0), (4, 4)

2. (0, 1), (3, 5)

3. (0, 2), (5, 10)

4. (3, 0), (2, 3)

5. (6, 4), (3, 4)

6. (5, 4), (6, 1)

7. (6, 3), (2, 1)

8. (5, 2), (1, 2)

23. Let M be the midpoint of A and B, where A  (a1, a2), B  (1, 3), and M  (2, 6). (A) Use the fact that 2 is the average of a1 and 1 to find a1. (B) Use the fact that 6 is the average of a2 and 3 to find a2. (C) Find d(A, M ) and d(M, B). 24. Let M be the midpoint of A and B, where A  (3, 5), B  (b1, b2), and M  (4, 2). (A) Use the fact that 4 is the average of 3 and b1 to find b1. (B) Use the fact that 2 is the average of 5 and b2 to find b2. (C) Find d(A, M ) and d(M, B).

In Problems 9–16, write the equation of a circle with the indicated center and radius. 9. C  (0, 0), r  7

10. C  (0, 0), r  5

25. Find x such that (x, 7) is 10 units from (4, 1).

11. C  (2, 3), r  6

12. C  (5, 6), r  2

26. Find x such that (x, 2) is 4 units from (3, 3).

13. C  (4, 1), r  17

14. C  (5, 6), r  111

27. Find y such that (2, y) is 3 units from (1, 4).

15. C  (3, 4), r  12

16. C  (4, 1), r  15

28. Find y such that (3, y) is 13 units from (9, 2).

In Problems 17–22, write an equation for the given set of points. Graph your equation.

In Problems 29–32, write a verbal description of the graph and then write an equation that would produce the graph.

17. The set of all points that are two units from the origin.

29.

y 5

18. The set of all points that are four units from the origin. 19. The set of all points that are one unit from (1, 0). 20. The set of all points that are one unit from (0, 1).

5

5

21. The set of all points that are three units from (2, 1). 22. The set of all points that are two units from (3, 2).

5

x

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y

30.

43. x2  6x  y2  16 44. x2  y2  8y  9

5

45. x2  y2  6x  4y  36 5

5

x

46. x2  y2  2x  10y  55 47. 3x2  3y2  24x  18y  24  0 48. 2x2  2y2  8x  20y  30  0

5

In Problems 49–52, solve for y, producing two equations, and then graph both of these equations in the same viewing window.

y

31. 5

49. x2  y2  3 5

5

x

50. x2  y2  5 51. (x  3)2  (y  1)2  2 52. (x  2)2  (y  1)2  3

5

In Problems 53 and 54, show that the given points are the vertices of a right triangle (see the Pythagorean theorem in Appendix B). Find the length of the line segment from the midpoint of the hypotenuse to the opposite vertex.

y

32. 5

53. (3, 2), (1, 2), (8, 5) 5

5

x

5

54. (1, 3), (3, 5), (5, 1) Find the perimeter (to two decimal places) of the triangle with the vertices indicated in Problems 55 and 56. 55. (3, 1), (1, 2), (4, 3)

In Problems 33–38, M is the midpoint of A and B. Find the indicated point. Verify that d(A, M)  d(M, B)  12d(A, B). 33. A  (4.3, 5.2), B  (9.6, 1.7), M  ? 34. A  (2.8, 3.5), B  (4.1, 7.6), M  ? 35. A  (25, 10), M  (5, 2), B  ? 36. M  (2.5, 3.5), B  (12, 10), A  ? 37. M  (8, 6), B  (2, 4), A  ? 38. A  (4, 2), M  (1.5, 4.5), B  ? In Problems 39–48, find the center and radius of the circle with the given equation. Graph the equation. 39. x2  ( y  2)2  9

56. (2, 4), (3, 1), (3, 2) x1  x2 y1  y2 , b, 2 2 1 show that d(P1, M )  d(M, P2)  2d(P1, P2). (This is one step in the proof of Theorem 2).

57. If P1  (x1, y1), P2  (x2, y2) and M  a

58. A parallelogram ABCD is shown in the figure. (A) Find the midpoint of the line segment joining A and C. (B) Find the midpoint of the line segment joining B and D. (C) What can you conclude about the diagonals of the parallelogram? y B  (a, b)

C  (a  c, b)

40. (x  5)2  y2  16 41. (x  4)2  ( y  2)2  7 42. (x  5)2  ( y  7)2  15

A  (0, 0)

x D  (c, 0)

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Distance in the Plane

Find the equation of a circle that has a diameter with the end points given in Problems 59–62.

191

y (2, r  1)

59. (0, 5), (0, 3)

r

60. (2, 0), (8, 0)

x

61. (7, 3), (1, 7) 62. (3, 2), (7, 4) 4 feet

In Problems 63–66, find the equation of circle with the given center whose graph passes through the given point.

Arched doorway

63. Center: (0, 0), point on the circle: (3, 9) 64. Center: (0, 0), point on the circle: (2, 6)

70. ENGINEERING The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 12 millimeters apart and the top is 4 millimeters above the ends, what is the radius of the circle containing the arc?

65. Center: (2, 2), point on the circle: (3, 5) 66. Center: (5, 4), point on the circle: (2, 3)

APPLICATIONS 67. SPORTS A singles court for lawn tennis is a rectangle 27 feet wide and 78 feet long (see the figure). Points B and F are the midpoints of the end lines of the court. Rivet 18 feet B

C

★71.

A

18 feet D 78 feet

27 feet

E F

CONSTRUCTION Town B is located 36 miles east and 15 miles north of town A (see the figure). A local telephone company wants to position a relay tower so that the distance from the tower to town B is twice the distance from the tower to town A. (A) Show that the tower must lie on a circle, find the center and radius of this circle, and graph. (B) If the company decides to position the tower on this circle at a point directly east of town A, how far from town A should they place the tower? Compute answer to one decimal place.

G

y 25

(A) Sketch a graph of the court with A at the origin of your coordinate system, C on the positive y axis, and G on the positive x axis. Find the coordinates of points A through G. (B) Find d(B, D) and d(F, C ) to the nearest foot.

Town B Tower

(36, 15)

(x, y)

68. SPORTS Refer to Problem 67. Find d(A, D) and d(C, G ) to the nearest foot. 69. ARCHITECTURE An arched doorway is formed by placing a circular arc on top of a rectangle (see the figure). If the doorway is 4 feet wide and the height of the arc above its ends is 1 foot, what is the radius of the circle containing the arc? [Hint: Note that (2, r  1) must satisfy x2  y2  r 2.]

Town A ★72.

x 25

CONSTRUCTION Repeat Problem 71 if the distance from the tower to town A is twice the distance from the tower to town B.

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Equations of a Line Z Graphing Lines Z Finding the Slope of a Line Z Determining Special Forms of the Equation of a Line Z Connecting Slope to Parallel and Perpendicular Lines

In this section, we consider one of the most basic geometric figures—a line. When we use the term line in this book, we mean straight line. We will learn how to recognize and graph a line and how to use information concerning a line to find its equation. Adding these important tools to our mathematical toolbox will enable us to use lines as an effective problem-solving tool, as evidenced by the application exercises at the end of this section.

Z Graphing Lines With your past experience in graphing equations in two variables, you probably remember that first-degree equations in two variables, such as y  3x  5

3x  4y  9

y  23 x

have graphs that are lines. This fact is stated in Theorem 1.

Z THEOREM 1 The Equation of a Line If A, B, and C are constants, with A and B not both 0, and x and y are variables, then the graph of the equation Ax  By  C

Standard Form

(1)

is a line. Any line in a rectangular coordinate system has an equation of this form.

Also, the graph of any equation of the form y  mx  b

(2)

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193

where m and b are constants, is a line. Form (2), which we will discuss in detail later, is simply a special case of form (1) for B  0. This can be seen by solving form (1) for y in terms of x: A C y x B B

B0

To graph either equation (1) or (2), we plot any two points from the solution set and use a straightedge to draw a line through these two points. The points where the line crosses the axes are convenient to use and easy to find. The y intercept* is the y coordinate of the point where the graph crosses the y axis, and the x intercept is the x coordinate of the point where the graph crosses the x axis. To find the y intercept, let x = 0 and solve for y; to find the x intercept, let y = 0 and solve for x. It is often advisable to find a third point as a checkpoint. All three points must lie on the same line or a mistake has been made.

EXAMPLE

1

Using Intercepts to Graph a Line Graph the equation 3x  4y  12. SOLUTION

Find intercepts, a third checkpoint (optional), and draw a line through the two (three) points (Fig. 1). y 5

(8, 3) Checkpoint

(4, 0) 5

5

x

0

4

8

y intercept is 3

y

3

0

3

5

Z Figure 1

MATCHED PROBLEM Graph the equation 4x  3y  12.

10

x

x intercept is 4 (0, 3)



1 

*If the x intercept is a and the y intercept is b, then the graph of the line passes through the points (a, 0) and (0, b). It is common practice to refer to both the numbers a and b and the points (a, 0) and (0, b) as the x and y intercepts of the line.

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Technology Connections To solve Example 1 on a graphing calculator, we first solve the equation for y:

3x  4y  12 4y  3x  12 y  0.75x  3

To find the y intercept of this line, we graph the preceding equation, press TRACE, and then enter 0 for x [Fig. 2(a)]. The displayed y value is the y intercept. The x intercept can be found by using the zero option on the CALC menu. After selecting the zero option, you will be asked to provide three x values: a left bound (a number less than the zero), a right bound (a number greater than the zero), and a guess (a number between the left and right bounds). You can enter the three values from the keypad, but most find it easier to use the cursor. The zero or x intercept is displayed at the bottom of the screen [Fig. 2(b)].

5

5

5

10

5

10

5

5

(a) y intercept

(b) x intercept

Z Figure 2

Z Finding the Slope of a Line If we take two different points P1(x1, y1) and P2(x2, y2) on a line, then the ratio of the change in y to the change in x as we move from point P1 to point P2 is called the slope of the line. Roughly speaking, slope is a measure of the “steepness” of a line. Sometimes the change in x is called the run and the change in y is called the rise. Z DEFINITION 1 Slope of a Line If a line passes through two distinct points P1(x1, y1) and P2(x2, y2), then its slope m is given by the formula y m

y2  y1 x2  x1

P2(x2, y2)

x1  x2

Vertical change (rise)  Horizontal change (run)

y2  y1 Rise x

P1(x1, y1) x2  x1 Run

(x2, y1)

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195

For a horizontal line, y doesn’t change as x changes; hence, its slope is 0. On the other hand, for a vertical line, x doesn’t change as y changes; hence, x1 = x2 and its slope is not defined: y2  y1 y2  y1  x2  x1 0

For a vertical line, slope is not defined.

In general, the slope of a line may be positive, negative, 0, or not defined. Each of these cases is interpreted geometrically as shown in Table 1.

Table 1 Geometric Interpretation of Slope Line

Slope

Example y

Rising as x moves from left to right y values are increasing

Positive

x

y

Falling as x moves from left to right y values are decreasing

Negative

x

y

Horizontal y values are constant

0

x

y

Vertical x values are constant

Not defined

x

In using the formula to find the slope of the line through two points, it doesn’t matter which point is labeled P1 or P2, because changing the labeling will change the sign in both the numerator and denominator of the slope formula: y2  y1 y1  y2  x2  x1 x1  x2 b a b a m

Z Figure 3

b b  a a

For example, the slope of the line through the points (3, 2) and (7, 5) is 52 3 3 25    73 4 4 37 In addition, it is important to note that the definition of slope doesn’t depend on the two points chosen on the line as long as they are distinct. This follows from the fact that the ratios of corresponding sides of similar triangles are equal (Fig. 3).

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EXAMPLE

2

Finding Slopes For each line in Figure 4, find the run, the rise, and the slope. (All the horizontal and vertical line segments have integer lengths.)

y

In Figure 4(a), the run is 3, the rise is 6 and the slope is 63  2. In Figure 4(b), 2 the run is 6, the rise is 4 and the slope is 4  6  3 . SOLUTION

5

5

5

x

MATCHED PROBLEM

2

For each line in Figure 5, find the run, the rise, and the slope. (All the horizontal and vertical line segments have integer lengths.)

5

(a) y

y

5

y

5

5

5

x

5

5

5

5

x

5

5

5

(b)

EXAMPLE

5

(a)

Z Figure 4

x

(b)



Z Figure 5

3

Finding Slopes Sketch a line through each pair of points and find the slope of each line. (A) (3, 4), (3, 2)

(B) (2, 3), (1, 3)

(C) (4, 2), (3, 2)

(D) (2, 4), (2, 3)

SOLUTIONS y

(A)

y

(B)

5

5

(2, 3)

(3, 2) 5

5

x

5

5

(1, 3)

(3, 4) 5

m

x

2  (4) 6  1 3  (3) 6

5

m

3  3 6   2 1  (2) 3

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(C)

197

y

(D) 5

5

(4, 2)

Equations of a Line

(2, 4)

(3, 2)

5

x

5

5

5

x

(2, 3) 5

5

m

22 0  0 3  (4) 7

MATCHED PROBLEM

3  4 7  ; 22 0 slope is not defined m



3

Find the slope of the line through each pair of points. Do not graph. (A) (3, 3), (2, 3)

(B) (2, 1), (1, 2)

(C) (0, 4), (2, 4)

(D) (3, 2), (3, 1)



Z Determining Special Forms of the Equation of a Line We start by investigating why y  mx  b is called the slope–intercept form for a line.

ZZZ EXPLORE-DISCUSS

1

(A) Graph y  x  b for b  5, 3, 0, 3, and 5 simultaneously in the same coordinate system. Verbally describe the geometric significance of b. (B) Graph y  mx  1 for m  2, 1, 0, 1, and 2 simultaneously in the same coordinate system. Verbally describe the geometric significance of m.

f (x)

(0, b)

(1, m  b) x

As you see from the preceding exploration, constants m and b in y = mx  b have special geometric significance, which we now explicitly state. If we let x = 0, then y = b and the graph of y = mx  b crosses the y axis at (0, b). Thus, the constant b is the y intercept. For example, the y intercept of the graph of y = 2x – 7 is 7. We have already seen that the point (0, b) is on the graph of y = mx  b. If we let x = 1, then it follows that the point (1, m  b) is also on the graph (Fig. 6). Because the graph of y = mx  b is a line, we can use these two points to compute the slope: Slope 

Z Figure 6

y2  y1 (m  b)  b  m x2  x1 10

(x1, y1)  (0, b) (x2, y2)  (1, m  b)

Thus, m is the slope of the line with equation y = mx  b.

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Z THEOREM 2 Slope–Intercept Form y  mx  b Rise m  Slope Run b  y intercept

y y  mx  b y intercept b

Rise Run

EXAMPLE

4

y

x

Using the Slope–Intercept Form (A) Write the slope–intercept equation of a line with slope (B) Find the slope and y intercept, and graph y 

5

3 4x

2 3

and y intercept 5.

 1.

SOLUTIONS 3

5

5

x

(A) Substitute m  23 and b  5 in y = mx  b to obtain y  23x  5.

4

(B) The y intercept of y  34 x  1 is 1 and the slope is 34. If we start at the point (0, 1) and move four units to the right (run), then the y coordinate of a point on the line must move up three units (rise) to the point (4, 2). Drawing a line through these two points produces the graph shown in Figure 7. 

5

Z Figure 7

MATCHED PROBLEM

4

Write the slope–intercept equation of the line with slope the equation.

y  y1 m x  x1

(x, y) x

Z Figure 8

and y intercept 2. Graph 

In Example 4 we found the equation of a line with a given slope and y intercept. It is also possible to find the equation of a line passing through a given point with a given slope or to find the equation of a line containing two given points. Suppose a line has slope m and passes through the point (x1, y1). If (x, y) is any other point on the line (Fig. 8), then

y

(x1, y 1)

5 4

(x, y 1)

that is, y  y1  m(x  x1)

(3)

Because the point (x1, y1) also satisfies equation (3), we can conclude that equation (3) is an equation of a line with slope m that passes through (x1, y1).

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199

Z THEOREM 3 Point–Slope Form An equation of a line with slope m that passes through (x1, y1) is y  y1  m(x  x1) which is called the point–slope form of an equation of a line.

If we are given the coordinates of two points on a line, we can use the given coordinates to find the slope and then use the point–slope form with either of the given points to find the equation of the line.

EXAMPLE

5

Point–Slope Form (A) Find an equation for the line that has slope 23 and passes through the point (2, 1). Write the final answer in the form Ax  By  C. (B) Find an equation for the line that passes through the two points (4, 1) and (8, 5). Write the final answer in the form y  mx  b. SOLUTIONS

(A) If m  23 and (x1, y1)  (2, 1), then y  y1  m(x  x1) 2 y  1  [x  (2)] 3 3( y  1)  2(x  2) 3y  3  2x  4 or 2x  3y  7

Substitute y1  1, x1  2, and m  23 . Multiply both sides by 3.

Remove parentheses. Write in standard form.

2x  3y  7

(B) First use the slope formula to find the slope of the line: m

y2  y1 5  (1) 6 1    x2  x1 8  4 12 2

Substitute x1  4, y1  1, x2  8, and y2  5 in the slope formula.

Now we choose (x1, y1)  (4, 1) and proceed as in part A: y  y1  m(x  x1) 1 y  (1)   (x  4) 2 1 y1 x2 2 1 y x1 2

1 Substitute x1  4, y1  1, and m   . 2 Remove parentheses.

Subtract 1 from both sides.

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Verify that choosing (x1, y1) = (8, 5), the other given point, produces the same equation. 

MATCHED PROBLEM

5

(A) Find an equation for the line that has slope 25 and passes through the point (3, 2). Write the final answer in the form Ax  By  C. (B) Find an equation for the line that passes through the two points (3, 1) and (7, 3). Write the final answer in the form y  mx  b.



The simplest equations of lines are those for horizontal and vertical lines. Consider the following two equations: x  0y  a 0x  y  b

or or

xa yb

(4) (5)

In equation (4), y can be any number as long as x  a. Thus, the graph of x  a is a vertical line crossing the x axis at (a, 0). In equation (5), x can be any number as long as y  b. Thus, the graph of y  b is a horizontal line crossing the y axis at (0, b). We summarize these results as follows:

Z THEOREM 4 Vertical and Horizontal Lines Equation xa (short for x  0y  a) yb

(short for 0x  y  b)

y

Graph Vertical line through (a, 0) (Slope is undefined.) Horizontal line through (0, b) (Slope is 0.) xa

yb

b a

x

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EXAMPLE

6

Equations of a Line

201

Graphing Horizontal and Vertical Lines Graph the line x  2 and the line y  3. SOLUTION y 5

5

y3

5

x

x  2 5

MATCHED PROBLEM



6

Graph the line x  4 and the line y  2.



The various forms of the equation of a line that we have discussed are summarized in Table 2 for convenient reference. Table 2 Equations of a Line Standard form

Ax  By  C

A and B not both 0

Slope–intercept form

y  mx  b

Slope: m; y intercept: b

Point–slope form

y  y1  m(x  x1)

Slope: m; Point: (x1, y1)

Horizontal line

yb

Slope: 0

Vertical line

xa

Slope: Undefined

ZZZ EXPLORE-DISCUSS

2

Determine conditions on A, B, and C so that the linear equation Ax  By = C can be written in each of the following forms, and discuss the possible number of x and y intercepts in each case. 1. y  mx  b, m  0 2. y  b 3. x  a

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Z Connecting Slope to Parallel and Perpendicular Lines From geometry, we know that two vertical lines are parallel to each other and that a horizontal line and a vertical line are perpendicular to each other. How can we tell when two nonvertical lines are parallel or perpendicular to each other? Theorem 5, which we state without proof, provides a convenient test.

Z THEOREM 5 Parallel and Perpendicular Lines Given two nonvertical lines L1 and L2 with slopes m1 and m2, respectively, then L1  L2 L1  L2

if and only if if and only if

m1  m2 m1m2  1

The symbols  and  mean, respectively, “is parallel to” and “is perpendicular to.” In the case of perpendicularity, the condition m1m2 = 1 also can be written as m2  

1 m1

or

m1  

1 m2

Thus, Two nonvertical lines are perpendicular if and only if their slopes are the negative reciprocals of each other.

EXAMPLE

7

Parallel and Perpendicular Lines Given the line: L: 3x  2y = 5 and the point P(3, 5), find an equation of a line through P that is (A) Parallel to L

(B) Perpendicular to L

Write the final answers in the slope–intercept form y = mx  b. SOLUTIONS

First, find the slope of L by writing 3x  2y = 5 in the equivalent slope–intercept form y = mx  b: 3x  2y  5 2y  3x  5 y  32 x  52 Thus, the slope of L is 32. The slope of a line parallel to L is the same, 32, and the slope of a line perpendicular to L is 23. We now can find the equations of the two lines in parts A and B using the point–slope form.

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(A) Parallel (m  32):

Equations of a Line

203

(B) Perpendicular (m  23):

y  y1  m(x  x1) y  5  32 (x  3) y  5  32 x  92 y  32 x  192

y  y1  m(x  x1) y  5  23 (x  3) y  5  23x  2 y  23x  3

MATCHED PROBLEM

Substitute for x1, y1, and m. Remove parentheses. Add 5 to both sides.



7

Given the Line L: 4x  2y = 3 and the point P(2, 3), find an equation of a line through P that is (A) Parallel to L

(B) Perpendicular to L

Write the final answers in the slope–intercept form y = mx  b.

EXAMPLE

8

Cost Analysis A hot dog vendor pays $25 per day to rent a pushcart and $1.25 for the ingredients in one hot dog. (A) Find the cost of selling x hot dogs in 1 day. (B) What is the cost of selling 200 hot dogs in 1 day? (C) If the daily cost is $355, how many hot dogs were sold that day? SOLUTIONS

(A) The rental charge of $25 is the vendor’s fixed cost—a cost that is accrued every day and does not depend on the number of hot dogs sold. The cost of ingredients does depend on the number sold. The cost of the ingredients for x hot dogs is $1.25x. This is the vendor’s variable cost—a cost that depends on the number of hot dogs sold. The total cost for selling x hot dogs is C(x)  1.25x  25

Total Cost  Variable Cost  Fixed Cost

(B) The cost of selling 200 hot dogs in 1 day is C(200)  1.25(200)  25  $275



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(C) The number of hot dogs that can be sold for $355 is the solution of the equation 1.25x  25  355 1.25x  330 330 x 1.25  264 hot dogs

MATCHED PROBLEM

Subtract 25 from each side. Divide both sides by 1.25. Simplify.



8

It costs a pretzel vendor $20 per day to rent a cart and $0.75 for each pretzel. (A) Find the cost of selling x pretzels in 1 day. (B) What is the cost of selling 150 pretzels in 1 day? 

(C) If the daily cost is $275, how many pretzels were sold that day?

Technology Connections A graphing calculator can be used to solve equations like 1.25x  25  355 (see Example 8). First enter both sides of the equation in the equation editor [Fig. 9(a)] and choose window variables [Fig. 9(b)] so that the graphs of both equations appear on the screen. There is no “right” choice for

the window variables. Any choice that displays the intersection point will do. (Here is how we chose our window variables: We chose Ymax  600 to place the graph of the horizontal line below the top of the window. We chose Ymin  200 to place the graph of the x axis above the text displayed at the bottom of the screen. Since x cannot be negative, we chose Xmin  0. We used trial and error to determine a reasonable choice for Xmax.) Now choose intersect on the CALC menu, and respond to the prompts from the calculator. The coordinates of the intersection point of the two graphs are shown at the bottom of the screen [Fig. 9(c)]. 600

0

400

200

(a)

Z Figure 9

(b)

(c)

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ANSWERS

Equations of a Line

TO MATCHED PROBLEMS 2. (A) Run  5, rise  4, slope  45  0.8 (B) Run  3, rise  6, slope  6 3  2 3. (A) m  0 (B) m  1 (C) m  4 (D) m is not defined

y

1.

5

5

x

5

5. (A) 2x  5y  4

4. y  54 x  2 y

y

6. 5

5

5 5

5

x

x4

5

5

y  2

4

5

5

7. (A) y  2x  1 (B) y  12 x  4 8. (A) C(x)  0.75x  20 (B) $132.50

2-3

(B) y  25 x  15

(C) 340 pretzels

Exercises

In Problems 1–6, use the graph of each line to find the rise, run, and slope. Write the equation of each line in the standard form Ax  By  C, A  0. (All the horizontal and vertical line segments have integer lengths.)

y

2. 5

y

1.

5

5

5

5

5

x

5

5

x

x

205

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y

3.

y

8. 5

5

5

5

x

5

5

x

5

5

y

4.

y

9.

6 5

6

6

x

5

5

x

5 6

y

5.

y

10. 5

5

5

5

x

5

5

x

5

5

y

6.

y

11.

5

5

5

5

x

5

5

x

5 5

y

12. In Problems 7–12, use the graph of each line to find the x intercept, y intercept, and slope, if they exist. Write the equation of each line, using the slope–intercept form whenever possible.

5

y

7.

5

5

5

5

5

x

5

5

x

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Graph each equation in Problems 13–28, and indicate the slope, if it exists. 13. y  35 x  4

14. y  32 x  6

15. y  34 x

16. y  23 x  3

17. 4x  2y  0

18. 6x  2y  0

19. 2x  3y  15

20. 4x  3y  24

21. 4x  5y  24

22. 6x  7y  49

23.

y x  1 8 4

24.

Equations of a Line

51. (1, 6); (5, 2)

52. (3, 4); (6, 1)

53. (4, 8); (2, 0)

54. (2, 1); (10, 5)

55. (3, 4); (5, 4)

56. (0, 2); (4, 2)

57. (4, 6); (4, 3)

58. (3, 1); (3, 4)

59. x intercept 4; y intercept 3

60. x intercept 4; y intercept 5

In Problems 61–72, write an equation of the line that contains the indicated point and meets the indicated condition(s). Write the final answer in the standard form Ax  By  C, A  0.

y x  1 6 5

25. x  3

26. y  2

61. (3, 4); parallel to y  3x  5

27. y  3.5

28. x  2.5

62. (4, 0); parallel to y  2x  1

In Problems 29–34, find the equation of the line with the indicated slope and y intercept. Write the final answer in the standard form Ax  By  C, A  0. 29. Slope  1; y intercept  0

69. (5, 0); parallel to 3x  2y  4

33. Slope  0; y intercept  2

70. (3, 5); parallel to 3x  4y  8

34. Slope undefined; x intercept  3

71. (0, 4); perpendicular to x  3y  9

In Problems 35–40, find the equation of the line passing through the given point with the given slope. Write the final answer in the slope–intercept form y  mx  b. Use a graphing calculator to find the intercepts of the line. 36. (4, 0); m  3 38. (2, 3); m 

39. (2, 3); m 

12

45

40. (2, 1); m  43

In Problems 41–46, find the equation of the line passing through the two given points. Write the final answer in the slope–intercept form y  mx  b or in the form x  c. 41. (2, 5); (4, 3)

42. (1, 4); (3, 2)

43. (3, 2); (3, 5)

44. (0, 5); (2, 5)

45. (4, 2); (0, 2)

46. (5, 4); (5, 6)

49. (5, 4); m 

25

73. Discuss the relationship between the graphs of the lines with equation y  mx  2, where m is any real number. 74. Discuss the relationship between the graphs of the lines with equation y  0.5x  b, where b is any real number. Problems 75–80 refer to the quadrilateral with vertices A(0, 2), B(4, 1), C(1, 5), and D(3, 2). 75. Show that AB  DC.

76. Show that DA  CB.

77. Show that AB  BC.

78. Show that AD  DC.

80. Find an equation of the perpendicular bisector of AB.

48. (2, 0); m  2 50. (4, 2); m 

72. (2, 4); perpendicular to 4x  5y  0

79. Find an equation of the perpendicular bisector* of AD.

In Problem 47–60, sketch a graph of the line that contains the indicated point(s) and/or has the indicated slope and/or has the indicated intercepts. Then write the equation of the line in the slope–intercept form y  mx  b or in the form x  c. 47. (0, 4); m  3

65. (2, 5); parallel to y axis

68. (2, 3); horizontal

32. Slope  53; y intercept  6

37. (5, 4); m 

64. (2, 4); perpendicular to y  23 x  5

67. (3, 2); vertical

31. Slope  23; y intercept  4

3 2

63. (2, 3); perpendicular to y  13 x

66. (7, 3); parallel to x axis

30. Slope  1; y intercept  7

35. (0, 3); m  2

207

1 2

81. (A) Graph the following equations in the same coordinate system: 3x  2y 

6

3x  2y  6

3x  2y 

3

3x  2y  3

*The perpendicular bisector of a line segment is a line perpendicular to the segment and passing through its midpoint.

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90. x2  y2  80, (4, 8)

(B) Graph all the equations in part A on a graphing calculator. (C) From your observations in part A, describe the family of lines obtained by varying C in Ax  By  C while holding A and B fixed. (D) Verify your conclusions in part B with a proof. 82. (A) Graph the following two equations in the same coordinate system: 3x  4y  12

4x  3y  12

Graph these equations in a squared viewing window on a graphing calculator. (B) Graph the following two equations in the same coordinate system: 2x  3y  12

3x  2y  12

Graph these equations in a squared viewing window on a graphing calculator. (C) From your observations in parts A and B, describe the apparent relationship of the graphs of Ax  By  C and Bx  Ay  C. (D) Verify your conclusions in part C with a proof. 83. Prove that if a line L has x intercept (a, 0) and y intercept (0, b), then the equation of L can be written in the intercept form y x  1 a b

a, b  0

84. Prove that if a line L passes through P1  (x1, y1) and P2  (x2, y2), then the equation of L can be written in the two-point form

91. (x  3)2  ( y  4)2  169, (8, 16) 92. (x  5)2  ( y  9)2  289, (13, 6)

APPLICATIONS 93. BOILING POINT OF WATER At sea level, water boils when it reaches a temperature of 212°F. At higher altitudes, the atmospheric pressure is lower and so is the temperature at which water boils. The boiling point B in degrees Fahrenheit at an altitude of x feet is given approximately by B  212  0.0018x (A) Complete Table 3.

Table 3 x

0

5,000

10,000

15,000

20,000

25,000

30,000

B (B) Based on the information in the table, write a brief verbal description of the relationship between altitude and the boiling point of water. 94. AIR TEMPERATURE As dry air moves upward, it expands and cools. The air temperature A in degrees Celsius at an altitude of x kilometers is given approximately by A  25  9x (A) Complete Table 4.

( y  y1)(x2  x1)  ( y2  y1)(x  x1)

Table 4 In Problems 85 and 86, write the equation of the line with the indicated intercepts in intercept form (see Problem 83) and then in the standard form Ax  By  C, A  0. 85. (3, 0) and (0, 5)

0

1

2

3

4

5

A

86. (2, 0) and (0, 7)

Problems 87–92 are calculus related. Recall that a line tangent to a circle at a point is perpendicular to the radius drawn to that point (see the figure). Find the equation of the line tangent to the circle at the indicated point. Write the final answer in the standard form Ax  By  C, A  0. Graph the circle and the tangent line on the same coordinate system. 87. x2  y2  25, (3, 4) 88. x2  y2  100, (8, 6) 89. x2  y2  50, (5, 5)

x

(B) Based on the information in the table, write a brief verbal description of the relationship between altitude and air temperature. 95. CAR RENTAL A car rental agency computes daily rental charges for compact cars with the equation c  25  0.25x where c is the daily charge in dollars and x is the daily mileage. Translate this algebraic statement into a verbal statement that can be used to explain the daily charges to a customer. 96. INSTALLATION CHARGES A telephone store computes charges for phone installation with the equation c  15  0.7x

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where c is the installation charge in dollars and x is the time in minutes spent performing the installation. Translate this algebraic statement into a verbal statement that can be used to explain the installation charges to a customer. 97. COST ANALYSIS A doughnut shop has a fixed cost of $124 per day and a variable cost of $0.12 per doughnut. Find the total daily cost of producing x doughnuts. How many doughnuts can be produced for a total daily cost of $250? 98. COST ANALYSIS A small company manufactures picnic tables. The weekly fixed cost is $1,200 and the variable cost is $45 per table. Find the total weekly cost of producing x picnic tables. How many picnic tables can be produced for a total weekly cost of $4,800? 99. PHYSICS Hooke’s law states that the relationship between the stretch s of a spring and the weight w causing the stretch is linear (a principle upon which all spring scales are constructed). For a particular spring, a 5-pound weight causes a stretch of 2 inches, while with no weight the stretch of the spring is 0. (A) Find a linear equation that expresses s in terms of w. (B) What is the stretch for a weight of 20 pounds? (C) What weight will cause a stretch of 3.6 inches? 100. PHYSICS The distance d between a fixed spring and the floor is a linear function of the weight w attached to the bottom of the spring. The bottom of the spring is 18 inches from the floor when the weight is 3 pounds and 10 inches from the floor when the weight is 5 pounds. (A) Find a linear equation that expresses d in terms of w. (B) Find the distance from the bottom of the spring to the floor if no weight is attached. (C) Find the smallest weight that will make the bottom of the spring touch the floor. (Ignore the height of the weight.) 101. PHYSICS The two most widespread temperature scales are Fahrenheit* (F) and Celsius† (C). It is known that water freezes at 32°F or 0°C and boils at 212°F or 100°C. (A) Find a linear equation that expresses F in terms of C. (B) If a European family sets its house thermostat at 20°C, what is the setting in degrees Fahrenheit? If the outside temperature in Milwaukee is 86°F, what is the temperature in degrees Celsius? 102. PHYSICS Two other temperature scales, used primarily by scientists, are Kelvin‡ (K) and Rankine** (R). Water freezes at

*Invented in 1724 by Daniel Gabriel Fahrenheit (1686–1736), a German physicist. † Invented in 1742 by Anders Celsius (1701–1744), a Swedish astronomer. ‡ Invented in 1848 by Lord William Thompson Kelvin (1824–1907), a Scottish mathematician and physicist. Note that the degree symbol “ ° ” is not used with degrees Kelvin. **Invented in 1859 by John Maquorn Rankine (1820–1872), a Scottish engineer and physicist.

Equations of a Line

209

273 K or 492°R and boils at 373 K or 672°R. Find a linear equation that expresses R in terms of K. 103. OCEANOGRAPHY After about 9 hours of a steady wind, the height of waves in the ocean is approximately linearly related to the duration of time the wind has been blowing. During a storm with 50-knot winds, the wave height after 9 hours was found to be 23 feet, and after 24 hours it was 40 feet. (A) If t is time after the 50-knot wind started to blow and h is the wave height in feet, write a linear equation that expresses height h in terms of time t. (B) How long will the wind have been blowing for the waves to be 50 feet high? 104. OCEANOGRAPHY Refer to Problem 103. A steady 25-knot wind produces a wave 7 feet high after 9 hours and 11 feet high after 25 hours. (A) Write a linear equation that expresses height h in terms of time t. (B) How long will the wind have been blowing for the waves to be 20 feet high? Express all calculated quantities to three significant digits. 105. DEMOGRAPHICS Life expectancy in the United States has increased from about 49.2 years in 1900 to about 77.3 years in 2000. The growth in life expectancy is approximately linear with respect to time. (A) If L represents life expectancy and t represents the number of years since 1900, write a linear equation that expresses L in terms of t. (B) What is the predicted life expectancy in the year 2020? Express all calculated quantities to three significant digits.

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106. DEMOGRAPHICS The average number of persons per household in the United States has been shrinking steadily for as long as statistics have been kept and is approximately linear with respect to time. In 1900, there were about 4.76 persons per household and in 1990, about 2.5. (A) If N represents the average number of persons per household and t represents the number of years since 1900, write a linear equation that expresses N in terms of t. (B) What is the predicted household size in the year 2000?

First Street

Distance in feet

Avenue A Avenue B

Avenue C

d2

d1 5,000

0

Express all calculated quantities to three significant digits. ★107.

5,000

Main Street

Distance in feet CITY PLANNING The design of a new subdivision calls for ★108.

CITY PLANNING Refer to Problem 107. Find the distance d2 (to the nearest foot) from Avenue B to Avenue C.

three parallel streets connecting First Street with Main Street (see the figure). Find the distance d1 (to the nearest foot) from Avenue A to Avenue B.

Linear Equations and Models Z Slope as a Rate of Change Z Linear Models Z Linear Regression

Mathematical modeling is the process of using mathematics to solve real-world problems. This process can be broken down into three steps (Fig. 1): Step 1. Construct the mathematical model, a problem whose solution will provide information about the real-world problem. Step 2. Solve the mathematical model. Step 3. Interpret the solution to the mathematical model in terms of the original real-world problem. Real-world problem

uc

In

str

te rp

re

n Co

t

1.

Mathematical solution

t

3.

2-4

2. Solve

Mathematical model

Z Figure 1

In more complex problems, this cycle may have to be repeated several times to obtain the required information about the real-world problem. In this section, we discuss one of the simplest mathematical models, a linear equation. With the aid of a graphing calculator, we also learn how to analyze a linear model based on real-world data.

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211

Z Slope as a Rate of Change If x and y are related by the equation y  mx  b, where m and b are constants with m  0, then x and y are linearly related. If (x1, y1) and (x2, y2) are two distinct points on this line, then the slope of the line is m

y2  y1 Change in y  x2  x1 Change in x

(1)

In applications, ratio (1) is called the rate of change of y with respect to x. Since the slope of a line is unique, the rate of change of two linearly related variables is constant. Here are some examples of familiar rates of change: miles per hour, revolutions per minute, price per pound, passengers per plane, etc. If y is distance and x is time, then the rate of change is also referred to as speed or velocity. If the relationship between x and y is not linear, ratio (1) is called the average rate of change of y with respect to x.

EXAMPLE

1

Estimating Body Surface Area Appropriate doses of medicine for both animals and humans are often based on body surface area (BSA). Since weight is much easier to determine than BSA, veterinarians use the weight of an animal to estimate BSA. The following linear equation expresses BSA for canines in terms of weight*: a  16.21w  375.6 where a is BSA in square inches and w is weight in pounds. (A) Interpret the slope of the BSA equation. (B) What is the effect of a 1-pound increase in weight? SOLUTIONS

(A) The rate of change BSA with respect to weight is 16.12 square inches per pound. (B) Since slope is the ratio of rise to run, increasing w by 1 pound (run) increases a by 16.12 square inches (rise). 

MATCHED PROBLEM

1

The following linear equation expresses BSA for felines in terms of weight: a  28.55w  118.7 where a is BSA in square inches and w is weight in pounds. (A) Interpret the slope of the BSA equation. (B) What is the effect of a 1-pound increase in weight? *Based on data from Veterinary Oncology Consultants, PTY LTD.



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ZZZ EXPLORE-DISCUSS

1

As illustrated in Example 1(A), the slope m of a line with equation y  mx  b has two interpretations: 1. m is the rate of change of y with respect to x. 2. Increasing x by one unit will change y by m units. How are these two interpretations related?

Z Linear Models We can use our experience with lines gained in Section 2-3 to construct linear models for applications involving linearly related quantities. This process is best illustrated through examples.

EXAMPLE

2

Business Markup Policy A sporting goods store sells a fishing rod that cost $60 for $82 and a pair of crosscountry ski boots that cost $80 for $106. (A) If the markup policy of the store for items that cost more than $30 is assumed to be linear, find a linear model that express the retail price P in terms of the wholesale cost C. (B) What is the effect on the price of a $1 increase in cost for any item costing over $30? (C) Use the model to find the retail price for a pair of running shoes that cost $40. SOLUTIONS

(A) If price P is linearly related to cost C, then we are looking for the equation of a line whose graph passes through (C1, P1)  (60, 82) and (C2, P2)  (80, 106). We find the slope, and then use the point–slope form to find the equation. m

P2  P1 106  82 24    1.2 C2  C1 80  60 20

P  P1  m(C  C1)

Substitute P1  82, C1  60, and m  1.2 in the point–slope formula.

P  82  1.2(C  60)

Remove parentheses.

P  82  1.2C  72 P  1.2C  10

Substitute C1  60, P1  82, C2  80, and P2  106 in the slope formula.

Add 82 to both sides.

C 7 30

Linear model

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213

(B) If the cost is increased by $1, then the price will increase by 1.2(1)  $1.20. (C) P  1.2(40)  10  $58.

MATCHED PROBLEM



2

The sporting goods store in Example 2 is celebrating its twentieth anniversary with a 20% off sale. The sale price of a mountain bike is $380. What was the presale price of the bike? How much did the bike cost the store? 

ZZZ EXPLORE-DISCUSS

2

The wholesale supplier for the sporting goods store in Example 2 offers the store a 15% discount on all items. The store decides to pass on the savings from this discount to the consumer. Which of the following markup policies is better for the consumer? 1. Apply the store’s markup policy to the discounted cost. 2. Apply the store’s markup policy to the original cost and then reduce this price by 15%. Support your choice with examples.

EXAMPLE

3

Mixing Antifreeze Ethylene glycol and propylene glycol are liquids used in antifreeze and deicing solutions. Ethylene glycol is listed as a hazardous chemical by the Environmental Protection Agency, while propylene glycol is generally regarded as safe. Table 1 lists solution concentration percentages and the corresponding freezing points for each chemical.

Table 1 Concentration

Ethylene Glycol

Propylene Glycol

20%

15°F

17°F

50%

36°F

28°F

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(A) Assume that the concentration and the freezing point for ethylene glycol are linearly related. Construct a linear model for the freezing point. (B) Interpret the slope in part (A). (C) What percentage (to one decimal place) of ethylene glycol will result in a freezing point of 10°F? SOLUTIONS

(A) We begin by defining appropriate variables: Let p  percentage of ethylene glycol in the antifreeze solution f  freezing point of the antifreeze solution From Table 1, we see that (20, 15) and (50, 36) are two points on the line relating p and f. The slope of this line is m

f2  f1 15  (36) 51    1.7 p2  p1 20  50 30

and its equation is f  15  1.7( p  20) f  1.7p  49

Linear model

(B) The rate of change of the freezing point with respect to the percentage of ethylene glycol in the antifreeze solution is 1.7 degrees per percentage of ethylene glycol. Increasing the amount of ethylene glycol by 1% will lower the freezing point by 1.7°F. (C) We must find p when f is 10°. f  1.7p  49  10  1.7p  49 1.7p  59 59 p  34.7% 1.7

MATCHED PROBLEM

Add 10  1.7p to both sides. Divide both sides by 1.7.



3

Refer to Table 1. (A) Assume that the concentration and the freezing point for propylene glycol are linearly related. Construct a linear model for the freezing point. (B) Interpret the slope in part (A). (C) What percentage (to one decimal place) of propylene glycol will result in a freezing point of 15°F?



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215

100

Technology Connections

0

100

Figure 2 shows the solution of Example 3(C) on a graphing calculator.

EXAMPLE

4

100

Z Figure 2 y1  1.7x  49, y2  10

Underwater Pressure The pressure at sea level is 14.7 pounds per square inch. As you descend into the ocean, the pressure increases linearly at a rate of about 0.445 pounds per square foot. (A) Find the pressure p at a depth of d feet. (B) If a diver’s equipment is rated to be safe up to a pressure of 40 pounds per square foot, how deep (to the nearest foot) is it safe to use this equipment? SOLUTIONS

(A) Let p  md  b. At the surface, d  0 and p  14.7, so b  14.7. The slope m is the given rate of change, m  0.445. Thus, the pressure at a depth of d feet is p  0.445d  14.7 (B) The safe depth is the solution of the equation 0.445d  14.7  40 0.445d  25.3 25.3 d 0.445  57 feet

Subtract 14.7 from each side. Divide both sides by 0.445. Simplify.



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MATCHED PROBLEM

4

The rate of change of pressure in fresh water is 0.432 pounds per square foot. Repeat Example 4 for a body of fresh water. 

80

Technology Connections

0

100

20

Figure 3 shows the solution of Example 4(B) on a graphing calculator.

Z Figure 3 y1  0.445x  14.7, y2  40

Z Linear Regression In real-world applications we often encounter numerical data in the form of a table. The very powerful mathematical tool, regression analysis, can be used to analyze numerical data. In general, regression analysis is a process for finding a function that provides a useful model for a set of data points. Graphs of equations are often called curves and regression analysis is also referred to as curve fitting. In Example 5, we use a linear model obtained by using linear regression on a graphing calculator.

EXAMPLE

5

Diamond Prices Prices for round-shaped diamonds taken from an online trader are given in Table 2. Table 2 Round-Shaped Diamond Prices Weight (Carats)

Price

0.5

$1,340

0.6

$1,760

0.7

$2,540

0.8

$3,350

0.9

$4,130

1.0

$4,920

Source: www.tradeshop.com

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S E C T I O N 2–4

Linear Equations and Models

217

(A) A linear model for the data in Table 2 is given by p  7,380c  2,530

(2)

where p is the price of a diamond weighing c carats. (We will discuss the source of models like this later in this section.) Plot the points in Table 2 on a Cartesian coordinate system, producing a scatter plot, and graph the model on the same axes. (B) Interpret the slope of the model in equation (2). (C) Use the model to estimate the cost of a 0.85-carat diamond and the cost of a 1.2-carat diamond. Round answers to the nearest dollar. (D) Use the model to estimate the weight of a diamond that sells for $3,000. Round the answer to two significant digits. SOLUTIONS

(A) A scatter plot is simply a plot of the points in Table 2 [Fig. 4(a)]. To add the graph of the model to the scatter plot, we find any two points that satisfy equation (2) [we choose (0.4, 422) and (1.4, 7,802)]. Plotting these points and drawing a line through them gives us Figure. 4(b). p $8,000

Price

11/02/06

$4,000

0.5

1

1.5

1

1.5

c

Carats (a) Scatter plot

p $8,000

Price

bar67388_ch02_192-230

$4,000

0.5

c

Carats (b) Linear model

Z Figure 4

(B) The rate of change of the price of a diamond with respect to its weight is 7,380. Increasing the weight by 1 carat will increase the price by about $7,380.

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(C) The graph of the model [Fig. 4(b)] does not pass through any of the points in the scatter plot, but it comes close to all of them. [Verify this by evaluating equation (2) at c  0.5, 0.6, . . . , 1.] Thus, we can use equation (2) to approximate points not in Table 2. c  0.85 p  7,380(0.85)  2,530  $3,743

c  1.2 p  7,380(1.2)  2,530  $6,326

A 0.85-carat diamond will cost about $3,743 and a 1.2-carat diamond will cost about $6,326. (D) To find the weight of a $3,000 diamond, we solve the following equation for c: 7,380c  2,530  3,000 7,380c  3,000  2,530  5,530 5,530 c  0.75 7,380

To two significant digits

A $3,000 diamond will weigh about 0.75 carats.

MATCHED PROBLEM



5

Prices for emerald-shaped diamonds taken from an online trader are given in Table 3. Repeat Example 5 for this data with the linear model p  7,270c  2,450 where p is the price of an emerald-shaped diamond weighing c carats. Table 3 Emerald-Shaped Diamond Prices Weight (Carats)

Price

0.5

$1,350

0.6

$1,740

0.7

$2,610

0.8

$3,320

0.9

$4,150

1.0

$4,850

Source: www.tradeshop.com



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Linear Equations and Models

219

The model we used in Example 5 was obtained by using a technique called linear regression and the model is called the regression line. This technique produces a line that is the best fit for a given data set. We will not discuss the theory behind this technique, nor the meaning of “best fit.” Although you can find a linear regression line by hand, we prefer to leave the calculations to a graphing calculator or a computer. Don’t be concerned if you don’t have either of these electronic devices. We will supply the regression model in the applications we discuss, as we did in Example 5.

If you want to use a graphing calculator to construct regression lines, you should consult your user’s manual.* The process varies greatly from one calculator to another. Figure 5 shows three of the screens related to the construction of the model in Example 5 on a Texas Instruments TI-83 Plus.

Technology Connections

8,000

0

1.5

1,000

(a) Entering the data.

(b) Finding the model.

(c) Graphing the data and the model.

Z Figure 5 *User’s manuals for the most popular graphing calculators are readily available on the Internet.

In Example 5, we used the regression line to approximate points that were not given in Table 2, but would fit between points in the table. This process is called interpolation. In the next example we use a regression model to approximate points outside the given data set. This process is called extrapolation and the approximations are often referred to as predictions.

EXAMPLE

6

New Car Sales Table 4 contains information about new car sales in recent years. The linear regression model for consumer sales is s  4.85  0.13t

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where s is the numbers of cars (in millions) sold to consumers each year and t is time in years with t  0 corresponding to 2000. (A) Interpret the slope of the regression line as a rate of change. (B) Use the regression line to predict the sales in 2015. Table 4 Sales* of New Cars (millions of vehicles) 2000

2001

2002

2003

2004

Consumer

4.7

4.6

4.5

4.3

4.2

Business

3.9

3.6

3.4

3.1

3.1

*Includes both purchases and leases. Source: U.S. Bureau of Transportation Statistics

SOLUTIONS

(A) The slope m = 0.13 is the rate of change of sales with respect to time. Since the slope is negative and the sales are given in millions of the units, the sales are decreasing at a rate of 0.13(1,000,000)  130,000 cars per year. (B) If t  15, then s  4.85  0.13(15)  2.9

or

2,900,000 cars

So approximately 2,900,000 new cars will be sold to comsumers in 2015.

MATCHED PROBLEM



6

Repeat Example 6 using the following linear regression model for the number of cars sold to business: b  4.05  0.21t where b is the numbers of cars (in millions) sold to business each year and t is time in years with t  0 corresponding to 2000. 

ZZZ EXPLORE-DISCUSS

3

Use the Internet or a library to expand the data in Table 5 to include years after 2004. Compare the actual values for the years you find to the model’s predicted values. Discuss the accuracy of predictions given by the regression model as time goes by.

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ANSWERS

Linear Equations and Models

221

TO MATCHED PROBLEMS

1. (A) The rate of change BSA with respect to weight is 28.55 square inches per pound. (B) Increasing w by 1 pound increases a by 28.55 square inches. 2. Presale price is $475. Cost is $387.50 3. (A) f  1.5p  47 (B) The rate of change of the freezing point with respect to the percentage of propylene glycol in the antifreeze solution is 1.5. Increasing the percentage of propylene glycol by 1% will lower the freezing point by 1.5°F. (C) 41.3% 4. (A) p  0.432d  14.7 (B) 59 ft p 5. (A)

Price

$8,000

$4,000

0.5

1

1.5

c

Carats

(B) The rate of change of the price of a diamond with respect to the size is 7,270. Increasing the size by 1 carat will increase the price by about $7,270. (C) $3,730; $6,274 (D) 0.75 carats 6. (A) The slope m  0.21 is the rate of change of sales with respect to time. Since the slope is negative and the sales are given in millions of the units, the sales are decreasing at a rate of 0.21(1,000,000)  210,000 cars per year. (B) Approximately 900,000 cars will be sold to businesses in 2015.

2-4

Exercises

APPLICATIONS 1. COST ANALYSIS A plant can manufacture 80 golf clubs per day for a total daily cost of $8,147 and 100 golf clubs per day for a total daily cost of $9,647. (A) Assuming that daily cost and production are linearly related, find the total daily cost of producing x golf clubs. (B) Interpret the slope of this cost equation. (C) What is the effect of a 1 unit increase in production? 2. COST ANALYSIS A plant can manufacture 50 tennis rackets per day for a total daily cost of $4,174 and 60 tennis rackets per day for a total daily cost of $4,634. (A) Assuming that daily cost and production are linearly related, find the total daily cost of producing x tennis rackets. (B) Interpret the slope of this cost equation. (C) What is the effect of a 1 unit increase in production? 3. FORESTRY Forest rangers estimate the height of a tree by measuring the tree’s diameter at breast height (DBH) and then

using a model constructed for a particular species.* A model for white spruce trees is h  4.06d  24.1 where d is the DBH in inches and h is the tree height in feet. (A) Interpret the slope of this model. (B) What is the effect of a 1-inch increase in DBH? (C) How tall is a white spruce with a DBH of 12 inches? Round answer to the nearest foot. (D) What is the DBH of a white spruce that is 100 feet tall? Round answer to the nearest inch. 4. FORESTRY A model for black spruce trees is h  2.27d  33.1 where d is the DBH in inches and h is the tree height in feet. *Models in Problems 3 and 4 are based on data found at http://flash.lakeheadu.ca/~fluckai/htdbh04.xls

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(A) Interpret the slope of this model. (B) What is the effect of a 1-inch increase in DBH? (C) How tall is a black spruce with a DBH of 12 inches? Round answer to the nearest foot. (D) What is the DBH of a black spruce that is 100 feet tall? Round answer to the nearest inch. 5. In 1983 Dr. J. D. Robinson and Dr. D. R. Miller published the following models for estimating the weight of a woman: Robinson: w  108  3.7h Miller: w  117  3.0h where w is weight (in pounds) and h is height over 5 feet (in inches). (A) Interpret the slope of each model. (B) If a women is 56 tall, what does each model predict her weight to be? (C) If a women weighs 140 pounds, what does each model predict her height to be? 6. Dr. J. D. Robinson and Dr. D. R. Miller also published the following models for estimating the weight of a man: Robinson: w  115  4.2h Miller: w  124  3.1h where w is weight (in pounds) and h is height over 5 feet (in inches). (A) Interpret the slope of each model. (B) If a man is 510 tall, what does each model predict his weight to be? (C) If a man weighs 160 pounds, what does each model predict his height to be? 7. SPEED OF SOUND The speed of sound through the air near sea level is linearly related to the temperature of the air. If sound travels at 741 mph at 32°F and at 771 mph at 72°F, construct a linear model relating the speed of sound (s) and the air temperature (t). Interpret the slope of this model. 8. SPEED OF SOUND The speed of sound through the air near sea level is linearly related to the temperature of the air. If sound travels at 337 mps (meters per second) at 10°C and at 343 mps at 20°C, construct a linear model relating the speed of sound (s) and the air temperature (t). Interpret the slope of this model. 9. CIGARETTE SMOKING The percentage of male cigarette smokers in the United States declined from 28.3% in 1990 to 25.2% in 2000. Find a linear model relating percentage of male smokers (m) to years since 1990 (t). Predict the year in which the population of male smokers will fall below 20%. 10. CIGARETTE SMOKING The percentage of female cigarette smokers in the United States declined from 22.9% in 1990 to 21.1% in 2000. Find a linear model relating percentage of female smokers ( f ) to years since 1990 (t). Predict the year in which the population of female smokers will fall below 18%.

11. BUSINESS—DEPRECIATION A farmer buys a new tractor for $142,000 and assumes that it will have a trade-in value of $67,000 after 10 years. The farmer uses a constant rate of depreciation (commonly called straight-line depreciation—one of several methods permitted by the IRS) to determine the annual value of the tractor. (A) Find a linear model for the depreciated value V of the tractor t years after it was purchased. (B) Interpret the slope of this model. (C) What is the depreciated value of the tractor after 6 years? 12. BUSINESS—DEPRECIATION A charter fishing company buys a new boat for $154,900 and assumes that it will have a trade-in value of $46,100 after 16 years. (A) Use straight-line depreciation to find a linear model for the depreciated value V of the boat t years after it was purchased. (B) Interpret the slope of this model. (C) In which year will the depreciated value of the boat fall below $100,000? 13. BUSINESS—MARKUP POLICY A drugstore sells a drug costing $85 for $112 and a drug costing $175 for $238. (A) If the markup policy of the drugstore is assumed to be linear, write an equation that expresses retail price R in terms of cost C (wholesale price). (B) What is the slope of the graph of the equation found in part A? Interpret verbally. (C) What does a store pay (to the nearest dollar) for a drug that retails for $185? 14. BUSINESS—MARKUP POLICY A clothing store sells a shirt costing $20 for $33 and a jacket costing $60 for $93. (A) If the markup policy of the store for items costing over $10 is assumed to be linear, write an equation that expresses retail price R in terms of cost C (wholesale price). (B) What is the slope of the equation found in part A? Interpret verbally. (C) What does a store pay for a suit that retails for $240? 15. FLIGHT CONDITIONS In stable air, the air temperature drops about 5 F for each 1,000-foot rise in altitude. (A) If the temperature at sea level is 70°F and a commercial pilot reports a temperature of 20 F at 18,000 feet, write a linear equation that expresses temperature T in terms of altitude A (in thousands of feet). (B) How high is the aircraft if the temperature is 0 F? 16. FLIGHT NAVIGATION An airspeed indicator on some aircraft is affected by the changes in atmospheric pressure at different altitudes. A pilot can estimate the true airspeed by observing the indicated airspeed and adding to it about 2% for every 1,000 feet of altitude. (A) If a pilot maintains a constant reading of 200 miles per hour on the airspeed indicator as the aircraft climbs from sea level to an altitude of 10,000 feet, write a linear equation that expresses true airspeed T (miles per hour) in terms of altitude A (thousands of feet). (B) What would be the true airspeed of the aircraft at 6,500 feet?

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17. RATE OF DESCENT—PARACHUTES At low altitudes, the altitude of a parachutist and time in the air are linearly related. A jump at 2,880 ft using the U.S. Army’s T-10 parachute system lasts 120 seconds. (A) Find a linear model relating altitude a (in feet) and time in the air t (in seconds). (B) The rate of descent is the speed at which the jumper falls. What is the rate of descent for a T-10 system?

223

Linear Equations and Models

where x is the state population and y is the number of licensed drivers in the state.

Table 6 Licensed Drivers in 2003 State

Population

Licensed Drivers

Alaska

6.5

4.8

18. RATE OF DESCENT—PARACHUTES The U.S. Army is considering a new parachute, the ATPS system. A jump at 2,880 ft using the ATPS system lasts 180 seconds. (A) Find a linear model relating altitude a (in feet) and time in the air t (in seconds). (B) What is the rate of descent for an ATPS system parachute?

Delaware

8.2

5.9

Montana

9.2

7

North Dakota

6.3

4.6

South Dakota

7.6

5.5

19. REAL ESTATE Table 5 contains recent average and median purchase prices for a house in Texas. A linear regression model for the average purchase price is

Vermont

6.2

5.4

Wyoming

5

3.8

Source: Bureau of Transportation Statistics

y  4.6x  146 where x is years since 2000 and y is average purchase price (in thousands of dollars). (A) Plot the average price data and the model on the same axes. (B) Predict the average price in 2010. (C) Interpret the slope of the model.

Table 5 Texas Real Estate Prices

(A) Plot the data in Table 6 and the model on the same axes. (B) If the population of Alabama in 2003 is about 4.5 million, use the model to estimate the number of licensed drivers in Alabama. (C) If the population of Arizona in 2003 is about 5.6 million, use the model to estimate the number of licensed drivers in Arizona.

Year

Average Price (in thousands)

Median Price (in thousands)

2000

$146

$112

22. LICENSED DRIVERS Table 7 contains the state population and the number of licensed drivers in the state (both in millions) for several states with population over 10 million. The regression model for this data is

2001

$150

$120

y  0.59x  1.06

2002

$156

$125

2003

$160

$128

where x is the state population and y is the number of licensed drivers in the state.

2004

$164

$130

Source: Real Estate Center at Texas A&M University

Table 7 Licensed Drivers in 2003 Population

Licensed Drivers

California

35

22

y  4.4x  114

Florida

17

13

where x is years since 2000 and y is median purchase price (in thousands of dollars). (A) Plot the median price data and the model on the same axes. (B) Predict the median price in 2010. (C) Interpret the slope of the model.

Illinois

13

8

Michigan

10

7

New York

19

11

Ohio

11

8

21. LICENSED DRIVERS Table 6 contains the state population and the number of licensed drivers in the state (both in millions) for the states with population under 10 million. The regression model for this data is

Pennsylvania

12

8

Texas

22

13

20. REAL ESTATE A linear regression model for the median purchase price in Table 5 is

y  0.69x  0.5

State

Source: Bureau of Transportation Statistics

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GRAPHS

(A) Plot the data in Table 7 and the model on the same axes. (B) If the population of California in 2003 is about 35.5 million, use the model to estimate the number of licensed drivers in California. (C) If the population of New York in 2003 is about 19.2 million, use the model to estimate the number of licensed drivers in New York. Problems 23–26 require a graphing calculator or a computer that can calculate the linear regression line for a given data set. 23. OLYMPIC GAMES Find a linear regression model for the men’s 100-meter freestyle data given in Table 8, where x is years since 1968 and y is winning time (in seconds). Do the same for the women’s 100-meter freestyle data. (Round regression coefficients to four significant digits.) Do these models indicate that the women will eventually catch up with the men? If so, when? Do you think this will actually occur?

Table 8 Winning Times in Olympic Swimming Events

25. SUPPLY AND DEMAND Table 9 contains price–supply data and price–demand data for corn. Find a linear regression model for the price–supply data where x is supply (in billions of bushels) and y is price (in dollars). Do the same for the price–demand data. (Round regression coefficients to three significant digits.) Find the price at which supply and demand are equal. (In economics, this price is referred to as the equilibrium price.)

Table 9 Supply and Demand for U.S. Corn Price ($/bu.)

Supply (Billion bu.)

Price ($/bu.)

Demand (Billion bu.)

2.15

6.29

2.07

9.78

2.29

7.27

2.15

9.35

2.36

7.53

2.22

8.47

2.48

7.93

2.34

8.12

2.47

8.12

2.39

7.76

2.55

8.24

2.47

6.98

100-Meter Men

Freestyle Women

200-Meter Men

Backstroke Women

1968

52.20

60.0

2:09.60

2:24.80

1972

51.22

58.59

2:02.82

2:19.19

1976

49.99

55.65

1:59.19

2:13.43

1980

50.40

54.79

2:01.93

2:11.77

1984

49.80

55.92

2:00.23

2:12.38

1988

48.63

54.93

1:59.37

2:09.29

1992

49.02

54.65

1:58.47

2:07.06

1996

48.74

54.50

1:58.54

2:07.83

Table 10 Supply and Demand for U.S. Soybeans

2000

48.30

53.83

1:56.76

2:08.16

Price ($/bu.)

Supply (Billion bu.)

Price ($/bu.)

Demand (Billion bu.)

2004

48.17

53.84

1:54.76

2:09.16 5.15

1.55

4.93

2.60

5.79

1.86

5.48

2.40

5.88

1.94

5.71

2.18

6.07

2.08

6.07

2.05

6.15

2.15

6.40

1.95

6.25

2.27

6.66

1.85

Source: www.usda.gov/nass/pubs/histdata.htm

Source: www.infoplease.com

24. OLYMPIC GAMES Find a linear regression model for the men’s 200-meter backstroke data given in Table 8 where x is years since 1968 and y is winning time (in seconds). Do the same for the women’s 200-meter backstroke data. (Round regression coefficients to five significant digits.) Do these models indicate that the women will eventually catch up with the men? If so, when? Do you think this will actually occur?

26. SUPPLY AND DEMAND Table 10 contains price–supply data and price–demand data for soybeans. Find a linear regression model for the price–supply data where x is supply (in billions of bushels) and y is price (in dollars). Do the same for the price–demand data. (Round regression coefficients to three significant digits.) Find the equilibrium price for soybeans.

Source: www.usda.gov/nass/pubs/histdata.htm

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CHAPTER 2-1

2

Cartesian Coordinate System

2-3

Distance in the Plane

The distance between the two points P1(x1, y1) and P2(x2, y2) is d(P1, P2)  2(x2  x1)2  ( y2  y1)2 and the midpoint of the line segment joining P1(x1, y1) and P2(x2, y2) is Ma

x1  x2 y1  y2 , b 2 2

The standard form for the equation of a circle with radius r and center at (h, k) is (x  h)2  ( y  k)2  r2,

225

Review

A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line and a vertical real number line at their origins. These lines are called the coordinate axes. The horizontal axis is often referred to as the x axis and the vertical axis as the y axis. These axes divide the plane into four quadrants. Each point in the plane corresponds to its coordinates—an ordered pair (a, b) determined by passing horizontal and vertical lines through the point. The abscissa or x coordinate a is the coordinate of the intersection of the vertical line with the horizontal axis, and the ordinate or y coordinate b is the coordinate of the intersection of the horizontal line with the vertical axis. The point (0, 0) is called the origin. A solution of an equation in two variables is an ordered pair of real numbers that makes the equation a true statement. The solution set of an equation is the set of all its solutions. The graph of an equation in two variables is the graph of its solution set formed using point-by-point plotting or with the aid of a graphing utility. The reflection of the point (a, b) through the y axis is the point (a, b), through the x axis is the point (a, b), and through the origin is the point (a, b). The reflection of a graph is the reflection of each point on the graph. If reflecting a graph through the y axis, x axis, or origin does not change its shape, the graph is said to be symmetric with respect to the y axis, x axis, or origin, respectively. To test an equation for symmetry, determine if the equation is unchanged when y is replaced with y (x axis symmetry), x is replaced with x ( y axis symmetry), or both x and y are replaced with x and y (origin symmetry).

2-2

Review

r 7 0

Equations of a Line

The standard form for the equation of a line is Ax  By  C, where A, B, and C are constants, A and B not both 0. The y intercept is the y coordinate of the point where the graph crosses the y axis, and the x intercept is the x coordinate of the point where the graph crosses the x axis. The slope of the line through the points (x1, y1) and (x2, y2) is m

y2  y1 x2  x1

if x1  x2

The slope is not defined for a vertical line where x1  x2. Two lines with slopes m1 and m2 are parallel if and only if m1  m2 and perpendicular if and only if m1m2  1. Equations of a Line Standard form Ax  By  C Slope–intercept form y  mx  b

A and B not both 0 Slope: m; y intercept: b

Point–slope form

y  y1  m(x  x1) Slope: m; Point: (x1, y1)

Horizontal line

yb

Slope: 0

Vertical line x

xa

Slope: Undefined

2-4

Linear Equations and Models

A mathematical model is a problem whose solution will provide information about a real-world problem. If y  mx  b, then the variables x and y are linearly related and the rate of change of y with respect to x is the constant m. If x and y are not linearly related, the ratio ( y2  y1)(x2  x1) is called the average rate of change of y with respect to x. Regression analysis produces a function whose graph is a curve that fits (approximates) a set of data points. A scatter plot is the graph of the points in a data set. Linear regression produces a regression line that is the best fit (in some sense) for a given data set. Graphing calculators or other electronic devices are frequently used to find regression lines.

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2

Review Exercises

Work through all the problems in this chapter review and check answers in the back of the book. Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that type of problem is discussed. Where weaknesses show up, review appropriate sections in the text.

9. (A) Graph the triangle with vertices A  (1, 2), B  (4, 3), and C  (1, 4). (B) Find the perimeter to two decimal places. (C) Use the Pythagorean theorem to determine if the triangle is a right triangle. (D) Find the midpoint of each side of the triangle.

1. Plot A  (4, 1), B  (2, 3), and C  (1, 2) in a rectangular coordinate system.

10. Use the graph of the linear function in the figure to find the rise, run, and slope. Write the equation of the line in the form Ax  By  C, where A, B, and C are integers with A 0. (The horizontal and vertical line segments have integer lengths.)

2. Refer to Problem 1. Plot the reflection of A through the x axis, the reflection of B through the y axis, and the reflection of C through the origin.

y

3. Test each equation for symmetry with respect to the x axis, y axis, and origin and sketch its graph. (A) y  2x (B) y  2x  1 (C) y  2|x| (D) x  2| y|

5

4. Use the following graph to estimate to the nearest integer the missing coordinates of the indicated points. (Be sure you find all possible answers.) (A) (0, ?) (B) (?, 0) (C) (?, 4)

5

x

5

y

11. Graph 3x  2y  9 and indicate its slope.

5

5

5

5

x

12. Write an equation of a line with x intercept 6 and y intercept 4. Write the final answer in the standard form Ax  By  C, where A, B, and C are integers. 13. Write the slope–intercept form of the equation of the line with slope 23 and y intercept 2.

5

5. Given the points A(2, 3) and B(4, 0), find: (A) Distance between A and B (B) Slope of the line through A and B (C) Slope of a line perpendicular to the line through A and B 6. Write the equation of a circle with radius 17 and center: (A) (0, 0) (B) (3, 2) 7. Find the center and radius of the circle given by (x  3)2  ( y  2)2  5 8. Let M be the midpoint of A and B, where A  (a1, a2), B  (2, 5), and M  (4, 3). (A) Use the fact that 4 is the average of a1 and 2 to find a1. (B) Use the fact that 3 is the average of a2 and 5 to find a2. (C) Find d(A, M ) and d(M, B).

14. Write the equations of the vertical and horizontal lines passing through the point (3, 4). What is the slope of each? Test each equation in Problems 15–18 for symmetry with respect to the x axis, y axis, and the origin. Sketch the graph of the equation. 15. y  x2  2

16. y2  x  2

17. 9y2  4x2  36

18. 9y2  4x2  36

19. Write a verbal description of the graph shown in the figure and then write an equation that would produce the graph.

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Problems 30 and 31 refer to a triangle with base b and height h (see the figure). Write a mathematical expression in terms of b and h for each of the verbal statements in Problems 30 and 31.

5

5

Review Exercises

5

h

x

b

30. The base is five times the height.

5

20. (A) Find an equation of the line through P(4, 3) and Q(0, 3). Write the final answer in the standard form Ax  By  C, where A, B, and C are integers with A 0. (B) Find d(P, Q). 21. Write the slope–intercept form of the equation of the line that passes through the point (2, 1) and is (A) parallel to the line 6x  3y  5 (B) perpendicular to the line 6x  3y  5 22. Find the equation of a circle that passes through the point (1, 4) with center at (3, 0). 23. Find the center and radius of the circle given by x2  y2  4x  6y  3 24. Find the equation of the set of points equidistant from (3, 3) and (6, 0). What is the name of the geometric figure formed by this set? 25. Are the graphs of mx  y  b and x  my  b parallel, perpendicular, or neither? Justify your answer.

31. The height is one-fourth of the base.

APPLICATIONS 32. LINEAR DEPRECIATION A computer system was purchased by a small company for $12,000 and is assumed to have a depreciated value of $2,000 after 8 years. If the value is depreciated linearly from $12,000 to $2,000: (A) Find the linear equation that relates value V (in dollars) to time t (in years). (B) What would be the depreciated value of the system after 5 years? 33. COST ANALYSIS A video production company is planning to produce an instructional CD. The producer estimates that it will cost $24,900 to produce the CD and $5 per unit to copy and distribute the CD. The budget for this project is $62,000. How many CDs can be produced without exceeding the budget?

26. Use completing the square to find the center and radius of the circle with equation:

34. FORESTRY Forest rangers estimate the height of a tree by measuring the tree’s diameter at breast height (DBH) and then using a model constructed for a particular species. A model for sugar maples is

x2  4x  y2  2y  3  0

h  2.9d  30.2

27. Refer to Problem 26. Find the equation of the line tangent to the circle at the point (4, 3). Graph the circle and the line on the same coordinate system.

where d is the DBH in inches and h is the tree height in feet. (A) Interpret the slope of this model. (B) What is the effect of a 1-inch increase in DBH? (C) How tall is a sugar maple with a DBH of 3 inches? Round answer to the nearest foot. (D) What is the DBH of a sugar maple that is 45 feet tall? Round answer to the nearest inch.

28. Find the equation of a circle with center (4, 3) whose graph passes through the point (1, 2). 29. Extend the following graph to one that exhibits the indicated symmetry: (A) x axis only (B) y axis only (C) origin only (D) x axis, y axis, and origin y

BSA  1,321  0.3433  Wt

5

5

5

5

35. ESTIMATING BODY SURFACE AREA An important criterion for determining drug dosage for children is the patient’s body surface area (BSA). John D. Current published the following useful model for estimating BSA*:

x

where BSA is given in square centimeters and Wt in grams. (A) Interpret the slope of this model. (B) What is the effect of a 100-gram increase in weight? (C) What is the BSA for a child that weighs 15 kilograms? *“Body Surface Area in Infants and Children,” The Internet Journal of Anesthesiology, 1998, Volume 2, Number 2.

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★36.

ARCHITECTURE A circular arc forms the top of an entryway with 6-foot vertical sides 8 feet apart. If the top of the arc is 2 feet above the ends, what is the radius of the arc?

37. SPORTS MEDICINE The following quotation was found in a sports medicine handout: “The idea is to raise and sustain your heart rate to 70% of its maximum safe rate for your age. One way to determine this is to subtract your age from 220 and multiply by 0.7.” (A) If H is the maximum safe sustained heart rate (in beats per minute) for a person of age A (in years), write a formula relating H and A. (B) What is the maximum safe sustained heart rate for a 20-year-old? (C) If the maximum safe sustained heart rate for a person is 126 beats per minute, how old is the person? 38. DATA ANALYSIS Winning times in the men’s Olympic 400-meter freestyle event in minutes for selected years are given in Table 1. A mathematical model for these data is f (x)  0.021x  5.57 where x is years since 1900. (A) Compare the model and the data graphically and numerically. (B) Estimate (to three decimal places) the winning time in 2008.

Table 1 Year

Time

1912

5.41

1932

4.81

1952

4.51

1972

4.00

1992

3.75

CHAPTER

ZZZ GROUP

39. POLITICAL SCIENCE Association of economic class and party affiliation did not start with Roosevelt’s New Deal; it goes back to the time of Andrew Jackson (1767–1845). Paul Lazarsfeld of Columbia University published an article in the November 1950 issue of Scientific American in which he discussed statistical investigations of the relationships between economic class and party affiliation. The data in Table 2 are taken from this article.

Table 2 Political Affiliations in 1836 Ward

Average Assessed Value per Person (in $100)

Democratic Votes (%)

12

1.7

51

3

2.1

49

1

2.3

53

5

2.4

36

2

3.6

65

11

3.7

35

10

4.7

29

4

6.2

40

6

7.1

34

9

7.4

29

8

8.7

20

7

11.9

23

(A) Find a linear regression model for the data in the second and third columns of the table, using the average assessed value as the independent variable. (B) Use the linear regression model to predict (to two decimal places) the percentage of votes for democrats in a ward with an average assessed value of $300.

2 ACTIVITY Rates of Change

In Section 2-4, we noted that the rate of change of linearly related quantities is always constant. In this activity, we consider some rates of change that are not constant.

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1. Average Rate. If you score 90 on your first math exam and 100 on the second exam, then your average exam score for the two exams is 12(90  100)  95. The number 95 is called the arithmetic average of 90 and 100. Now suppose you walk uphill at a rate of 3 mph for 5 hours and then turn around and return to your starting point by walking downhill at 6 mph for 2.5 hours. The arithmetic average of the rates for each leg of the trip is 12(3  6)  4.5 mph. On the other hand, you walked a total distance of 30 miles in 7.5 hours so that the rate for the round-trip is 307.5  4 mph. Which is your average rate? The basic formula D  RT is valid whenever an object travels a distance D at a constant rate R for a fixed time T. If the rate is not constant, then this formula can still be used but must be interpreted differently. To be precise, for objects moving at nonconstant rates, average rate is total distance divided by total time. Thus, your average rate for the total trip up and down the hill is 4 mph, not 4.5 mph. The formula R  DT now has two interpretations: R  DT is the rate for an object moving at a constant rate and the average rate for an object whose rate is not always the same. (A) If r is the rate for one leg of a round-trip and s is the rate for the return trip, express the average rate for the round-trip in terms of r and s. (B) A boat can travel 10 mph in still water. The boat travels 60 miles up a river with a 5 mph current and then returns to its starting point. Find the average rate for the round-trip using the definition of average rate and then check with the formula you found in part A. (C) Referring to the hill-climbing example discussed earlier, if you walk up the hill at 3 mph, how fast must you walk downhill to average 6 mph for the round-trip? (This is similar to a famous problem communicated to Albert Einstein by Max Wertheimer. See Abraham S. Luchins and Edith H. Luchins, The Einstein–Wertheimer Correspondence on Geometric Proofs and Mathematical Puzzles, Mathematical Intelligencer 2, Spring 1990, pp. 40–41. For a discussion of this and other interesting rate–time problems, see Lawrence S. Braden, My Favorite Rate–Time Problems, Mathematics Teacher, November 1991, pp. 635–638.) 2. Instantaneous Rate. One of the fundamental concepts of calculus is the instantaneous rate of a moving object, which is closely related to the average rate. To introduce this concept, consider the following problem. A small steel ball dropped from a tower will fall a distance of y feet in x seconds, as given approximately by the formula (from physics) y  16x2 Figure 1 shows the position of the ball on a number line (positive direction down) at the end of 0, 1, 2, and 3 seconds. Clearly, the ball is not falling at a constant rate.

Position at start (x  0 seconds) Position at x  1 second [y  16(12)  16 feet] y

Position at x  2 seconds [y  16(22)  64 feet]

Position at x  3 seconds [y  16(32)  144 feet] Ground y

Z Figure 1 Position of a falling object [Note: Positive direction is down.]

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(A) What is the average rate that the ball falls during the first second (from x = 0 to x = 1 second)? During the second second? During the third second? By definition, average rate involves the distance an object travels over an interval of time, as in part A. How can we determine the rate of an object at a given instant of time? For example, how fast is the ball falling at exactly 2 seconds after it was released? We will approach this problem from two directions, numerically and algebraically. (B) Complete the following table of average rates. What number do these average rates appear to approach? Time interval

[1.9, 2]

[1.99, 2]

[1.999, 2]

[1.9999, 2]

Distance fallen Average rate

(C) Show that the average rate over the time interval [t, 2] is

64  16t2 . Simplify this algebraic expression 2t

and discuss its values for t very close to 2. (D) Based on the results of parts B and C, how fast do you think the ball is falling at 2 seconds?

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3

SECTIONS 3-1

Functions

3-2

Graphing Functions

3-3

Transformations of Functions

3-4

Quadratic Functions

3-5

Operations on Functions; Composition

3-6

Inverse Functions Chapter 3 Review Chapter 3 Group Activity: Mathematical Modeling: Choosing a Long-Distance Calling Plan Cumulative Review Exercises Chapters 1–3

CHAPTER

THE function concept is one of the most important ideas in mathematics. The study of mathematics beyond the most elementary level requires a firm understanding of a basic list of elementary functions, their properties, and their graphs. In this chapter we introduce the important concept of a function, discuss basic properties, and consider operations that can be performed with functions. As we progress through this chapter, and subsequent chapters of the book, we will encounter a number of different types of elementary functions. A thorough understanding of the definitions, graphs, and properties of these elementary functions will provide you with a set of tools that should become a part of your mathematical toolbox for use in this course and most future courses or activities that involve mathematics.

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Functions Z Definition of Function Z Functions Defined by Equations Z Function Notation Z Application Z A Brief History of the Function Concept

The idea of correspondence plays a central role in the formulation of the function concept. You have already had experiences with correspondences in everyday life. For example: To To To To To

each each each each each

person there corresponds an age. item in a store there corresponds a price. automobile there corresponds a license number. circle there corresponds an area. number there corresponds its cube.

One of the most important aspects of any science (managerial, life, social, physical, computer, etc.) is the establishment of correspondences among various types of phenomena. Once a correspondence is known, predictions can be made. A chemist can use a gas law to predict the pressure of an enclosed gas, given its temperature. An engineer can use a formula to predict the deflections of a beam subject to different loads. A computer scientist can use formulas to compare the efficiency of algorithms for sorting data stored in a computer. An economist would like to be able to predict interest rates, given the rate of change of the money supply. And so on.

Z Definition of Function What do all the preceding examples have in common? Each describes the matching of elements from one set with elements in a second set. Let’s consider the correspondences between the sets A and B listed in Tables 1–3. Table 1

Table 2

Table 3

A

B

A

B

A

B

2

5

2

5

2

5

1

3

1

3

1

3

0

0

0

0

0

0

1

3

1

3

1

3

2

5

2

5

2

5

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Tables 1 and 2 specify functions, but Table 3 does not. Why not? The definition of the concept of a function will explain.

Z DEFINITION 1 Definition of Function A function is a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set. The first set is called the domain and the set of all corresponding elements in the second set is called the range.

Table 1 specifies a function with domain A  {2, 1, 0, 1, 2} and range B  {5, 3, 0, 3, 5} because to each domain value in A there corresponds exactly one element in B. Table 2 also specifies a function with domain A and range C  {5, 3, 5}. Notice that the range value 3 corresponds to three different domain values, 2, 0, and 2. This does not violate the conditions in Definition 1. On the other hand, Table 3 does not specify a function, because two different range values, 3 and 5, correspond to the domain value 2.

ZZZ EXPLORE-DISCUSS

1

Consider the set of students enrolled in a college and the set of faculty members of that college. Define a correspondence between the two sets by saying that a student corresponds to a faculty member if the student is currently enrolled in a course taught by the faculty member. Is this correspondence a function? Discuss.

Every function can be specified by using ordered pairs of elements, where the first component represents a domain element and the second element represents the corresponding range element. Thus, the functions specified in Tables 1 and 2 can be written as follows: F  5(2, 5), (1, 3), (0, 0), (1, 3), (2, 5)6 G  5(2, 3), (1, 5), (0, 3), (1, 5), (2, 3)6 In both cases, notice that no two ordered pairs have the same first component and different second components. On the other hand, if we list the set H of ordered pairs determined by Table 3, we have H  5(2, 3), (1, 5), (0, 3), (1, 5), (2, 3), (2, 5)6

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Notice that the last two ordered pairs, (2, 3) and (2, 5), have the same first component and different second components. Thus, set H does not specify a function. This ordered pair approach suggests an alternative but equivalent way of defining the concept of function that provides additional insight into this concept.

Z DEFINITION 2 Set Form of the Definition of Function A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components. The set of all first components in a function is called the domain of the function, and the set of all second components is called the range.

EXAMPLE

1

Functions Specified as Sets of Ordered Pairs Determine whether each set specifies a function. If it does, then state the domain and range. (A) S  5(1, 4), (2, 3), (3, 2), (4, 3), (5, 4)6 (B) T  5(1, 4), (2, 3), (3, 2), (2, 4), (1, 5)6 SOLUTIONS

(A) Because all the ordered pairs in S have distinct first components, this set specifies a function. The domain and range are Domain  51, 2, 3, 4, 56 Range  52, 3, 46

Set of first components Set of second components

(B) Because there are ordered pairs in T with the same first component [for example, (1, 4) and (1, 5)], this set does not specify a function. 

MATCHED PROBLEM

1

Determine whether each set defines a function. If it does, then state the domain and range. (A) S  5(2, 1), (1, 2), (0, 0), (1, 1), (2, 2)6 (B) T  5(2, 1), (1, 2), (0, 0), (1, 2), (2, 1)6



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Z Functions Defined by Equations We can specify or define a particular function in various ways: (1) by a verbal description; (2) by a table; (3) by a set of ordered pairs; or, if the domain and range are sets of numbers, (4) by an equation, or (5) by a graph. If the domain of a function is a large or infinite set, it may be impractical or impossible to actually list all of the ordered pairs that belong to the function, or to display the function in a table. Such a function can often be defined by a verbal description of the “rule of correspondence” (for example, “to each real number corresponds its square”), that clearly specifies the element of the range that corresponds to each element of the domain. When the domain and range are sets of numbers, the algebraic and graphical analogs of the verbal description are the equation and graph, respectively. We will find it valuable to be able to view a particular function from multiple perspectives— algebraic (in terms of an equation), graphical (in terms of a graph), and numeric (in terms of a table). Both versions of the definition of function are quite general, with no restriction on the type of elements that make up the domain or range, and no restriction on the numbers of elements in the domain or range. We are primarily interested, however, in functions with real number domains and ranges. In this text, unless otherwise indicated, the domain and range of a function will be sets of real numbers. For such a function we often use an equation in two variables to specify both the rule of correspondence and the set of ordered pairs. Consider the equation y  x2  2x

x any real number

(1)

This equation assigns to each domain value x exactly one range value y. For example, If x  4, If x  13,

then then

y  (4)2  2(4)  24 y  (13)2  2(13)  59

Thus, we can view equation (1) as a function with rule of correspondence y  x2  2x

x corresponds to x2  2x

or, equivalently, as a function with set of ordered pairs 5(x, y) ƒ y  x2  2x, x a real number6 The variable x is called an independent variable, indicating that values can be assigned “independently” to x from the domain. The variable y is called a dependent variable, indicating that the value of y “depends” on the value assigned to x and on the given equation. In general, any variable used as a placeholder for domain values is called an independent variable; any variable used as a placeholder for range values is called a dependent variable. Which equations can be used to define functions?

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Z FUNCTIONS DEFINED BY EQUATIONS In an equation in two variables, if to each value of the independent variable there corresponds exactly one value of the dependent variable, then the equation defines a function. If there is any value of the independent variable to which there corresponds more than one value of the dependent variable, then the equation does not define a function.

Notice that we have used the phrase “an equation defines a function” rather than “an equation is a function.”