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Precalculus: Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

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This bestselling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking. Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. The authors' attention to detail and clarity--the same as found in James Stewart's market-leading Calculus text--is what makes this text the proven market leader.

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          exponents and radicals x 5 x m2n xn 1 x 2n 5 n x x n xn a b 5 n y y  x m x n 5 x m1n 1 x m 2 n 5 x mn  1 xy2 n 5 x n y n  n x 1/n 5 ! x n  n  n m n x m/n 5 ! x 5 1! x2 m  n  !xy 5 !x !y m  geometric formulas m  n m  n x ! x 5 n Åy !y n  mn  n "! x 5 " !x 5 !x  special products 1 x 1 y 2 2 5 x2 1 2 x y 1 y2 1 x 2 y 2 2 5 x2 2 2 x y 1 y2  Formulas for area A, perimeter P, circumference C, volume V : Rectangle  Box  A 5 l„  V 5 l„ h  P 5 2l 1 2„  h  „  Triangle  Pyramid  A 5 12 bh  1 x 1 y 2 3 5 x 3 1 3x 2 y 1 3x y 2 1 y 3  h  FACtORING formulas  x 2 1 2xy 1 y 2 5 1 x 1 y 2 2 2  2  x 2 2xy 1 y 5 1 x 2 y 2  2  V 5 13 ha 2  h  1 x 2 y 2 3 5 x 3 2 3x 2 y 1 3x y 2 2 y 3 x2 2 y2 5 1 x 1 y 2 1 x 2 y 2  „  l  l  a  a  b  Circle  Sphere 2  V 5 43 pr 3  C 5 2pr  A 5 4pr 2  A 5 pr  x 3 1 y 3 5 1 x 1 y 2 1 x 2 2 xy 1 y 2 2 x 3 2 y 3 5 1 x 2 y 2 1 x 2 1 xy 1 y 2 2  r  r  QUADRATIC FORMULA If ax 2 1 bx 1 c 5 0, then x5  2b 6 "b 2 2 4ac 2a  inequalities and absolute value  Cylinder  Cone V 5 13 pr 2h  2  V 5 pr h r h  h r  If a , b and b , c, then a , c. If a , b, then a 1 c , b 1 c. If a , b and c . 0, then ca , cb. If a , b and c , 0, then ca . cb.  heron's formula  If a . 0, then 0 x 0 5 a  means  x 5 a  or  x 5 2a. 0 x 0 , a  means  2a , x , a. 0 x 0 . a  means  x . a  or  x , 2a.  B  Area 5 !s1s 2 a2 1s 2 b2 1s 2 c2 a1b1c where s 5 2  c A  a b  C  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  distance and midpoint formulas  Graphs of Functions  Distance between P1 1 x 1 , y 1 2 and P2 1 x 2 , y 2 2 :  Linear functions:   f1x2 5 mx 1 b y  d 5 "1 x2 2 x1 2 2 1 1y2 2 y1 2 2  Midpoint of P1P2:;    a lines  x1 1 x2 y1 1 y2 , b 2 2  y  b b x  x  Ï=b  y2 2 y1 m5 x2 2 x1  Slope of line through P1 1 x 1 , y 1 2 and P2 1 x 2 , y 2 2       Ï=mx+b  Power functions:   f1x2 5 x n y 2 y1 5 m 1 x 2 x1 2  Point-slope equation of line through P1 1 x 1, y 1 2 with slope m  Slope-intercept equation of line with slope m and y-intercept b  y 5 mx 1 b  Two-intercept equation of line with x-intercept a and y-intercept b  y x 1 51 a b  y  y  x  Ï=≈  x        n Root functions:   f1x2 5 ! x  logarithms  y  y  y 5 log a x  means  a y 5 x log a a x 5 x  a log a x 5 x  log a 1 5 0  log a a 5 1  log x 5 log 10 x  ln x 5 log e x  log a x y 5 log a x 1 log a y  log a a}x}b 5 log a x 2 log a y y loga x log b x 5 loga b  log a x b 5 b log a x  Ï=x£  x  Ï=œ∑ x  x        Ï=£œx ∑  Reciprocal functions:   f1x2 5 1/x n y  y  exponential and logarithmic functions y  y  y=a˛ a>1 1 0 y  Ï=  1 0  x y  y=log a x a>1  x  x  y=a˛ 0<a<1  x  1 x        Absolute value function  1 ≈  Greatest integer function  y  y=log a x 0<a<1  Ï=  y  1  0  1  x  0  1  1  x  x  Ï=|x |        Ï=" x'  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  x  Complex Numbers  polar coordinates y  For the complex number z 5 a 1 bi   the modulus is 0 z 0 5 "a2 1 b 2  r    the argument is u, where tan u 5 b/a Im bi  0  a+bi  | z|  ¨  y 5 r sin u r2 5 x2 1 y2 y tan u 5 x  y x  x  Sums of powers of integers  ¨  0  x 5 r cos u  P (x, y) P (r, ¨)    the conjugate is z 5 a 2 bi  a  Re  a15n  ak5  k51  2 ak 5  3 ak 5  n  n  k51  Polar form of a complex number  n  For z 5 a 1 bi, the polar form is  k51  z 5 r 1 cos u 1 i sin u2  where r 5 0 z 0 is the modulus of z and u is the argument of z  De Moivre's Theorem  zn 5 3 r 1 cos u 1 i sin u2 4 n 5 r n 1 cos nu 1 i sin nu2 !z 5 3 r 1 cos u 1 i sin u 24 n  5r  1/n  1/n  u 1 2kp u 1 2kp b a cos 1 i sin n n  n1n 1 12 12n 1 12  n  6  k51  b2a  The derivative of f at a is fr1a2 5 lim xSa  rotation of axes  f 1 x2 2 f 1 a2 x2a  f 1 a 1 h2 2 f 1 a2 h  area under the graph of f  P (x, y) P (X, Y )  The area under the graph of f on the interval 3a, b4 is the limit of the sum of the areas of approximating rectangles  X  0  4  f 1 b2 2 f 1 a2  hS0  ƒ  n 1n 1 12 2  The average rate of change of f between a and b is  fr1a2 5 lim y  2 2  the derivative  where k 5 0, 1, 2, . . . , n 2 1  Y  n1n 1 12  A 5 lim a f 1xk 2 Dx nS` n  k51  x  where  Dx 5 Rotation of axes formulas x 5 X cos f 2 Y sin f     y 5 X sin f 1 Y cos f  b2a n  xk 5 a 1 k Dx y  Îx  Angle-of-rotation formula for conic sections To eliminate the xy-term in the equation f(xk)  Ax 2 1 Bxy 1 Cy 2 1 Dx 1 Ey 1 F 5 0 rotate the axis by the angle f that satisfies A2C cot 2f 5 }} B  0  a  x⁄  x¤  x‹  x k-1 x k  b  x  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  seventh edition  Precalculus  mathematics for calculus  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  about the authors  J ames S tewart received his MS  L othar R edlin grew up on Van-  S aleem W atson received his  from Stanford University and his PhD from the University of Toronto. He did research at the University of London and was influenced by the famous mathematician George Polya at Stanford University. Stewart is Professor Emeritus at McMaster University and is currently Professor of Mathematics at the University of Toronto. His research field is harmonic analysis and the connections between mathematics and music. James Stewart is the author of a bestselling calculus textbook series published by Cengage Learning, including Calculus, Calculus: Early Transcendentals, and Calculus: Concepts and Contexts; a series of precalculus texts; and a series of highschool mathematics textbooks.  couver Island, received a Bachelor of Science degree from the University of Victoria, and received a PhD from McMaster University in 1978. He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach. He is currently Professor of Mathematics at The Pennsylvania State University, Abington Campus. His research field is topology.  Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his PhD in 1978. He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland. He also taught at The Pennsylvania State University. He is currently Professor of Mathematics at California State University, Long Beach. His research field is functional analysis.  Stewart, Redlin, and Watson have also published College Algebra, Trigonometry, Algebra and Trigonometry, and (with Phyllis Panman) College Algebra: Concepts and Contexts.  A bout  the  C over  The cover photograph shows a bridge in Valencia, Spain, designed by the Spanish architect Santiago Calatrava. The bridge leads to the Agora Stadium, also designed by Calatrava, which was completed in 2009 to host the Valencia Open tennis tournament. Calatrava has always been very interested in how mathematics can help him realize the buildings he imagines. As a young student, he taught himself descriptive geometry from  books in order to represent three-dimensional objects in two dimensions. Trained as both an engineer and an architect, he wrote a doctoral thesis in 1981 entitled "On the Foldability of Space Frames," which is filled with mathematics, especially geometric transformations. His strength as an engineer enables him to be daring in his architecture.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SEVENTH edition  Precalculus  mathematics for calculus  James Stewart M c Master University and University of Toronto  Lothar Redlin The Pennsylvania State University  Saleem Watson California State University, Long Beach  With the assistance of Phyllis Panman  Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Precalculus: Mathematics for Calculus, Seventh Edition James Stewart, Lothar Redlin, Saleem Watson Product Director: Richard Stratton Product Manager: Gary Whalen Content Developer: Stacy Green Associate Content Developer: Samantha Lugtu Product Assistant: Katharine Werring  © 2016, 2012 Cengage Learning WCN: 02-200-203  ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher.  Media Developer: Lynh Pham Senior Marketing Manager: Mark Linton Content Project Manager: Jennifer Risden Art Director: Vernon Boes Manufacturing Planner: Rebecca Cross Production Service: Martha Emry BookCraft Photo Researcher: Lumina Datamatics Ltd. Text Researcher: Lumina Datamatics Ltd. Copy Editor: Barbara Willette Illustrator: Precision Graphics; Graphic World, Inc. Text Designer: Diane Beasley Cover Designer: Cheryl Carrington Cover Image: AWL Images/Masterfile Compositor: Graphic World, Inc.  For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to permissionrequest@cengage.com.  Library of Congress Control Number: 2014948805 Student Edition: ISBN: 978-1-305-07175-9 Loose-leaf Edition: ISBN: 978-1-305-58602-4 Cengage Learning 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Cengage Learning Solutions, visit www.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com.  Printed in the United States of America Print Number: 01 Print Year: 2014  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  contents  Preface x To the Student xvii Prologue: Principles of Problem Solving P1  		  chapter 1  Fundamentals  1  		 Chapter Overview 1 1.1 Real Numbers 2 1.2 Exponents and Radicals 13 1.3 Algebraic Expressions 25 1.4 Rational Expressions 36 1.5 Equations 45 1.6 Complex Numbers 59 1.7 Modeling with Equations 65 1.8 Inequalities 81 1.9 The Coordinate Plane; Graphs of Equations; Circles 92 1.10 Lines 106 1.11 Solving Equations and Inequalities Graphically 117 1.12 Modeling Variation 122 Chapter 1 Review 130 Chapter 1 Test 137 ■  		  chapter 2  FOCUS ON MODELING Fitting Lines to Data 139  Functions  147  		 Chapter Overview 147 2.1 Functions 148 2.2 Graphs of Functions 159 2.3 Getting Information from the Graph of a Function 170 2.4 Average Rate of Change of a Function 183 2.5 Linear Functions and Models 190 2.6 Transformations of Functions 198 2.7 Combining Functions 210 2.8 One-to-One Functions and Their Inverses 219 		 Chapter 2 Review 229 		 Chapter 2 Test 235 ■  FOCUS ON MODELING Modeling with Functions 237  v  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  vi Contents  		  chapter 3  Polynomial and Rational Functions  245  		 Chapter Overview 245 3.1 Quadratic Functions and Models 246 3.2 Polynomial Functions and Their Graphs 254 3.3 Dividing Polynomials 269 3.4 Real Zeros of Polynomials 275 3.5 Complex Zeros and the Fundamental Theorem of Algebra 287 3.6 Rational Functions 295 3.7 Polynomial and Rational Inequalities 311 		 Chapter 3 Review 317 		 Chapter 3 Test 323 ■  		  chapter 4  FOCUS ON MODELING Fitting Polynomial Curves to Data 325  Exponential and Logarithmic Functions  329  		 Chapter Overview 329 4.1 Exponential Functions 330 4.2 The Natural Exponential Function 338 4.3 Logarithmic Functions 344 4.4 Laws of Logarithms 354 4.5 Exponential and Logarithmic Equations 360 4.6 Modeling with Exponential Functions 370 4.7 Logarithmic Scales 381 		 Chapter 4 Review 386 		 Chapter 4 Test 391 ■  FOCUS ON MODELING Fitting Exponential and Power Curves to Data 392 Cumulative Review Test: Chapters 2, 3, and 4 (Website)  		  chapter 5  Trigonometric Functions: Unit Circle Approach 401  		 Chapter Overview 401 5.1 The Unit Circle 402 5.2 Trigonometric Functions of Real Numbers 409 5.3 Trigonometric Graphs 419 5.4 More Trigonometric Graphs 432 5.5 Inverse Trigonometric Functions and Their Graphs 439 5.6 Modeling Harmonic Motion 445 		 Chapter 5 Review 460 		 Chapter 5 Test 465 ■  FOCUS ON MODELING Fitting Sinusoidal Curves to Data 466  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Contents vii  		  chapter 6  Trigonometric Functions: Right Triangle Approach  471  		 Chapter Overview 471 6.1 Angle Measure 472 6.2 Trigonometry of Right Triangles 482 6.3 Trigonometric Functions of Angles 491 6.4 Inverse Trigonometric Functions and Right Triangles 501 6.5 The Law of Sines 508 6.6 The Law of Cosines 516 		 Chapter 6 Review 524 		 Chapter 6 Test 531 ■  		  chapter 7  FOCUS ON MODELING Surveying 533  Analytic Trigonometry  537  		 Chapter Overview 537 7.1 Trigonometric Identities 538 7.2 Addition and Subtraction Formulas 545 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas 553 7.4 Basic Trigonometric Equations 564 7.5 More Trigonometric Equations 570 		 Chapter 7 Review 576 		 Chapter 7 Test 580 ■  FOCUS ON MODELING Traveling and Standing Waves 581 Cumulative Review Test: Chapters 5, 6, and 7 (Website)  		  chapter 8  Polar Coordinates and Parametric Equations  		 Chapter Overview 587 8.1 Polar Coordinates 588 8.2 Graphs of Polar Equations 594 8.3 Polar Form of Complex Numbers; De Moivre's Theorem 602 8.4 Plane Curves and Parametric Equations 611 		 Chapter 8 Review 620 		 Chapter 8 Test 624 ■  FOCUS ON MODELING The Path of a Projectile 625  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  587  viii Contents  		  chapter 9  Vectors in Two and Three Dimensions  629  		 Chapter Overview 629 9.1 Vectors in Two Dimensions 630 9.2 The Dot Product 639 9.3 Three-Dimensional Coordinate Geometry 647 9.4 Vectors in Three Dimensions 653 9.5 The Cross Product 659 9.6 Equations of Lines and Planes 666 		 Chapter 9 Review 670 		 Chapter 9 Test 675 ■  FOCUS ON MODELING Vector Fields 676 Cumulative Review Test: Chapters 8 and 9 (Website)  		  chapter 10  Systems of Equations and Inequalities  679  		 Chapter Overview 679 10.1 Systems of Linear Equations in Two Variables 680 10.2 Systems of Linear Equations in Several Variables 690 10.3 Matrices and Systems of Linear Equations 699 10.4 The Algebra of Matrices 712 10.5 Inverses of Matrices and Matrix Equations 724 10.6 Determinants and Cramer's Rule 734 10.7 Partial Fractions 745 10.8 Systems of Nonlinear Equations 751 10.9 Systems of Inequalities 756 		 Chapter 10 Review 766 		 Chapter 10 Test 773 ■  		  chapter 11  FOCUS ON MODELING Linear Programming 775  Conic Sections  781  		 Chapter Overview 781 11.1 Parabolas 782 11.2 Ellipses 790 11.3 Hyperbolas 799 11.4 Shifted Conics 807 11.5 Rotation of Axes 816 11.6 Polar Equations of Conics 824 		 Chapter 11 Review 831 		 Chapter 11 Test 835 ■  FOCUS ON MODELING Conics in Architecture 836 Cumulative Review Test: Chapters 10 and 11 (Website)  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Contents ix  		  chapter 12  Sequences and Series  841  		 Chapter Overview 841 12.1 Sequences and Summation Notation 842 12.2 Arithmetic Sequences 853 12.3 Geometric Sequences 858 12.4 Mathematics of Finance 867 12.5 Mathematical Induction 873 12.6 The Binomial Theorem 879 		 Chapter 12 Review 887 		 Chapter 12 Test 892 ■  		  chapter 13  FOCUS ON MODELING Modeling with Recursive Sequences 893  Limits: A Preview of Calculus  		 Chapter Overview 897 13.1 Finding Limits Numerically and Graphically 898 13.2 Finding Limits Algebraically 906 13.3 Tangent Lines and Derivatives 914 13.4 Limits at Infinity; Limits of Sequences 924 13.5 Areas 931 		 Chapter 13 Review 940 		 Chapter 13 Test 943 ■  FOCUS ON MODELING Interpretations of Area 944 Cumulative Review Test: Chapters 12 and 13 (Website) APPENDIX A Geometry Review 949 APPENDIX B Calculations and Significant Figures (Website) APPENDIX C Graphing with a Graphing Calculator (Website) APPENDIX D Using the TI-83/84 Graphing Calculator (Website) ANSWERS A1 INDEX I1  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  897  PREFACE What do students really need to know to be prepared for calculus? What tools do instructors really need to assist their students in preparing for calculus? These two questions have motivated the writing of this book. To be prepared for calculus a student needs not only technical skill but also a clear understanding of concepts. Indeed, conceptual understanding and technical skill go hand in hand, each reinforcing the other. A student also needs to gain an appreciation for the power and utility of mathematics in modeling the real world. Every feature of this textbook is devoted to fostering these goals. In this Seventh Edition our objective is to further enhance the effectiveness of the book as an instructional tool for teachers and as a learning tool for students. Many of the changes in this edition are a result of suggestions we received from instructors and students who are using the current edition; others are a result of insights we have gained from our own teaching. Some chapters have been reorganized and rewritten, new sections have been added (as described below), the review material at the end of each chapter has been substantially expanded, and exercise sets have been enhanced to further focus on the main concepts of precalculus. In all these changes and numerous others (small and large) we have retained the main features that have contributed to the success of this book.  New to the Seventh Edition ■  ■  ■  ■  ■  ■  ■  Exercises More than 20% of the exercises are new, and groups of exercises now have headings that identify the type of exercise. New Skills Plus exercises in most sections contain more challenging exercises that require students to extend and synthesize concepts. Review Material The review material at the end of each chapter now includes a summary of Properties and Formulas and a new Concept Check. Each Concept Check provides a step-by-step review of all the main concepts and applications of the chapter. Answers to the Concept Check questions are on tear-out sheets at the back of the book. Discovery Projects References to Discovery Projects, including brief descriptions of the content of each project, are located in boxes where appropriate in each chapter. These boxes highlight the applications of precalculus in many different real-world contexts. (The projects are located at the book companion website: www.stewartmath.com.) Geometry Review A new Appendix A contains a review of the main concepts of geometry used in this book, including similarity and the Pythagorean Theorem. CHAPTER 1 Fundamentals This chapter now contains two new sections. Section 1.6, "Complex Numbers" (formerly in Chapter 3), has been moved here. Section 1.12, "Modeling Variation," is now also in this chapter. CHAPTER 2 Functions This chapter now includes the new Section 2.5, "Linear Functions and Models." This section highlights the connection between the slope of a line and the rate of change of a linear function. These two interpretations of slope help prepare students for the concept of the derivative in calculus. CHAPTER 3 Polynomial and Rational Functions This chapter now includes the new Section 3.7, "Polynomial and Rational Inequalities." Section 3.6, "Rational Functions," has a new subsection on rational functions with "holes." The sections on complex numbers and on variation have been moved to Chapter 1.  x  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Preface xi ■  ■  ■  CHAPTER 4 Exponential and Logarithmic Functions The chapter now includes two sections on the applications of these functions. Section 4.6, "Modeling with Exponential Functions," focuses on modeling growth and decay, Newton's Law of Cooling, and other such applications. Section 4.7, "Logarithmic Scales," covers the concept of a logarithmic scale with applications involving the pH, Richter, and decibel scales. CHAPTER 5 Trigonometric Functions: Unit Circle Approach This chapter includes a new subsection on the concept of phase shift as used in modeling harmonic motion. CHAPTER 10 Systems of Equations and Inequalities The material on systems of inequalities has been rewritten to emphasize the steps used in graphing the solution of a system of inequalities.  Teaching with the Help of This Book We are keenly aware that good teaching comes in many forms and that there are many different approaches to teaching and learning the concepts and skills of precalculus. The organization and exposition of the topics in this book are designed to accommodate different teaching and learning styles. In particular, each topic is presented algebraically, graphically, numerically, and verbally, with emphasis on the relationships between these different representations. The following are some special features that can be used to complement different teaching and learning styles:  Exercise Sets The most important way to foster conceptual understanding and hone technical skill is through the problems that the instructor assigns. To that end we have provided a wide selection of exercises. ■  ■  ■  ■  ■  ■  ■  Concept Exercises  These exercises ask students to use mathematical language to state fundamental facts about the topics of each section. Skills Exercises These exercises reinforce and provide practice with all the learning objectives of each section. They comprise the core of each exercise set. Skills Plus Exercises The Skills Plus exercises contain challenging problems that often require the synthesis of previously learned material with new concepts. Applications Exercises We have included substantial applied problems from many different real-world contexts. We believe that these exercises will capture students' interest. Discovery, Writing, and Group Learning Each exercise set ends with a block of exercises labeled Discuss ■ Discover ■ Prove ■ Write. These exercises are designed to encourage students to experiment, preferably in groups, with the concepts developed in the section and then to write about what they have learned rather than simply looking for the answer. New Prove exercises highlight the importance of deriving a formula. Now Try Exercise . . . At the end of each example in the text the student is directed to one or more similar exercises in the section that help to reinforce the concepts and skills developed in that example. Check Your Answer Students are encouraged to check whether an answer they obtained is reasonable. This is emphasized throughout the text in numerous Check Your Answer sidebars that accompany the examples (see, for instance, pages 54 and 71).  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xii Preface  A Complete Review Chapter We have included an extensive review chapter primarily as a handy reference for the basic concepts that are preliminary to this course. ■  ■  Chapter 1 Fundamentals This is the review chapter; it contains the fundamental concepts from algebra and analytic geometry that a student needs in order to begin a precalculus course. As much or as little of this chapter can be covered in class as needed, depending on the background of the students. Chapter 1 Test The test at the end of Chapter 1 is designed as a diagnostic test for determining what parts of this review chapter need to be taught. It also serves to help students gauge exactly what topics they need to review.  Flexible Approach to Trigonometry The trigonometry chapters of this text have been written so that either the right triangle approach or the unit circle approach may be taught first. Putting these two approaches in different chapters, each with its relevant applications, helps to clarify the purpose of each approach. The chapters introducing trigonometry are as follows. ■  ■  Chapter 5 Trigonometric Functions: Unit Circle Approach This chapter introduces trigonometry through the unit circle approach. This approach emphasizes that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar. Chapter 6 Trigonometric Functions: Right Triangle Approach This chapter introduces trigonometry through the right triangle approach. This approach builds on the foundation of a conventional high-school course in trigonometry.  Another way to teach trigonometry is to intertwine the two approaches. Some instructors teach this material in the following order: Sections 5.1, 5.2, 6.1, 6.2, 6.3, 5.3, 5.4, 5.5, 5.6, 6.4, 6.5, and 6.6. Our organization makes it easy to do this without obscuring the fact that the two approaches involve distinct representations of the same functions.  Graphing Calculators and Computers We make use of graphing calculators and computers in examples and exercises throughout the book. Our calculator-oriented examples are always preceded by examples in which students must graph or calculate by hand so that they can understand precisely what the calculator is doing when they later use it to simplify the routine, mechanical part of their work. The graphing calculator sections, subsections, examples, and exercises, all marked with the special symbol , are optional and may be omitted without loss of continuity. ■  ■  ■  Using a Graphing Calculator General guidelines on using graphing calculators and a quick reference guide to using TI-83/84 calculators are available at the book companion website: www.stewartmath.com. Graphing, Regression, Matrix Algebra Graphing calculators are used throughout the text to graph and analyze functions, families of functions, and sequences; to calculate and graph regression curves; to perform matrix algebra; to graph linear inequalities; and other powerful uses. Simple Programs We exploit the programming capabilities of a graphing calculator to simulate real-life situations, to sum series, or to compute the terms of a recursive sequence (see, for instance, pages 628, 896, and 939).  Focus on Modeling The theme of modeling has been used throughout to unify and clarify the many applications of precalculus. We have made a special effort to clarify the essential process of translating problems from English into the language of mathematics (see pages 238 and 686). ■  Constructing Models There are many applied problems throughout the book in which students are given a model to analyze (see, for instance, page 250). But the material on modeling, in which students are required to construct mathematical models, has been organized into clearly defined sections and subsections (see, for instance, pages 370, 445, and 685).  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Preface xiii ■  Focus on Modeling Each chapter concludes with a Focus on Modeling section. The first such section, after Chapter 1, introduces the basic idea of modeling a real-life situation by fitting lines to data (linear regression). Other sections pre­ sent ways in which polynomial, exponential, logarithmic, and trigonometric functions, and systems of inequalities can all be used to model familiar phenomena from the sciences and from everyday life (see, for instance, pages 325, 392, and 466).  Review Sections and Chapter Tests Each chapter ends with an extensive review section that includes the following. ■  ■  ■  ■  ■  ■  Properties and Formulas The Properties and Formulas at the end of each chapter contains a summary of the main formulas and procedures of the chapter (see, for instance, pages 386 and 460). Concept Check and Concept Check Answers The Concept Check at the end of each chapter is designed to get the students to think about and explain each concept presented in the chapter and then to use the concept in a given problem. This provides a step-by-step review of all the main concepts in a chapter (see, for instance, pages 230, 319, and 769). Answers to the Concept Check questions are on tear-out sheets at the back of the book. Review Exercises The Review Exercises at the end of each chapter recapitulate the basic concepts and skills of the chapter and include exercises that combine the different ideas learned in the chapter. Chapter Test Each review section concludes with a Chapter Test designed to help students gauge their progress. Cumulative Review Tests Cumulative Review Tests following selected chapters are available at the book companion website. These tests contain problems that combine skills and concepts from the preceding chapters. The problems are designed to highlight the connections between the topics in these related chapters. Answers Brief answers to odd-numbered exercises in each section (including the review exercises) and to all questions in the Concepts exercises and Chapter Tests, are given in the back of the book.  Mathematical Vignettes Throughout the book we make use of the margins to provide historical notes, key insights, or applications of mathematics in the modern world. These serve to enliven the material and show that mathematics is an important, vital activity and that even at this elementary level it is fundamental to everyday life. ■  ■  Mathematical Vignettes These vignettes include biographies of interesting mathematicians and often include a key insight that the mathematician discovered (see, for instance, the vignettes on Viète, page 50; Salt Lake City, page 93; and radiocarbon dating, page 367). Mathematics in the Modern World This is a series of vignettes that emphasize the central role of mathematics in current advances in technology and the sciences (see, for instance, pages 302, 753, and 784).  Book Companion Website A website that accompanies this book is located at www.stewartmath.com. The site includes many useful resources for teaching precalculus, including the following. ■  Discovery Projects Discovery Projects for each chapter are available at the book companion website. The projects are referenced in the text in the appropriate sections. Each project provides a challenging yet accessible set of activities that enable students (perhaps working in groups) to explore in greater depth an interesting aspect of the topic they have just learned (see, for instance, the Discovery Projects Visualizing a Formula, Relations and Functions, Will the Species Survive?, and Computer Graphics I and II, referenced on pages 29, 163, 719, 738, and 820).  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xiv Preface ■  ■  ■  ■  ■  Focus on Problem Solving Several Focus on Problem Solving sections are available on the website. Each such section highlights one of the problem-solving principles introduced in the Prologue and includes several challenging problems (see for instance Recognizing Patterns, Using Analogy, Introducing Something Extra, Taking Cases, and Working Backward). Cumulative Review Tests Cumulative Review Tests following Chapters 4, 7, 9, 11, and 13 are available on the website. Appendix B: Calculations and Significant Figures This appendix, available at the book companion website, contains guidelines for rounding when working with approximate values. Appendix C: Graphing with a Graphing Calculator This appendix, available at the book companion website, includes general guidelines on graphing with a graphing calculator as well as guidelines on how to avoid common graphing pitfalls. Appendix D: Using the TI-83/84 Graphing Calculator In this appendix, available at the book companion website, we provide simple, easy-to-follow, step-by-step instructions for using the TI-83/84 graphing calculators.  Acknowledgments We feel fortunate that all those involved in the production of this book have worked with exceptional energy, intense dedication, and passionate interest. It is surprising how many people are essential in the production of a mathematics textbook, including content editors, reviewers, faculty colleagues, production editors, copy editors, permissions editors, solutions and accuracy checkers, artists, photo researchers, text designers, typesetters, compositors, proofreaders, printers, and many more. We thank them all. We particularly mention the following.  Reviewers for the Sixth Edition Raji Baradwaj, UMBC; Chris Herman, Lorain County Community College; Irina Kloumova, Sacramento City College; Jim McCleery, Skagit Valley College, Whidbey Island Campus; Sally S. Shao, Cleveland State University; David Slutzky, Gainesville State College; Edward Stumpf, Central Carolina Community College; Ricardo Teixeira, University of Texas at Austin; Taixi Xu, Southern Polytechnic State University; and Anna Wlodarczyk, Florida International University. Reviewers for the Seventh Edition Mary Ann Teel, University of North Texas; Natalia Kravtsova, The Ohio State University; Belle Sigal, Wake Technical Community College; Charity S. Turner, The Ohio State University; Yu-ing Hargett, Jefferson State Community College–Alabama; Alicia Serfaty de Markus, Miami Dade College; Cathleen Zucco-Teveloff, Rider University; Minal Vora, East Georgia State College; Sutandra Sarkar, Georgia State University; Jennifer Denson, Hillsborough Community College; Candice L. Ridlon, University of Maryland Eastern Shore; Alin Stancu, Columbus State University; Frances Tishkevich, Massachusetts Maritime Academy; Phil Veer, Johnson County Community College; Cathleen Zucco-Teveloff, Rider University; Phillip Miller, Indiana University–Southeast; Mildred Vernia, Indiana University– Southeast; Thurai Kugan, John Jay College–CUNY. We are grateful to our colleagues who continually share with us their insights into teaching mathematics. We especially thank Robert Mena at California State University, Long Beach; we benefited from his many insights into mathematics and its history. We thank Cecilia McVoy at Penn State Abington for her helpful suggestions. We thank Andrew Bulman-Fleming for writing the Solutions Manual and Doug Shaw at the University of Northern Iowa for writing the Instructor Guide and the Study Guide. We are very grateful to Frances Gulick at the University of Maryland for checking the accuracy of the entire manuscript and doing each and every exercise; her many suggestions and corrections have contributed greatly to the accuracy and consistency of the contents of this book.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Preface xv  We thank Martha Emry, our production service and art editor; her energy, devotion, and experience are essential components in the creation of this book. We are grateful for her remarkable ability to instantly recall, when needed, any detail of the entire manuscript as well as her extraordinary ability to simultaneously manage several interdependent editing tracks. We thank Barbara Willette, our copy editor, for her attention to every detail in the manuscript and for ensuring a consistent, appropriate style throughout the book. We thank our designer, Diane Beasley, for the elegant and appropriate design for the interior of the book. We thank Graphic World for their attractive and accurate graphs and Precision Graphics for bringing many of our illustrations to life. We thank our compositors at Graphic World for ensuring a balanced and coherent look for each page of the book. At Cengage Learning we thank Jennifer Risden, content project manager, for her professional management of the production of the book. We thank Lynh Pham, media developer, for his expert handling of many technical issues, including the creation of the book companion website. We thank Vernon Boes, art director, for his capable administration of the design of the book. We thank Mark Linton, marketing manager, for helping bring the book to the attention of those who may wish to use it in their classes. We particularly thank our developmental editor, Stacy Green, for skillfully guiding and facilitating every aspect of the creation of this book. Her interest in the book, her familiarity with the entire manuscript, and her almost instant responses to our many queries have made the writing of the book an even more enjoyable experience for us. Above all we thank our acquisitions editor, Gary Whalen. His vast editorial experience, his extensive knowledge of current issues in the teaching of mathematics, his skill in managing the resources needed to enhance this book, and his deep interest in mathematics textbooks have been invaluable assets in the creation of this book.  Ancillaries Instructor Resources Instructor Companion Site Everything you need for your course in one place! This collection of book-specific lecture and class tools is available online via www.cengage.com/login. Access and download PowerPoint presentations, images, instructor's manual, and more. Complete Solutions Manual The Complete Solutions Manual provides worked-out solutions to all of the problems in the text. Located on the companion website. Test Bank The Test Bank provides chapter tests and final exams, along with answer keys. Located on the companion website. Instructor's Guide The Instructor's Guide contains points to stress, suggested time to allot, text discussion topics, core materials for lecture, workshop/discussion suggestions, group work exercises in a form suitable for handout, and suggested homework problems. Located on the companion website. Lesson Plans The Lesson Plans provides suggestions for activities and lessons with notes on time allotment in order to ensure timeliness and efficiency during class. Located on the companion website. Cengage Learning Testing Powered by Cognero (ISBN-10: 1-305-25853-3; ISBN-13: 978-1-305-25853-2) CLT is a flexible online system that allows you to author, edit, and manage test bank content; create multiple test versions in an instant; and deliver tests from your LMS, your classroom or wherever you want. This is available online via www.cengage.com/login.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  xvi Preface Enhanced WebAssign Printed Access Card: 978-1-285-85833-3 Instant Access Code: 978-1-285-85831-9 Enhanced WebAssign combines exceptional mathematics content with the most powerful online homework solution, WebAssign®. Enhanced WebAssign engages students with immediate feedback, rich tutorial content, and an interactive, fully customizable eBook, Cengage YouBook, to help students to develop a deeper conceptual understanding of their subject matter.  Student Resources Student Solutions Manual (ISBN-10: 1-305-25361-2; ISBN-13: 978-1-305-25361-2) The Student Solutions Manual contains fully worked-out solutions to all of the oddnumbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. Study Guide (ISBN-10: 1-305-25363-9; ISBN-13: 978-1-305-25363-6) The Study Guide reinforces student understanding with detailed explanations, worked-out examples, and practice problems. It also lists key ideas to master and builds problemsolving skills. There is a section in the Study Guide corresponding to each section in the text. Note-Taking Guide (ISBN-10: 1-305-25383-3; ISBN-13: 978-1-305-25383-4) The Note-Taking Guide is an innovative study aid that helps students develop a sectionby-section summary of key concepts. Text-Specific DVDs (ISBN-10: 1-305-25400-7; ISBN-13: 978-1-305-25400-8) The Text-Specific DVDs include new learning objective–based lecture videos. These DVDs provide comprehensive coverage of the course—along with additional explanations of concepts, sample problems, and applications—to help students review essential topics. CengageBrain.com To access additional course materials, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found. Enhanced WebAssign Printed Access Card: 978-1-285-85833-3 Instant Access Code: 978-1-285-85831-9 Enhanced WebAssign combines exceptional mathematics content with the most powerful online homework solution, WebAssign. Enhanced WebAssign engages students with immediate feedback, rich tutorial content, and an interactive, fully customizable eBook, Cengage YouBook, helping students to develop a deeper conceptual understanding of the subject matter.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  TO THE STUDENT  This textbook was written for you to use as a guide to mastering precalculus mathematics. Here are some suggestions to help you get the most out of your course. First of all, you should read the appropriate section of text before you attempt your homework problems. Reading a mathematics text is quite different from reading a novel, a newspaper, or even another textbook. You may find that you have to reread a passage several times before you understand it. Pay special attention to the examples, and work them out yourself with pencil and paper as you read. Then do the linked exercises referred to in "Now Try Exercise . . ." at the end of each example. With this kind of preparation you will be able to do your homework much more quickly and with more understanding. Don't make the mistake of trying to memorize every single rule or fact you may come across. Mathematics doesn't consist simply of memorization. Mathematics is a problem-solving art, not just a collection of facts. To master the subject you must solve problems—lots of problems. Do as many of the exercises as you can. Be sure to write your solutions in a logical, step-by-step fashion. Don't give up on a problem if you can't solve it right away. Try to understand the problem more clearly—reread it thoughtfully and relate it to what you have learned from your teacher and from the examples in the text. Struggle with it until you solve it. Once you have done this a few times you will begin to understand what mathematics is really all about. Answers to the odd-numbered exercises, as well as all the answers (even and odd) to the concept exercises and chapter tests, appear at the back of the book. If your answer differs from the one given, don't immediately assume that you are wrong. There may be a calculation that connects the two answers and makes both correct. For example, if you get 1/1 !2 2 12 but the answer given is 1 1 !2, your answer is correct, because you can multiply both numerator and denominator of your answer by !2 1 1 to change it to the given answer. In rounding approximate answers, follow the guidelines in Appendix B: Calculations and Significant Figures. The symbol is used to warn against committing an error. We have placed this symbol in the margin to point out situations where we have found that many of our students make the same mistake.  Abbreviations The following abbreviations are used throughout the text. cm dB F ft g gal h H Hz in. J kcal kg km  centimeter decibel farad foot gram gallon hour henry Hertz inch Joule kilocalorie kilogram kilometer  kPa kilopascal L liter lb pound lm lumen M mole of solute 	 per liter of solution m meter mg milligram MHz megahertz mi mile min minute milliliter mL mm millimeter  N qt oz s V V W yd yr °C °F K ⇒ ⇔  Newton quart ounce second ohm volt watt yard year degree Celsius degree Fahrenheit Kelvin implies is equivalent to  xvii  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  prologue  principles of problem solving The ability to solve problems is a highly prized skill in many aspects of our lives; it is certainly an important part of any mathematics course. There are no hard and fast rules that will ensure success in solving problems. However, in this Prologue we outline some general steps in the problem-solving process and we give principles that are useful in solving certain types of problems. These steps and principles are just common sense made explicit. They have been adapted from George Polya's insightful book How To Solve It.  AP Images  1. Understand the Problem George Polya (1887–1985) is famous among mathematicians for his ideas on problem solving. His lectures on problem solving ­at Stanford University attracted overflow crowds whom he held on the edges of their seats, leading them to discover solutions for themselves. He was able to do this because of his deep insight into the psychology of problem solving. His well-known book How To Solve It has been translated into 15 languages. He said that Euler (see page 63) was unique among great mathematicians because he explained how he found his results. Polya often said to his students and colleagues, "Yes, I see that your proof is correct, but how did you discover it?" In the preface to How To Solve It, Polya writes, "A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery."  The first step is to read the problem and make sure that you understand it. Ask yourself the following questions: What is the unknown? What are the given quantities? What are the given conditions? For many problems it is useful to draw a diagram and identify the given and required quantities on the diagram. Usually, it is necessary to introduce suitable notation In choosing symbols for the unknown quantities, we often use letters such as a, b, c, m, n, x, and y, but in some cases it helps to use initials as suggestive symbols, for instance, V for volume or t for time.  2. Think of a Plan Find a connection between the given information and the unknown that enables you to calculate the unknown. It often helps to ask yourself explicitly: "How can I relate the given to the unknown?" If you don't see a connection immediately, the following ideas may be helpful in devising a plan. ■ Try to Recognize Something Familiar  Relate the given situation to previous knowledge. Look at the unknown and try to recall a more familiar problem that has a similar unknown. ■ Try to Recognize Patterns  Certain problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, numerical, or algebraic. If you can see regularity or repetition in a problem, then you might be able to guess what the pattern is and then prove it. ■ Use Analogy  Try to think of an analogous problem, that is, a similar or related problem but one that is easier than the original. If you can solve the similar, simpler problem, then it might give you the clues you need to solve the original, more difficult one. For instance, if a problem involves very large numbers, you could first try a similar problem with smaller numbers. Or if the problem is in three-dimensional geometry, you could look for something similar in two-dimensional geometry. Or if the problem you start with is a general one, you could first try a special case. P1  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  P2 Prologue ■ Introduce Something Extra  You might sometimes need to introduce something new—an auxiliary aid—to make the connection between the given and the unknown. For instance, in a problem for which a ­diagram is useful, the auxiliary aid could be a new line drawn in the diagram. In a more algebraic problem the aid could be a new unknown that relates to the original unknown. ■ Take Cases  You might sometimes have to split a problem into several cases and give a different argument for each case. For instance, we often have to use this strategy in dealing with absolute value. ■ Work Backward  Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you might be able to reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x  5  7, we suppose that x is a number that satisfies 3x  5  7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x  4. Since each of these steps can be reversed, we have solved the problem. ■ Establish Subgoals  In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulfilled). If you can attain or accomplish these subgoals, then you might be able to build on them to reach your final goal. ■ Indirect Reasoning  Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q, we assume that P is true and Q is false and try to see why this cannot happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true. ■ Mathematical Induction  In proving statements that involve a positive integer n, it is frequently helpful to use the Principle of Mathematical Induction, which is discussed in Section 12.5.  3. Carry Out the Plan In Step 2, a plan was devised. In carrying out that plan, you must check each stage of the plan and write the details that prove that each stage is correct.  4. Look Back Having completed your solution, it is wise to look back over it, partly to see whether any errors have been made and partly to see whether you can discover an easier way to solve the problem. Looking back also familiarizes you with the method of solution, which may be useful for solving a future problem. Descartes said, "Every problem that I solved became a rule which served afterwards to solve other problems." We illustrate some of these principles of problem solving with an example.  Problem  ■  Average Speed  A driver sets out on a journey. For the first half of the distance, she drives at the leisurely pace of 30 mi/h; during the second half she drives 60 mi/h. What is her average speed on this trip?  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Prologue P3  Thinking About the Problem It is tempting to take the average of the speeds and say that the average speed for the entire trip is 30  60  45 mi/h 2 Try a special case. ▶  But is this simple-minded approach really correct? Let's look at an easily calculated special case. Suppose that the total distance traveled is 120 mi. Since the first 60 mi is traveled at 30 mi/h, it takes 2 h. The second 60 mi is traveled at 60 mi/h, so it takes one hour. Thus, the total time is 2  1  3 hours and the average speed is 120  40 mi/h 3 So our guess of 45 mi/h was wrong. SOLUTION  Understand the problem. ▶  We need to look more carefully at the meaning of average speed. It is defined as average speed   Introduce notation. ▶ State what is given. ▶  distance traveled time elapsed  Let d be the distance traveled on each half of the trip. Let t1 and t2 be the times taken for the first and second halves of the trip. Now we can write down the information we have been given. For the first half of the trip we have 30   d t1  60   d t2  and for the second half we have  Identify the unknown. ▶  Now we identify the quantity that we are asked to find: average speed for entire trip   Connect the given with the unknown. ▶  total distance 2d  total time t1  t2  To calculate this quantity, we need to know t1 and t2, so we solve the above equations for these times: t1   d 30  t2   d 60  Now we have the ingredients needed to calculate the desired quantity: average speed       2d 2d  t1  t2 d d  30 60 601 2d 2    Multiply numerator and d d denominator by 60 60 a  b 30 60  120d 120d   40 2d  d 3d  So the average speed for the entire trip is 40 mi/h.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  ■  P4 Prologue  Problems 1. Distance, Time, and Speed   An old car has to travel a 2-mile route, uphill and down. Because it is so old, the car can climb the first mile—the ascent—no faster than an average speed of 15 mi/h. How fast does the car have to travel the second mile—on the descent it can go faster, of course—to achieve an average speed of 30 mi/h for the trip?  2. Comparing Discounts   Which price is better for the buyer, a 40% discount or two successive discounts of 20%? Bettmann/Corbis  3. Cutting up a Wire   A piece of wire is bent as shown in the figure. You can see that one cut through the wire produces four pieces and two parallel cuts produce seven pieces. How many pieces will be produced by 142 parallel cuts? Write a formula for the number of pieces produced by n parallel cuts.  Don't feel bad if you can't solve these problems right away. Problems 1 and 4 were sent to Albert Einstein by his friend Wertheimer. Einstein (and his friend Bucky) enjoyed the problems and wrote back to Wertheimer. Here is part of his reply: Your letter gave us a lot of amusement. The first intelligence test fooled both of us (Bucky and me). Only on working it out did I notice that no time is available for the downhill run! Mr. Bucky was also taken in by the second example, but I was not. Such drolleries show us how stupid we are! (See Mathematical Intelligencer, Spring 1990, page 41.)  4. Amoeba Propagation   An amoeba propagates by simple division; each split takes 3 minutes to complete. When such an amoeba is put into a glass container with a nutrient fluid, the container is full of amoebas in one hour. How long would it take for the container to be filled if we start with not one amoeba, but two?  5. Batting Averages   Player A has a higher batting average than player B for the first half of the baseball season. Player A also has a higher batting average than player B for the second half of the season. Is it necessarily true that player A has a higher batting average than player B for the entire season?  6. Coffee and Cream   A spoonful of cream is taken from a pitcher of cream and put into a cup of coffee. The coffee is stirred. Then a spoonful of this mixture is put into the pitcher of cream. Is there now more cream in the coffee cup or more coffee in the pitcher of cream?  7. Wrapping the World   A ribbon is tied tightly around the earth at the equator. How much more ribbon would you need if you raised the ribbon 1 ft above the equator everywhere? (You don't need to know the radius of the earth to solve this problem.)  8. Ending Up Where You Started   A woman starts at a point P on the earth's surface and walks 1 mi south, then 1 mi east, then 1 mi north, and finds herself back at P, the starting point. Describe all points P for which this is possible.   [Hint: There are infinitely many such points, all but one of which lie in Antarctica.]  Many more problems and examples that highlight different problem-solving principles are available at the book companion website: www.stewartmath.com. You can try them as you progress through the book.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  1 1.1 Real Numbers 1.2 Exponents and Radicals 1.3 Algebraic Expressions 1.4 Rational Expressions 1.5 Equations 1.6 Complex Numbers 1.7 Modeling with Equations 1.8 Inequalities 1.9	The Coordinate Plane; Graphs of Equations; Circles 1.10 Lines 1.11	Solving Equations and Inequalities Graphically 1.12	Modeling Variation  Blend Images/Alamy  Fundamentals In this first chapterwe review the real numbers, equations, and the coordinate plane. You are probably already familiar with these concepts, but it is helpful to get a fresh look at how these ideas work together to solve problems and model (or describe) real-world situations. In the Focus on Modeling at the end of the chapter we learn how to find linear trends in data and how to use these trends to make predictions about the future.  Focus on Modeling Fitting Lines to Data  1  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  2 CHAPTER 1  ■  Fundamentals  1.1 Real Numbers ■ Real Numbers ■ Properties of Real Numbers ■ Addition and Subtraction ■ Multiplication and Division ■ The Real Line ■ Sets and Intervals ■ Absolute Value and Distance  Count  © Aleph Studio/Shutterstock.com  © Oleksiy Mark/Shutterstock.com  © Monkey Business Images/ Shutterstock.com  © bikeriderlondon/Shutterstock.com  In the real world we use numbers to measure and compare different quantities. For example, we measure temperature, length, height, weight, blood pressure, distance, speed, acceleration, energy, force, angles, age, cost, and so on. Figure 1 illustrates some situations in which numbers are used. Numbers also allow us to express relationships between different quantities—for example, relationships between the radius and volume of a ball, between miles driven and gas used, or between education level and starting salary.  Length  Weight  Speed  Figure 1 Measuring with real numbers  ■ Real Numbers Let's review the types of numbers that make up the real number system. We start with the natural numbers: 1, 2, 3, 4, . . . The different types of real numbers were invented to meet specific needs. For example, natural numbers are needed for counting, negative numbers for describing debt or below-zero temperatures, rational numbers for concepts like "half a gallon of milk," and irrational numbers for measuring certain distances, like the diagonal of a square.  The integers consist of the natural numbers together with their negatives and 0: . . . , 3, 2, 1, 0, 1, 2, 3, 4, . . . We construct the rational numbers by taking ratios of integers. Thus any rational number r can be expressed as m r n where m and n are integers and n ? 0. Examples are 1 2  Rational numbers –21 , -–37 , 46, 0.17, 0.6, 0.317 Integers  Irrational numbers 3  3 œ3 , œ5 , œ2 , π , — 2 π  Figure 2 The real number system  46  461  17 0.17  100  (Recall that division by 0 is always ruled out, so expressions like 30 and 00 are undefined.) There are also real numbers, such as !2, that cannot be expressed as a ratio of integers and are therefore called irrational numbers. It can be shown, with varying degrees of difficulty, that these numbers are also irrational: !3  Natural numbers  . . . , 3, 2, 1, 0, 1, 2, 3, . . .  37  !5  3 ! 2  p  3 p2  The set of all real numbers is usually denoted by the symbol . When we use the word number without qualification, we will mean "real number." Figure 2 is a diagram of the types of real numbers that we work with in this book. Every real number has a decimal representation. If the number is rational, then its corresponding decimal is repeating. For example, 1 2 157 495   0.5000. . .  0.50     23  0.66666. . .  0.6   0.3171717. . .  0.317   97  1.285714285714. . .  1.285714  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SECTION 1.1 A repeating decimal such as x  3.5474747. . . is a rational number. To convert it to a ratio of two integers, we write 1000x  3547.47474747. . . 10x  35.47474747. . . 990x  3512.0 Thus x  3512 990 . (The idea is to multiply x by appropriate powers of 10 and then subtract to eliminate the repeating part.)  ■  Real Numbers 3  (The bar indicates that the sequence of digits repeats forever.) If the number is irrational, the decimal representation is nonrepeating: !2  1.414213562373095. . .  p  3.141592653589793. . .  If we stop the decimal expansion of any number at a certain place, we get an approximation to the number. For instance, we can write p  3.14159265 where the symbol  is read "is approximately equal to." The more decimal places we retain, the better our approximation.  ■ Properties of Real Numbers We all know that 2  3  3  2, and 5  7  7  5, and 513  87  87  513, and so on. In algebra we express all these (infinitely many) facts by writing abba where a and b stand for any two numbers. In other words, "a  b  b  a" is a concise way of saying that "when we add two numbers, the order of addition doesn't matter." This fact is called the Commutative Property of addition. From our experience with numbers we know that the properties in the following box are also valid.  Properties of Real Numbers Property  Example  Description  Commutative Properties abba  7337  When we add two numbers, order doesn't matter.  ab  ba Associative Properties 1 a  b2  c  a  1 b  c2 1 ab2 c  a1 bc2  Distributive Property  3 # 5  5 # 3	When we multiply two numbers, order doesn't matter. 1 2  42  7  2  1 4  72 	When we add three numbers, it doesn't matter which two we add first. 1 3 # 72 # 5  3 # 1 7 # 52 	When we multiply three numbers, it doesn't matter which two we multiply first.  a1 b  c2  ab  ac 2 # 1 3  52  2 # 3  2 # 5 When we multiply a number by a sum of two numbers, we get the same result as we get if we # # # 1 b  c2 a  ab  ac 1 3  52 2  2 3  2 5 			multiply the number by each of the terms and then add the results. The Distributive Property applies whenever we multiply a number by a sum. Figure 3 explains why this property works for the case in which all the numbers are positive integers, but the property is true for any real numbers a, b, and c. 2(3+5)  The Distributive Property is crucial because it describes the way addition and multiplication interact with each other.  2#3  2#5  Figure 3 The Distributive Property  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  4 CHAPTER 1  ■  Fundamentals  Example 1  ■  Using the Distributive Property  (a) 21 x  32  2 # x  2 # 3  Distributive Property   2x  6 s  Simplify  (b) 1 a  b2 1 x  y2  1 a  b2 x  1 a  b2 y  Distributive Property  		  Associative Property of Addition   1 ax  bx2  1 ay  by2  		   ax  bx  ay  by  Distributive Property  	 	In the last step we removed the parentheses because, according to the Associative Property, the order of addition doesn't matter. Now Try Exercise 15  ■  ■ Addition and Subtraction Don't assume that a is a negative number. Whether a is negative or positive depends on the value of a. For example, if a  5, then a  5, a negative number, but if a  5, then a  152  5 (Property 2), a positive number.  The number 0 is special for addition; it is called the additive identity because a  0  a for any real number a. Every real number a has a negative, a, that satisfies a  1 a2  0. Subtraction is the operation that undoes addition; to subtract a number from another, we simply add the negative of that number. By definition a  b  a  1 b2  To combine real numbers involving negatives, we use the following properties.  Properties of Negatives Property  Example  1. 1 12 a  a  1 12 5  5  2. 1 a2  a  1 52  5  3. 1 a2 b  a1 b2  1 ab2 4. 1 a2 1 b2  ab  1 52 7  51 72  1 5 # 72  5. 1 a  b2  a  b  1 3  52  3  5  6. 1 a  b2  b  a  1 5  82  8  5  1 42 1 32  4 # 3  Property 6 states the intuitive fact that a  b and b  a are negatives of each other. Property 5 is often used with more than two terms: 1 a  b  c2  a  b  c  Example 2  ■  Using Properties of Negatives  Let x, y, and z be real numbers. (a) 1 x  22  x  2 (b) 1 x  y  z 2  x  y  1 z 2  x  y  z  Now Try Exercise 23  Property 5: (a  b)  a  b Property 5: (a  b)  a  b Property 2: (a)  a  ■  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SECTION 1.1  ■  Real Numbers 5  ■ Multiplication and Division The number 1 is special for multiplication; it is called the multiplicative identity because a # 1  a for any real number a. Every nonzero real number a has an inverse, 1/a, that satisfies a # 1 1/a2  1. Division is the operation that undoes multiplication; to divide by a number, we multiply by the inverse of that number. If b ? 0, then, by definition, a4ba#  1 b  We write a # 1 1/b2 as simply a/b. We refer to a/b as the quotient of a and b or as the fraction a over b; a is the numerator and b is the denominator (or divisor). To combine real numbers using the operation of division, we use the following properties.  Properties of Fractions Property  Example  Description  1.  a c #  ac b d bd  2 5 #  2 ## 5  10 3 7 3 7 21  2.  a c a d 4  # b d b c  2 5 2 7 14 When dividing fractions, invert the divisor and 4  #  	 multiply. 3 7 3 5 15  3.  a b ab   c c c  When adding fractions with the same denomina2 7 27 9    	 tor, add the numerators. 5 5 5 5  4.  a c ad  bc   b d bd  2 3 2#73#5 29    5 7 35 35  5.  ac a  bc b  2#5 2  # 3 5 3  6. If  a c  , then ad  bc b d  When multiplying fractions, multiply numerators and denominators.  When adding fractions with different denominators, find a common denominator. Then add the numerators. Cancel numbers that are common factors in numerator and denominator.  2 6  , so 2 # 9  3 # 6 3 9  Cross-multiply.  When adding fractions with different denominators, we don't usually use Prop­erty 4. Instead we rewrite the fractions so that they have the smallest possible common denominator (often smaller than the product of the denominators), and then we use Property 3. This denominator is the Least Common Denominator (LCD) described in the next example.  Example 3 Evaluate:  ■  Using the LCD to Add Fractions  5 7  36 120  Solution  Factoring each denominator into prime factors gives  36  22 # 32  and  120  23 # 3 # 5  We find the least common denominator (LCD) by forming the product of all the prime factors that occur in these factorizations, using the highest power of each prime factor. Thus the LCD is 23 # 32 # 5  360. So 5 7 5 # 10 7#3        Use common denominator # 36 120 36 10 120 # 3   50 21 71     Property 3: Adding fractions with the   same denominator 360 360 360  Now Try Exercise 29  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  ■  6 CHAPTER 1  ■  Fundamentals  ■ The Real Line The real numbers can be represented by points on a line, as shown in Figure 4. The positive direction (toward the right) is indicated by an arrow. We choose an arbitrary reference point O, called the origin, which corresponds to the real number 0. Given any convenient unit of measurement, each positive number x is represented by the point on the line a distance of x units to the right of the origin, and each negative number x is represented by the point x units to the left of the origin. The number associated with the point P is called the coordinate of P, and the line is then called a coordinate line, or a real number line, or simply a real line. Often we identify the point with its coordinate and think of a number as being a point on the real line. _3.1725 _2.63  _4.9 _4.7 _5 _4 _4.85  _3  1 _ 16  _ œ∑2 _2  _1  1 1 8 4 1 2  0  0.3 ∑  œ∑2  œ∑3 œ∑5  1  4.2 4.4 4.9999  π  2  3  4 5 4.3 4.5  Figure 4 The real line  The real numbers are ordered. We say that a is less than b and write a  b if b  a is a positive number. Geometrically, this means that a lies to the left of b on the number line. Equivalently, we can say that b is greater than a and write b  a. The symbol a  b 1 or b  a2 means that either a  b or a  b and is read "a is less than or equal to b." For instance, the following are true inequalities (see Figure 5): 7  7.4  7.5  _π _4  _3  !2  2  p  3 œ∑2  _2  _1  0  1  2  3  4  5  22  7.4 7.5 6  7  8  Figure 5  ■ Sets and Intervals A set is a collection of objects, and these objects are called the elements of the set. If S is a set, the notation a  S means that a is an element of S, and b o S means that b is not an element of S. For example, if Z represents the set of integers, then 3  Z but p o Z. Some sets can be described by listing their elements within braces. For instance, the set A that consists of all positive integers less than 7 can be written as A  51, 2, 3, 4, 5, 66  We could also write A in set-builder notation as  A  5x 0 x is an integer and 0  x  76  which is read "A is the set of all x such that x is an integer and 0  x  7."  © Monkey Business Images/Shutterstock.com  Discovery Project Real Numbers in the Real World Real-world measurements always involve units. For example, we usually measure distance in feet, miles, centimeters, or kilometers. Some measurements involve different types of units. For example, speed is measured in miles per hour or meters per second. We often need to convert a measurement from one type of unit to another. In this project we explore different types of units used for different purposes and how to convert from one type of unit to another. You can find the project at www.stewartmath.com.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SECTION 1.1  ■  Real Numbers 7  If S and T are sets, then their union S  T is the set that consists of all elements that are in S or T (or in both). The intersection of S and T is the set S  T consisting of all elements that are in both S and T. In other words, S  T is the common part of S and T. The empty set, denoted by , is the set that contains no element.  Example 4  ■  Union and Intersection of Sets  If S  51, 2, 3, 4, 5 6, T  54, 5, 6, 7 6, and V  56, 7, 8 6, find the sets S  T, S  T, and S  V. Solution  T 64748 1, 2, 3, 4, 5, 6, 7, 8 14243 123  S  V  S  T  51, 2, 3, 4, 5, 6, 76    All elements in S or T S  T  54, 56  SV      Elements common to both S and T     S and V have no element in common  Now Try Exercise 41  a  b  Figure 6 The open interval 1 a, b 2 a  b  Figure 7 The closed interval 3 a, b 4  Certain sets of real numbers, called intervals, occur frequently in calculus and correspond geometrically to line segments. If a  b, then the open interval from a to b consists of all numbers between a and b and is denoted 1 a, b2 . The closed interval from a to b includes the endpoints and is denoted 3a, b4 . Using set-builder notation, we can write 1 a, b2  5x 0 a  x  b6  3a, b4  5x 0 a  x  b6  Note that parentheses 1 2 in the interval notation and open circles on the graph in Figure 6 indicate that endpoints are excluded from the interval, whereas square brackets 3 4 and solid circles in Figure 7 indicate that the endpoints are included. Intervals may also include one endpoint but not the other, or they may extend infinitely far in one direction or both. The following table lists the possible types of intervals. Notation  Set description  1a, b2  5x 0 a  x  b6  a  b  3a, b2  5x 0 a  x  b6  a  b  a  b  a  b  3a, b4  5x 0 a  x  b6  1a, ` 2  5x 0 a  x6  3a, ` 2  a  5x 0 a  x6  1`, b2  a  5x 0 x  b6  b  5x 0 x  b6  1`, b4  1`, ` 2  Example 5  Graph  5x 0 a  x  b6  1a, b4  The symbol q ("infinity") does not stand for a number. The notation 1 a, ` 2 , for instance, simply indicates that the interval has no endpoint on the right but extends infinitely far in the positive direction.  ■  b  R (set of all real numbers) ■  Graphing Intervals  Express each interval in terms of inequalities, and then graph the interval. (a) 31, 22  5x 0 1  x  26 (b) 31.5, 44  5x 0 1.5  x  46 (c) 1 3, ` 2  5x 0 3  x6  _1  0 0  _3  2 1.5  4  0  Now Try Exercise 47  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  ■  8 CHAPTER 1  ■  Fundamentals  No Smallest or Largest Number in an Open Interval  Any interval contains infinitely many numbers—every point on the graph of an interval corresponds to a real number. In the closed interval 3 0 , 1 4 , the smallest number is 0 and the largest is 1, but the open interval 10 , 1 2 contains no small­ est or largest number. To see this, note that 0.01 is close to zero, but 0.001 is closer, 0.0001 is closer yet, and so on. We can always find a number in the interval 10 , 1 2 closer to zero than any given number. Since 0 itself is not in the inter­ val, the interval contains no smallest number. Similarly, 0.99 is close to 1, but 0.999 is closer, 0.9999 closer yet, and so on. Since 1 itself is not in the interval, the interval has no largest number.  Example 6  ■  Finding Unions and Intersections of Intervals  Graph each set. (a) 1 1, 32  32, 74    (b) 1 1, 32  32, 74 Solution  (a) T  he intersection of two intervals consists of the numbers that are in both intervals. Therefore 1 1, 32  32, 74  5x 0 1  x  3 and 2  x  76  5x 0 2  x  36  32, 32  This set is illustrated in Figure 8. (b) The union of two intervals consists of the numbers that are in either one interval or the other (or both). Therefore 1 1, 32  32, 74  5x 0 1  x  3 or 2  x  76  5x 0 1  x  76  1 1, 74  This set is illustrated in Figure 9. 0  0.01  (1, 3)  (1, 3)  0.1  0  1  0  3  1  3 [2, 7]  [2, 7] 0  0.001  0  0.01  2  0  7  2 (1, 7]  [2, 3) 0 0.0001  0.001  0  2  0  3  _3  Figure 10  ■  ■ Absolute Value and Distance  | 5 |=5 0  7  1  Figure 9 1 1, 3 2  3 2, 7 4  1 1, 7 4  Figure 8 1 1, 32  3 2, 74  3 2, 3 2 Now Try Exercise 61  | _3 |=3  7  5  The absolute value of a number a, denoted by 0 a 0 , is the distance from a to 0 on the real number line (see Figure 10). Distance is always positive or zero, so we have 0 a 0  0 for every number a. Remembering that a is positive when a is negative, we have the following definition.  Definition of Absolute Value If a is a real number, then the absolute value of a is 0a0  e  Example 7 (a) (b) (c) (d)  0 0 0 0  ■  a a  if a  0 if a  0  Evaluating Absolute Values of Numbers  30 3 3 0  1 32  3 00 0 3  p 0  1 3  p2  p  3  Now Try Exercise 67  1 since 3  p  1  3  p  02 ■  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SECTION 1.1  ■  Real Numbers 9  When working with absolute values, we use the following properties.  Properties of Absolute Value Property  Example  1. 0 a 0  0  0 3 0  3  0	The absolute value of a number is always positive or zero.  3. 0 ab 0  0 a 0 0 b 0  0 2 # 5 0  0 2 0 0 5 0 	The absolute value of a product is the product of the absolute values.  2. 0 a 0  0 a 0  4. `  Description  0 5 0  0 5 0 	A number and its negative have the same absolute value.  0a0 a `  b 0b0  5. 0 a  b 0  0 a 0  0 b 0  `  0 12 0 12 The absolute value of a quotient is the quotient of the `  	 absolute values. 3 0 3 0  0 3  5 0  0 3 0  0 5 0  Triangle Inequality  What is the distance on the real line between the numbers 2 and 11? From Figure 11 we see that the distance is 13. We arrive at this by finding either 0 11  1 22 0  13 or 0 1 22  11 0  13. From this observation we make the following definition (see Figure 12). 13 _2  0  | b-a | 11  Figure 11  a  b  Figure 12 Length of a line segment is 0 b  a 0  Distance between Points on the Real Line If a and b are real numbers, then the distance between the points a and b on the real line is d1 a, b2  0 b  a 0 From Property 6 of negatives it follows that 0ba0  0ab0  This confirms that, as we would expect, the distance from a to b is the same as the distance from b to a.  Example 8  Figure 13  Distance Between Points on the Real Line  The distance between the numbers 8 and 2 is  10 _8  ■  0  d1 a, b2  0 2  1 82 0  0 10 0  10  2  We can check this calculation geometrically, as shown in Figure 13. Now Try Exercise 75  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  ■  10 CHAPTER 1  ■  Fundamentals  1.1 Exercises Concepts  14. 21A  B 2  2A  2B 15. 15x  123  15x  3  1. Give an example of each of the following: (a) A natural number (b) An integer that is not a natural number (c) A rational number that is not an integer (d) An irrational number  16. 1x  a2 1x  b2  1x  a2x  1x  a2b 17. 2x13  y2  13  y22x  18. 71a  b  c2  71a  b2  7c  2. Complete each statement and name the property of real ­numbers you have used. (a) ab     ;  (b) a  1b  c2   (c) a 1b  c2   19–22 ■ Properties of Real Numbers   Rewrite the expression using the given property of real numbers.  Property   ;   ;  19. Commutative Property of Addition,   x  3  Property  20. Associative Property of Multiplication,   713x2   Property  21. Distributive Property,  3. Express the set of real numbers between but not including 2 and 7 as follows. (b) In interval notation: of the number x.  If x is not 0, then the sign of 0 x 0 is always    .  5. The distance between a and b on the real line is d 1a, b2    . So the distance between 5 and 2 is    .  6–8 ■ Yes or No? If No, give a reason. Assume that a and b are nonzero real numbers. 6. (a)	Is the sum of two rational numbers always a rational number? (b) Is the sum of two irrational numbers always an irrational number? 7. (a)	Is a  b equal to b  a? (b) Is 21 a  52 equal to 2a  10? 8. (a)	Is the distance between any two different real numbers always positive? (b)	Is the distance between a and b the same as the distance between b and a?  ■  Real Numbers   List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers  9. E1.5, 0, 52, !7, 2.71, p, 3.14, 100, 8F  20 10. E1.3, 1.3333. . . , !5, 5.34, 500, 123, !16, 246 579 ,  5 F  11–18 ■ Properties of Real Numbers   State the property of real numbers being used. 11. 3  7  7  3 12. 41 2  32  1 2  32 4  13. 1x  2y2  3z  x  1 2y  3z2  24. 1a  b28  23. 31x  y2  26. 43 16y2  25. 412m 2  27.  52 12x  4y2  28. 13a2 1b  c  2d2  29–32 ■ Arithmetic Operations   Perform the indicated operations. 29. (a)  3 10  30. (a)  2 3  31. (a)  2 3 A6   154  (b)   35  2  32. (a)  2 3   15  (b) 1  58  16   32 B    1 4  (b) A3  14 B A1  45 B  2 3  (b)  2  2 5 1 10   12  153  33–34 ■ Inequalities   Place the correct symbol (, , or ) in the space. 7 2  33. (a) 3  34. (a) 23 0.67 (c) 0 0.67 0  35–38 false.  Skills 9–10  5x  5y   23–28 ■ Properties of Real Numbers   Use properties of real numbers to write the expression without parentheses.  (a) In set-builder notation: 4. The symbol 0 x 0 stands for the  22. Distributive Property,  41A  B 2   ■   72  (b) 3 (b) 0 0.67 0  2 3  (c) 3.5  7 2  0.67  Inequalities   State whether each inequality is true or  35. (a) 3  4  (b) 3  4  36. (a) !3  1.7325  (b) 1.732  !3  37. (a)  10 2  5  (b)  6 10   56  38. (a)  7 11   138  (b) 35  34  39–40 ■ Inequalities   Write each statement in terms of inequalities. 39. (a) (b) (c) (d) (e)  x is positive. t is less than 4. a is greater than or equal to p. x is less than 13 and is greater than 25. The distance from p to 3 is at most 5.  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SECTION 1.1 40. (a) (b) (c) (d) (e) 41–44  y is negative. z is greater than 1. b is at most 8. „ is positive and is less than or equal to 17. y is at least 2 units from p. ■  Sets   Find the indicated set if A  51, 2, 3, 4, 5, 6, 76    B  52, 4, 6, 86 C  57, 8, 9, 106  69. (a) @ 0 6 0  0 4 0 @ 70. (a) @ 2  0 12 0 @ 71. (a)  42. (a) B  C  (b) B  C  74.  43. (a) A  C  (b) A  C  44. (a) A  B  C  (b) A  B  C  Sets   Find the indicated set if A  5x 0 x  26    B  5x 0 x  46  45. (a) B  C  C  5x 0 1  x  56  46. (a) A  C  (b) A  B  (b) B  C  (b) 0 A 13 B 1152 0  6 ` 24  (b) `  _3 _2 _1  0  1  2  3  _3 _2 _1  0  1  2  3  75. (a) 2 and 17 76. (a)  ■  1 0 1 0  7  12 ` 12  7  73–76 ■ Distance   Find the distance between the given numbers. 73.  45–46  Real Numbers 11  (b) 1  @ 1  0 1 0 @  0 122 # 6 0  72. (a) `  (b) A  B  41. (a) A  B  (b)  ■  7 15  and  211  11 8  and  103  (b) 3 and 21  (c)  (b) 38 and 57  (c) 2.6 and 1.8  SKILLS Plus 77–78 ■ Repeating Decimal   Express each repeating decimal as a fraction. (See the margin note on page 3.) 77. (a) 0.7  (b) 0.28  (c) 0.57  47–52 ■ Intervals   Express the interval in terms of inequalities, and then graph the interval.  78. (a) 5.23  (b) 1.37  (c) 2.135  47. 13, 02  79–82 ■ Simplifying Absolute Value   Express the quantity without using absolute value.  48. 1 2, 84  50. C6,  12 D  49. 32, 82  51. 32, ` 2  52. 1`, 12  53–58 ■ Intervals   Express the inequality in interval notation, and then graph the corresponding interval.  79. 0 p  3 0  81. 0 a  b 0 , where a  b  80. 0 1  !2 0  82. a  b  0 a  b 0 , where a  b  53. x  1  54. 1  x  2  55. 2  x  1  56. x  5  83–84 ■ Signs of Numbers  Let a, b, and c be real numbers such that a  0, b  0, and c  0. Find the sign of each expression.  57. x  1  58. 5  x  2  83. (a) a  (b) bc  (c) a  b  (d) ab  ac  84. (a) b  (b) a  bc  (c) c  a  (d) ab 2  59–60  ■  59. (a) (b)  Intervals   Express each set in interval notation. _3  0  5  −3  0  5  60. (a)  0  (b) 61–66  −2 ■  Applications  2  0  Intervals   Graph the set.  61. 12, 02  11, 12  62. 12, 02  11, 12  65. 1`, 42  14, ` 2  66. 1`, 64  12, 102  63. 34, 64  30, 82  67–72  ■  68. (a) 0 !5  5 0  30 ft  64. 34, 62  30, 82  Absolute Value   Evaluate each expression.  67. (a) 0 100 0  85. Area of a Garden   Mary's backyard vegetable garden measures 20 ft by 30 ft, so its area is 20  30  600 ft 2. She decides to make it longer, as shown in the figure, so that the area increases to A  201 30  x2 . Which property of real numbers tells us that the new area can also be written A  600  20x? x  20 ft  (b) 0 73 0  (b) 0 10  p 0  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  12 CHAPTER 1  ■  Fundamentals  Daily high temperature (*F)  86. Temperature Variation   The bar graph shows the daily high temperatures for Omak, Washington, and Geneseo, New York, during a certain week in June. Let TO represent the temperature in Omak and TG the temperature in Geneseo. Calculate TO  TG and 0 TO  TG 0 for each day shown. Which of these two values gives more information? Omak, WA Geneseo, NY  80 75 70 65  Sun  Mon  Tue  Wed Day  Thu  Fri  L  21 x  y2  108 (a) Will the post office accept a package that is 6 in. wide, 8 in. deep, and 5 ft long? What about a package that measures 2 ft by 2 ft by 4 ft? (b) What is the greatest acceptable length for a package that has a square base measuring 9 in. by 9 in.?  x  DiSCUSS  5 ft=60 in. y  6 in.  ■ DISCOVER  8 in.  ■  PROVE  x     1    2   10 100 1000  1/x  1.0 0.5 0.1 0.01 0.001  91. Discover: Locating Irrational Numbers on the Real Line   Using the figures below, explain how to locate the point !2 on a number line. Can you locate !5 by a similar method? How can the circle shown in the figure help us to locate p on a number line? List some other irrational numbers that you can locate on a number line.  Sat  87. Mailing a Package   The post office will accept only packages for which the length plus the "girth" (distance around) is no more than 108 in. Thus for the package in the figure, we must have  L  1/x  x  ■ WRITE  88. DISCUSS: Sums and Products of Rational and Irrational Numbers   Explain why the sum, the difference, and the product of two rational numbers are rational numbers. Is the product of two irrational numbers necessarily irrational? What about the sum? 89. DISCOVER ■ PROVE: Combining Rational and Irrational Numbers  Is 12  !2 rational or irrational? Is 12 # !2 rational or irrational? Experiment with sums and products of other rational and irrational numbers. Prove the following. (a) The sum of a rational number r and an irrational number t is irrational. (b) The product of a rational number r and an irrational number t is irrational. [Hint: For part (a), suppose that r  t is a rational number q, that is, r  t  q. Show that this leads to a contradiction. Use similar reasoning for part (b).]  90. DISCOVER: Limiting Behavior of Reciprocals  Complete the tables. What happens to the size of the fraction 1/x as x gets large? As x gets small?  œ∑2 0  1  1 1  2  0  π  92. Prove: Maximum and Minimum Formulas  Let max1a, b2 denote the maximum and min1 a, b2 denote the minimum of the real numbers a and b. For example, max12, 52  5 and min1 1, 22  2. ab 0ab0 . (a) Prove that max1a, b2  2 ab 0ab0 (b) Prove that min1a, b2  . 2 [Hint: Take cases and write these expressions without absolute values. See Exercises 81 and 82.] 93. Write: Real Numbers in the Real World   Write a paragraph describing different real-world situations in which you would use natural numbers, integers, rational numbers, and irrational numbers. Give examples for each type of situation. 94. Discuss: Commutative and Noncommutative Operations   We have learned that addition and multiplication are both commutative operations. (a) Is subtraction commutative? (b) Is division of nonzero real numbers commutative? (c) Are the actions of putting on your socks and putting on your shoes commutative? (d) Are the actions of putting on your hat and putting on your coat commutative? (e) Are the actions of washing laundry and drying it commutative? 95. PROVE: Triangle Inequality   We prove Property 5 of absolute values, the Triangle Inequality: 0xy0  0x0  0y0  (a) Verify that the Triangle Inequality holds for x  2 and y  3, for x  2 and y  3, and for x  2 and y  3. (b) Prove that the Triangle Inequality is true for all real numbers x and y. [Hint: Take cases.]  Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  SECTION 1.2  ■  Exponents and Radicals 13  1.2 Exponents and Radicals ■ Integer Exponents ■ Rules for Working with Exponents ■ Scientific Notation ■ Radicals ■ Rational Exponents ■ Rationalizing the Denominator; Standard Form In this section we give meaning to expressions such as a m/n in which the exponent m/n is a rational number. To do this, we need to recall some facts about integer exponents, radicals, and nth roots.  ■ Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5 # 5 # 5 is written as 53. In general, we have the following definition.  Exponential Notation If a is any real number and n is a positive integer, then the nth power of a is  #a#...#a a n  a1442443 n factors  The number a is called the base, and n is called the exponent.  Example 1 Note the distinction between 132 4 and 34. In 132 4 the exponent applies to 3, but in 34 the exponent applies only to 3.  ■  Exponential Notation  5 A 12 B  (a)  A 12 B A 12 B A 12 B A 12 B A 12 B  321 (b) 1 32 4  1 32 # 1 32 # 1 32 # 1 32  81 (c) 34  1 3 # 3 # 3 # 32  81 Now Try Exercise 17  ■  We can state several useful rules for working with exponential notation. To discover the rule for multiplication, we multiply 54 by 52: 54 # 52  15 # 5 # 5 # 5215 # 52  5 # 5 # 5 # 5 # 5 # 5  56  542 144424443 123  4 factors  1444442444443  2 factors  6 factors  It appears that to multiply two powers of the same base, we add their exponent

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