College Algebra and Trigonometry SEVENTH EDITION
Margaret L. Lial American River College John Hornsby University of New Orleans David I. Schneider University of Maryland Callie J. Daniels St. Charles Community College College Algebra and Trigonometry SEVENTH EDITION
Please contact https://support.pearson.com/getsupport/s/contactsupport with any queries on this content. Copyright © 2021, 2017, 2013 by Pearson Education, Inc. or its affiliates, 221 River Street, Hoboken, NJ 07030. All Rights Reserved. Manufactured in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights and Permissions department, please visit www.pearsoned.com/permissions/. Acknowledgments of third-party content appear on page C-1, which constitutes an extension of this copyright page. PEARSON, ALWAYS LEARNING, and MYMATHLAB FOR SCHOOL are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the U.S. and/or other countries. Unless otherwise indicated herein, any third-party trademarks, logos, or icons that may appear in this work are the property of their respective owners, and any references to third-party trademarks, logos, icons, or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc., or its affiliates, authors, licensees, or distributors. Library of Congress Cataloging-in-Publication Data Names: Lial, Margaret L., author. | Hornsby, John, author. | Schneider, David I., author. | Daniels, Callie J., author. Title: College algebra and trigonometry / Margaret L. Lial, American River College, John Hornsby University of New Orleans, David I. Schneider, University of Maryland, Callie J. Daniels, St. Charles Community College. Description: Seventh edition. | Hoboken, NJ : Pearson, [2021] | Includes index. Identifiers: LCCN 2019053777 | ISBN 9780135924549 (hardback) Subjects: LCSH: Algebra. | Trigonometry. Classification: LCC QA154.3 .L54 2021 | DDC 512/.13--dc23 LC record available at https://lccn.loc.gov/2019053777 ScoutAutomatedPrintCode ISBN 13: 978-0-13-673483-3 (High School Binding) ISBN 10: 0-13-673483-9 (High School Binding)
This text is dedicated to you—the student.We hope that it helps you achieve your goals. Remember to show up, work hard, and stay positive. Everything else will take care of itself. The Lial Author Team
vii Contents Preface xvii Resources for Success xxiii R Review of Basic Concepts 1 R.1 Fractions, Decimals, and Percents 2 Lowest Terms of a Fraction ■ Improper Fractions and Mixed Numbers ■ Operations with Fractions ■ Decimals as Fractions ■ Operations with Decimals ■ Fractions as Decimals ■ Percents as Decimals and Decimals as Percents ■ Percents as Fractions and Fractions as Percents R.2 Sets and Real Numbers 15 Basic Definitions ■ Operations on Sets ■ Sets of Numbers and the Number Line R.3 Real Number Operations and Properties 24 Order on the Number Line ■ Absolute Value ■ Operations on Real Numbers ■ Exponents ■ Order of Operations ■ Properties of Real Numbers R.4 Integer and Rational Exponents 40 Product Rule for Exponents ■ Power Rules for Exponents ■ Zero as an Exponent ■ Negative Exponents and the Quotient Rule ■ Rational Exponents R.5 Polynomials 51 Polynomials ■ Addition and Subtraction ■ Multiplication ■ Division R.6 Factoring Polynomials 61 Factoring Out the Greatest Common Factor ■ Factoring by Grouping ■ Factoring Trinomials ■ Factoring Binomials ■ Factoring by Substitution ■ Factoring Expressions with Negative or Rational Exponents R.7 Rational Expressions 72 Rational Expressions ■ Lowest Terms of a Rational Expression ■ Multiplication and Division ■ Addition and Subtraction ■ Complex Fractions R.8 Radical Expressions 82 Radical Notation ■ Simplified Radicals ■ Operations with Radicals ■ Rationalizing Denominators Test Prep 94 ■ Review Exercises 99 ■ Test 103 1 Equations and Inequalities 105 1.1 Linear Equations 106 Basic Terminology of Equations ■ Linear Equations ■ Identities, Conditional Equations, and Contradictions ■ Solving for a Specified Variable (Literal Equations)
viii CONTENTS 1.2 Applications and Modeling with Linear Equations 112 Solving Applied Problems ■ Geometry Problems ■ Motion Problems ■ Mixture Problems ■ Modeling with Linear Equations 1.3 Complex Numbers 123 Basic Concepts of Complex Numbers ■ Operations on Complex Numbers 1.4 Quadratic Equations 131 The Zero-Factor Property ■ The Square Root Property ■ Completing the Square ■ The Quadratic Formula ■ Solving for a Specified Variable ■ The Discriminant Chapter 1 Quiz (Sections 1.1–1.4) 141 1.5 Applications and Modeling with Quadratic Equations 142 Geometry Problems ■ The Pythagorean Theorem ■ Height of a Projected Object ■ Modeling with Quadratic Equations 1.6 OtherTypes of Equations and Applications 154 Rational Equations ■ Work Rate Problems ■ Equations with Radicals ■ Equations with Rational Exponents ■ Equations Quadratic in Form Summary Exercises on Solving Equations 167 1.7 Inequalities 168 Linear Inequalities ■ Three-Part Inequalities ■ Quadratic Inequalities ■ Rational Inequalities 1.8 Absolute Value Equations and Inequalities 180 Basic Concepts ■ Absolute Value Equations ■ Absolute Value Inequalities ■ Special Cases ■ Absolute Value Models for Distance and Tolerance Test Prep 188 ■ Review Exercises 193 ■ Test 199 2 Graphs and Functions 201 2.1 Rectangular Coordinates and Graphs 202 Ordered Pairs ■ The Rectangular Coordinate System ■ The Distance Formula ■ The Midpoint Formula ■ Equations in Two Variables 2.2 Circles 213 Center-Radius Form ■ General Form ■ An Application 2.3 Functions 221 Relations and Functions ■ Domain and Range ■ Determining Whether Relations Are Functions ■ Function Notation ■ Increasing, Decreasing, and Constant Functions 2.4 Linear Functions 238 Basic Concepts of Linear Functions ■ Standard Form Ax + By = C ■ Slope ■ Average Rate of Change ■ Linear Models Chapter 2 Quiz (Sections 2.1–2.4) 252
ix CONTENTS 2.5 Equations of Lines and Linear Models 253 Point-Slope Form ■ Slope-Intercept Form ■ Vertical and Horizontal Lines ■ Parallel and Perpendicular Lines ■ Modeling Data ■ Graphical Solution of Linear Equations in One Variable Summary Exercises on Graphs, Circles, Functions, and Equations 266 2.6 Graphs of Basic Functions 267 Continuity ■ The Identity, Squaring, and Cubing Functions ■ The Square Root and Cube Root Functions ■ The Absolute Value Function ■ Piecewise-Defined Functions ■ The Relation x = y 2 2.7 Graphing Techniques 279 Stretching and Shrinking ■ Reflecting ■ Symmetry ■ Even and Odd Functions ■ Translations Chapter 2 Quiz (Sections 2.5–2.7) 296 2.8 Function Operations and Composition 297 Arithmetic Operations on Functions ■ The Difference Quotient ■ Composition of Functions and Domain Test Prep 312 ■ Review Exercises 316 ■ Test 321 3 Polynomial and Rational Functions 323 3.1 Quadratic Functions and Models 324 Polynomial Functions ■ Quadratic Functions ■ Graphing Techniques ■ Completing the Square ■ The Vertex Formula ■ Quadratic Models 3.2 Synthetic Division 340 Synthetic Division ■ Remainder Theorem ■ Potential Zeros of Polynomial Functions 3.3 Zeros of Polynomial Functions 347 Factor Theorem ■ Rational Zeros Theorem ■ Number of Zeros ■ Conjugate Zeros Theorem ■ Zeros of a Polynomial Function ■ Descartes’ Rule of Signs 3.4 Polynomial Functions: Graphs, Applications, and Models 359 Graphs of ƒ1x2 = axn ■ Graphs of General Polynomial Functions ■ Behavior at Zeros ■ Turning Points and End Behavior ■ Graphing Techniques ■ Intermediate Value and Boundedness Theorems ■ Approximations of Real Zeros ■ Polynomial Models Summary Exercises on Polynomial Functions, Zeros, and Graphs 378 3.5 Rational Functions: Graphs, Applications, and Models 380 The Reciprocal Function ƒ1x 2 = 1 x ■ The Function ƒ1x 2 = 1 x 2 ■ Asymptotes ■ Graphing Techniques ■ Rational Models Chapter 3 Quiz (Sections 3.1–3.5) 401
x CONTENTS 3.6 Polynomial and Rational Inequalities 402 Polynomial Inequalities ■ Rational Inequalities Summary Exercises on Solving Equations and Inequalities 410 3.7 Variation 411 Direct Variation ■ Inverse Variation ■ Combined and Joint Variation Test Prep 420 ■ Review Exercises 425 ■ Test 431 Test Prep 516 ■ Review Exercises 519 ■ Test 523 4 Inverse, Exponential, and Logarithmic Functions 433 4.1 Inverse Functions 434 One-to-One Functions ■ Inverse Functions ■ Equations of Inverses ■ An Application of Inverse Functions to Cryptography 4.2 Exponential Functions 447 Exponents and Properties ■ Exponential Functions ■ Exponential Equations ■ Compound Interest ■ The Number e and Continuous Compounding ■ Exponential Models 4.3 Logarithmic Functions 463 Logarithms ■ Logarithmic Equations ■ Logarithmic Functions ■ Properties of Logarithms Summary Exercises on Inverse, Exponential, and Logarithmic Functions 476 4.4 Evaluating Logarithms and the Change-of-BaseTheorem 477 Common Logarithms ■ Applications and Models with Common Logarithms ■ Natural Logarithms ■ Applications and Models with Natural Logarithms ■ Logarithms with Other Bases Chapter 4 Quiz (Sections 4.1–4.4) 489 4.5 Exponential and Logarithmic Equations 489 Exponential Equations ■ Logarithmic Equations ■ Applications and Models 4.6 Applications and Models of Exponential Growth and Decay 501 The Exponential Growth or Decay Function ■ Growth Function Models ■ Decay Function Models Summary Exercises on Functions: Domains and Defining Equations 513 5 Trigonometric Functions 525 5.1 Angles 526 Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles
xi CONTENTS Test Prep 583 ■ Review Exercises 587 ■ Test 590 5.2 Trigonometric Functions 534 Trigonometric Functions ■ Quadrantal Angles ■ Reciprocal Identities ■ Signs and Ranges of Function Values ■ Pythagorean Identities ■ Quotient Identities 5.3 Trigonometric Function Values and Angle Measures 549 Right-Triangle-Based Definitions of the Trigonometric Functions ■ Cofunctions ■ Trigonometric Function Values of Special Angles ■ Reference Angles ■ Special Angles as Reference Angles ■ Determination of Angle Measures with Special Reference Angles ■ Calculator Approximations of Trigonometric Function Values ■ Calculator Approximations of Angle Measures ■ An Application Chapter 5 Quiz (Sections 5.1–5.3) 564 5.4 Solutions and Applications of Right Triangles 565 Historical Background ■ Significant Digits ■ Solving Triangles ■ Angles of Elevation or Depression ■ Bearing ■ Further Applications 6 The Circular Functions and Their Graphs 593 6.1 Radian Measure 594 Radian Measure ■ Conversions between Degrees and Radians ■ Arc Length on a Circle ■ Area of a Sector of a Circle 6.2 The Unit Circle and Circular Functions 607 Circular Functions ■ Values of the Circular Functions ■ Determining a Number with a Given Circular Function Value ■ Linear and Angular Speed 6.3 Graphs of the Sine and Cosine Functions 620 Periodic Functions ■ Graph of the Sine Function ■ Graph of the Cosine Function ■ Techniques for Graphing, Amplitude, and Period ■ Connecting Graphs with Equations ■ A Trigonometric Model 6.4 Translations of the Graphs of the Sine and Cosine Functions 633 Horizontal Translations ■ Vertical Translations ■ Combinations of Translations ■ A Trigonometric Model Chapter 6 Quiz (Sections 6.1–6.4) 644 6.5 Graphs of theTangent and Cotangent Functions 644 Graph of the Tangent Function ■ Graph of the Cotangent Function ■ Techniques for Graphing ■ Connecting Graphs with Equations 6.6 Graphs of the Secant and Cosecant Functions 653 Graph of the Secant Function ■ Graph of the Cosecant Function ■ Techniques for Graphing ■ Connecting Graphs with Equations ■ Addition of Ordinates Summary Exercises on Graphing Circular Functions 661
xii CONTENTS Test Prep 668 ■ Review Exercises 671 ■ Test 678 6.7 Harmonic Motion 661 Simple Harmonic Motion ■ Damped Oscillatory Motion 8 Applications of Trigonometry 771 8.1 The Law of Sines 772 Congruency and Oblique Triangles ■ Derivation of the Law of Sines ■ Using the Law of Sines ■ Description of the Ambiguous Case ■ Area of a Triangle 7 Trigonometric Identities and Equations 681 7.1 Fundamental Identities 682 Fundamental Identities ■ Uses of the Fundamental Identities 7.2 Verifying Trigonometric Identities 688 Strategies ■ Verifying Identities by Working with One Side ■ Verifying Identities by Working with Both Sides 7.3 Sum and Difference Identities 697 Cosine Sum and Difference Identities ■ Cofunction Identities ■ Sine and Tangent Sum and Difference Identities ■ Applications of the Sum and Difference Identities ■ Verifying an Identity Chapter 7 Quiz (Sections 7.1–7.3) 711 7.4 Double-Angle and Half-Angle Identities 711 Double-Angle Identities ■ An Application ■ Product-to-Sum and Sum-toProduct Identities ■ Half-Angle Identities ■ Verifying an Identity Summary Exercises on Verifying Trigonometric Identities 724 7.5 Inverse Circular Functions 724 Review of Inverse Functions ■ Inverse Sine Function ■ Inverse Cosine Function ■ Inverse Tangent Function ■ Other Inverse Circular Functions ■ Inverse Function Values 7.6 Trigonometric Equations 740 Linear Methods ■ Zero-Factor Property Method ■ Quadratic Methods ■ Trigonometric Identity Substitutions ■ Equations with Half-Angles ■ Equations with Multiple Angles ■ Applications Chapter 7 Quiz (Sections 7.5–7.6) 753 7.7 Equations Involving Inverse Trigonometric Functions 753 Solution for x in Terms of y Using Inverse Functions ■ Solution of Inverse Trigonometric Equations Test Prep 760 ■ Review Exercises 764 ■ Test 768
xiii CONTENTS 8.2 The Law of Cosines 787 Derivation of the Law of Cosines ■ Using the Law of Cosines ■ Heron’s Formula for the Area of a Triangle ■ Derivation of Heron’s Formula Chapter 8 Quiz (Sections 8.1–8.2) 800 8.3 Geometrically Defined Vectors and Applications 801 Basic Terminology ■ The Equilibrant ■ Incline Applications ■ Navigation Applications 8.4 Algebraically Defined Vectors and the Dot Product 811 Algebraic Interpretation of Vectors ■ Operations with Vectors ■ The Dot Product and the Angle between Vectors Summary Exercises on Applications of Trigonometry and Vectors 820 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 821 The Complex Plane and Vector Representation ■ Trigonometric (Polar) Form ■ Converting between Rectangular and Trigonometric Forms ■ An Application of Complex Numbers to Fractals ■ Products of Complex Numbers in Trigonometric Form ■ Quotients of Complex Numbers in Trigonometric Form 8.6 De Moivre’sTheorem; Powers and Roots of Complex Numbers 832 Powers of Complex Numbers (De Moivre’s Theorem) ■ Roots of Complex Numbers Chapter 8 Quiz (Sections 8.3–8.6) 839 8.7 Polar Equations and Graphs 839 Polar Coordinate System ■ Graphs of Polar Equations ■ Conversion from Polar to Rectangular Equations ■ Classification of Polar Equations 8.8 Parametric Equations, Graphs, and Applications 852 Basic Concepts ■ Parametric Graphs and Their Rectangular Equivalents ■ The Cycloid ■ Applications of Parametric Equations 9 Systems and Matrices 873 9.1 Systems of Linear Equations 874 Linear Systems ■ Substitution Method ■ Elimination Method ■ Special Systems ■ Application of Systems of Equations ■ Linear Systems with Three Unknowns (Variables) ■ Application of Systems to Model Data 9.2 Matrix Solution of Linear Systems 892 The Gauss-Jordan Method ■ Special Systems ■ The Gaussian Elimination Method 9.3 Determinant Solution of Linear Systems 907 Determinants ■ Cofactors ■ n * n Determinants ■ Determinant Theorems ■ Cramer’s Rule Test Prep 860 ■ Review Exercises 865 ■ Test 871
xiv CONTENTS 9.4 Partial Fractions 920 Decomposition of Rational Expressions ■ Distinct Linear Factors ■ Repeated Linear Factors ■ Distinct Linear and Quadratic Factors ■ Repeated Quadratic Factors Chapter 9 Quiz (Sections 9.1–9.4) 927 9.5 Nonlinear Systems of Equations 928 Nonlinear Systems with Real Solutions ■ Nonlinear Systems with Nonreal Complex Solutions ■ An Application of Nonlinear Systems Summary Exercises on Systems of Equations 938 9.6 Systems of Inequalities and Linear Programming 939 Linear Inequalities in Two Variables ■ Nonlinear Inequalities in Two Variables ■ Systems of Inequalities ■ Linear Programming 9.7 Properties of Matrices 951 Basic Definitions ■ Matrix Addition ■ Special Matrices ■ Matrix Subtraction ■ Scalar Multiplication ■ Matrix Multiplication ■ An Application of Matrix Algebra 9.8 Matrix Inverses 965 Identity Matrices ■ Multiplicative Inverses ■ Solution of Systems Using Inverse Matrices Test Prep 977 ■ Review Exercises 982 ■ Test 988 11 Further Topics in Algebra 1039 11.1 Sequences and Series 1040 Sequences ■ Series and Summation Notation ■ Summation Properties and Rules 10 Analytic Geometry 991 10.1 Parabolas 992 Conic Sections ■ Horizontal Parabolas ■ Geometric Definition and Equations of Parabolas ■ An Application of Parabolas 10.2 Ellipses 1001 Equations and Graphs of Ellipses ■ Translated Ellipses ■ Eccentricity ■ Applications of Ellipses Chapter 10 Quiz (Sections 10.1–10.2) 1014 10.3 Hyperbolas 1014 Equations and Graphs of Hyperbolas ■ Translated Hyperbolas ■ Eccentricity 10.4 Summary of the Conic Sections 1025 Characteristics ■ Identifying Conic Sections ■ Geometric Definition of Conic Sections Test Prep 1032 ■ Review Exercises 1034 ■ Test 1037
xv CONTENTS 11.2 Arithmetic Sequences and Series 1051 Arithmetic Sequences ■ Arithmetic Series 11.3 Geometric Sequences and Series 1061 Geometric Sequences ■ Geometric Series ■ Infinite Geometric Series ■ Annuities Summary Exercises on Sequences and Series 1072 11.4 The BinomialTheorem 1073 A Binomial Expansion Pattern ■ Pascal’s Triangle ■ n-Factorial ■ Binomial Coefficients ■ The Binomial Theorem ■ kth Term of a Binomial Expansion 11.5 Mathematical Induction 1080 Principle of Mathematical Induction ■ Proofs of Statements ■ Generalized Principle of Mathematical Induction ■ Proof of the Binomial Theorem Chapter 11 Quiz (Sections 11.1–11.5) 1087 11.6 Basics of CountingTheory 1088 Fundamental Principle of Counting ■ Permutations ■ Combinations ■ Characteristics That Distinguish Permutations from Combinations 11.7 Basics of Probability 1099 Basic Concepts ■ Complements and Venn Diagrams ■ Odds ■ Compound Events ■ Binomial Probability Test Prep 1111 ■ Review Exercises 1115 ■ Test 1119 Appendices 1121 Appendix A Polar Form of Conic Sections 1121 Equations and Graphs ■ Conversion from Polar to Rectangular Form Appendix B Rotation of Axes 1125 Derivation of Rotation Equations ■ Application of a Rotation Equation Appendix C Geometry Formulas 1129 Answers to Selected Exercises A-1 Photo Credits C-1 Index I-1
xvii WELCOMETOTHE 7TH EDITION In the seventh edition of College Algebra and Trigonometry, we continue our ongoing commitment to providing the best possible text to help instructors teach and students succeed. In this edition, we have remained true to the pedagogical style of the past while staying focused on the needs of today’s students. Support for all classroom types (traditional, corequisite, flipped, hybrid, and online) may be found in this classic text and its supplements backed by the power of Pearson’s MyMathLab for School. In this edition, we have drawn on the extensive teaching experience of the Lial team, with special consideration given to reviewer suggestions. General updates include enhanced readability as we continually strive to make math understandable for students, updates to our extensive list of applications and real-world mathematics problems, use of color in displays and side comments, and coordination of exercises and their related examples. The authors understand that teaching and learning mathematics today can be a challenging task. Some students are prepared for the challenge, while other students require more review and supplemental material. This text is written so that students with varying abilities and backgrounds will all have an opportunity for a successful learning experience. The Lial team believes this to be our best edition of College Algebra and Trigonometry yet, and we sincerely hope that you enjoy using it as much as we have enjoyed writing it. Additional textbooks in this series are College Algebra, Thirteenth Edition Trigonometry, Twelfth Edition Precalculus, Seventh Edition. HIGHLIGHTS OF NEW CONTENT ■ Chapter R has been expanded to include more of the basic concepts many students struggle with. It begins with new Section R.1 Fractions, Decimals, and Percents. Additional new topics have been inserted throughout the chapter, including operations with signed numbers (Section R.3), dividing a polynomial by a monomial (Section R.5), and factoring expressions with negative and rational exponents (Section R.6). Topics throughout the chapter have been reorganized for improved flow. Instructors may choose to cover review topics from Chapter R at the beginning of a course or to insert these topics as-needed in a just-in-time fashion. Either way, students who are under-prepared for the demands of college algebra and trigonometry, as well as those who need a quick review, will benefit from the material contained here. ■ The exercise sets were a key focus of this revision, and Chapters 1 and 2 are among the chapters that have benefitted. Specifically, Section 1.7 Inequalities has new exercises on solving quadratic and rational inequalities, and Section 1.8 Absolute Value Equations and Inequalities contains new exercises that involve the absolute value of a quadratic polynomial. Section 2.3 Functions has new exercises that use analytic methods to determine maximum and minimum values of a function. Preface
xviii PREFACE Section 2.6 Graphs of Basic Functions contains new exercises and applications using the greatest integer function. Section 2.4 Linear Functions includes enhanced discussion of the average rate of change of a linear function. This topic is then related to the difference quotient and the average rate of change of a nonlinear function in Section 2.8 Function Operations and Composition. ■ Chapter 3 includes new Section 3.6 Polynomial and Rational Inequalities. This section features a visual approach to solving such inequalities by interpreting the graphs of related functions. ■ In response to reviewer suggestions, Section 4.3 Logarithmic Functions has new exercises that relate exponential and logarithmic functions as inverses. Chapter 6 includes additional exercises devoted to finding arc length and area of a sector of a circle (Section 6.1), as well as new applications of linear and angular speed (Section 6.2) and harmonic motion (Section 6.7). ■ Proofs of identities in Chapter 7 now feature a drop-down style for increased clarity and student understanding. Based on reviewer requests, Section 7.7 Equations Involving Inverse Trigonometric Functions includes new exercises in which solutions of inverse trigonometric equations are found. ■ Based on reviewer feedback, Section 8.4 Algebraically Defined Vectors and the Dot Product has new exercises on finding the angle between two vectors, determining magnitude and direction angle for a vector, and identifying orthogonal vectors. Additionally, Chapter 8 contains new exercises requiring students to graph polar and parametric equations (Section 8.7) and give parametric representations of plane curves (Section 8.8). ■ Section 9.2 Matrix Solution of Linear Systems now includes a new example and related exercises that use Gaussian elimination to solve linear systems of equations. Section 10.2 Ellipses and Section 10.3 Hyperbolas include new examples and exercises in which completing the square is used to find the standard form of an ellipse or a hyperbola. FEATURES OF THIS TEXT SUPPORT FOR LEARNING CONCEPTS We provide a variety of features to support students’ learning of the essential topics of college algebra and trigonometry. Explanations that are written in understandable terms, figures and graphs that illustrate examples and concepts, graphing technology that supports and enhances algebraic manipulations, and real-life applications that enrich the topics with meaning all provide opportunities for students to deepen their understanding of mathematics. These features help students make mathematical connections and expand their own knowledge base. ■ Examples Numbered examples that illustrate the techniques for working exercises are found in every section. We use traditional explanations, side comments, and pointers to describe the steps taken—and to warn students about common pitfalls. Some examples provide additional graphing calculator solutions, although these can be omitted if desired. ■ Now Try Exercises Following each numbered example, the student is directed to try a corresponding odd-numbered exercise (or exercises). This feature allows for quick feedback to determine whether the student understands the principles illustrated in the example.
xix PREFACE ■ Real-Life Applications We have included hundreds of real-life applications, many with data updated from the previous edition. They come from fields such as business, entertainment, sports, biology, astronomy, geology, and environmental studies. ■ Function Boxes Beginning in Chapter 2, functions provide a unifying theme throughout the text. Special function boxes offer a comprehensive, visual introduction to each type of function and also serve as an excellent resource for reference and review. Each function box includes a table of values, traditional and calculator-generated graphs, the domain, the range, and other special information about the function. These boxes are assignable in MyMathLab for School. ■ Figures and Photos Today’s students are more visually oriented than ever before, and we have updated the figures and photos in this edition to promote visual appeal. Guided Visualizations with accompanying exercises and explorations are available and assignable in MyMathLab for School. ■ Cautions and Notes Text that is marked CAUTION warns students of common errors, and NOTE comments point out explanations that should receive particular attention. ■ Looking Ahead to Calculus These margin notes offer glimpses of how the topics currently being studied are used in calculus. ■ Use of Graphing Technology We have integrated the use of graphing calculators where appropriate, although this technology is completely optional and can be omitted without loss of continuity. We continue to stress that graphing calculators support understanding but that students must first master the underlying mathematical concepts. Exercises that require the use of a graphing calculator are marked with the icon . SUPPORT FOR PRACTICING CONCEPTS This text offers a wide variety of exercises to help students master college algebra and trigonometry. The extensive exercise sets provide ample opportunity for practice and increase in difficulty so that students at every level of understanding are challenged. The variety of exercise types promotes mastery of the concepts and reduces the need for rote memorization. ■ Concept Preview Each exercise set begins with a group of CONCEPT PREVIEW exercises designed to promote understanding of vocabulary and basic concepts of each section. These new exercises are assignable in MyMathLab for School and provide support, especially for hybrid, online, and flipped courses. ■ Exercise Sets In addition to traditional drill exercises, this text includes writing exercises, optional graphing calculator exercises , and multiple- choice, matching, true/false, and completion exercises. Those marked Concept Check focus on conceptual thinking. Connecting Graphs with Equations exercises challenge students to write equations that correspond to given graphs. Video solutions for select problems are available in MyMathLab for School. ■ Relating Concepts Exercises Appearing at the end of selected exercise sets, these groups of exercises are designed so that students who work them in numerical order will follow a line of reasoning that leads to an understanding of how various topics and concepts are related. All answers to these exercises appear in the student answer section, and these exercises are assignable in MyMathLab for School.
xx PREFACE SUPPORT FOR REVIEW ANDTEST PREP Ample opportunities for review are found both within the chapters and at the ends of chapters. Quizzes and Summary Exercises, interspersed within chapters, provide a quick assessment of students’ understanding of the material presented up to that point in the chapter. Chapter Test Preps provide comprehensive study aids to help students prepare for tests. ■ Quizzes Students can periodically check their progress with in-chapter quizzes that appear in all chapters, beginning with Chapter 1. All answers, with corresponding section references, appear in the student answer section. These quizzes are assignable in MyMathLab for School. ■ Summary Exercises These sets of in-chapter exercises give students the all-important opportunity to work mixed review exercises, requiring them to synthesize concepts and select appropriate solution methods. ■ End-of-Chapter Test Prep Following the final numbered section in each chapter, the Test Prep provides a list of Key Terms, a list of New Symbols (if applicable), and a two-column Quick Review that includes a section-bysection summary of concepts with corresponding examples. This feature concludes with a comprehensive set of Review Exercises and a Chapter Test. The Test Prep, Review Exercises, and Chapter Test are assignable in MyMathLab for School.
mymathlabforschool.com Get the most out of MyMathLab for School Integrated Review in MyMathLab for School Integrated Review can be used in corequisite courses or simply to help students who enter a course without a full understanding of prerequisite skills and concepts. Premade, editable Integrated Review assignments are available to assign in the Assignment Manager. Integrated Review landing pages (shown below) are visible by default at the start of most chapters, providing objective-level review. • Students begin each chapter by completing a Skills Check to pinpoint which topics, if any, they need to review. • Personalized review homework provides extra support for students who need it on just the topics they didn’t master in the preceding Skills Check. • Additional review materials, including worksheets and videos, are available. Preparedness 3UHSDUHGQHVV LV RQH RI WKH ELJJHVW FKDOOHQJHV LQ PDQ\ PDWK FRXUVHV 3HDUVRQ RHUV D variety of content and course options to support students with just-in-time remediation and key-concept review as needed. MyMathLab for School for College Algebra and Trigonometry 7e (access code required) 0\0DWK/DE IRU 6FKRRO LV WLJKWO\ LQWHJUDWHG ZLWK DXWKRU VW\OH RHULQJ D UDQJH RI author-created resources, to give students a consistent experience. xxi PREFACE
mymathlabforschool.com Get the most out of MyMathLab for School New! Enhanced Sample Assignments Author Callie Daniels makes course set-up easier by giving instructors a starting point for each section. Following Callie’s best practices in the classroom, Enhanced Sample Assignments maximize students’ performance. • Section Prep Assignments include Example Videos with assessment questions. This assignment pairs with MyNotes. Students actively participate while taking notes from the Example Video and then work the related exercises. • Section Homework LQFOXGHV DXWKRU VHOHFWHG SUREOHPV DQG LQFUHDVHV LQ GLɝFXOW\ • Cumulative Review Homework Assignments draw from section homework questions FRYHUHG WR WKDW SRLQW LQ WKH FRXUVHȃKHOSLQJ VWXGHQWV SUHSDUH IRU D ȴQDO H[DP Updated! Videos Updated videos cover all topics in the text to support students outside of the classroom. Quick Review videos FRYHU NH\ GHȴQLWLRQV DQG SURFHGXUHV Example Solution YLGHRV RHU D GHWDLOHG VROXWLRQ SURFHVV IRU HYHU\ example in the textbook. Updated! MyNotes and MyClassroomExamples MyNotes give students a note-taking structure to use while they read the text or watch the MyMathLab for School videos. MyClassroomExamples RHU VWUXFWXUH IRU QRWHV WDNHQ GXULQJ lecture and are for use with the ClassroomExamples found in the Annotated Instructor Edition. Both sets of notes are available in MyMathLab for School and can be customized by the instructor. xxii PREFACE
mymathlabforschool.com Resources for Success Instructor Resources Online resources can be downloaded at mymathlabforschool.com or from www.pearson.com. Annotated Instructor’s Edition ISBN: 0135924499 / 9780135924495 Answers are included on the same page beside the text exercises where possible for quick reference. Helpful Teaching Tips and Classroom Examples are also provided. Online Instructor’s Solution Manual By Beverly Fusfield Provides complete solutions to all text exercises Online Instructor’s Testing Manual Includes diagnostic pretests, grouped by section, with answers provided Testgen® TestGen (www.pearsoned.com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. PowerPoint® Lecture Slides and Classroom Example PowerPoints • The PowerPoint Lecture Slides feature presentations written and designed specifically for this text, including figures and examples from the text. • Classroom Example PowerPoints include fully worked-out solutions to all Classroom Examples. Learning Catalytics™ With MyMathLab for School, instructors and students have access to Learning Catalytics, which instructors can use to generate class discussion, guide lectures, and actively engage students. Prebuilt Learning Catalytics questions have been created specifically for this text. Simply search the tag “LialPrecalculus” within the Learning Catalytics Question Library. Student Resources Additional resources enhance student success. Student’s Solution Manual By Beverly Fusfield Provides detailed solutions to all odd-numbered text exercises Video Lectures • Quick Reviews cover key definitions and procedures from each section. • Example Solutions walk students through the detailed solution process for every example in the textbook. MyNotes with Integrated Review Worksheets MyNotes offer structure for students as they watch videos or read the text. These are available as a printed supplement and in MyMathLab for School. • Includes textbook examples along with ample space for students to write solutions and notes • Includes key concepts along with prompts for students to read, write, and reflect on what they have just learned • Customizable—instructors can add their own examples or remove examples that are not covered in their course. Integrated Review Worksheets prepare students for the College Algebra and Trigonometry material. • Includes key terms, guided examples with ample space for students to work, and references to extra help in MyMathLab for School MyClassroomExamples • Available in MyMathLab for School and offer structure for classroom lecture • Includes Classroom Examples along with ample space for students to write solutions and notes • Includes key concepts along with fill-in-the-blank opportunities to keep students engaged • Customizable—instructors can add their own examples or remove Classroom Examples that are not covered in their course. xxiii PREFACE
xxiv ACKNOWLEDGMENTS ACKNOWLEDGMENTS We wish to thank the following individuals who provided valuable input into this edition of the text. Zalmond Abbondanza – Palm Beach State College Beyza Aslan – University of North Florida Kathy Autrey – Northwestern State University, University of Louisiana Shirley Brown – Weatherford College Jolina Cadilli – Cypress College Betty Collins – Hinds Community College Daniela Johnson – Valencia College Catelin Peay-Britt – Coahoma Community College Leslie Plumlee – Western Kentucky University Anthony Precella – Del Mar College Luminita Razaila – University of North Florida Jiahui Yao – Mt. San Antonio College Paula Young – Mt. San Antonio College Robert Young – Eastern Florida State College Our sincere thanks to those individuals at Pearson Education who have supported us throughout this revision: Dawn Murrin, Anne Kelly, Lauren Morse, Joe Vetere, Mary Catherine Connors, and Jonathan Krebs. Terry McGinnis continues to provide behind-the-scenes guidance for both content and production. We have come to rely on her expertise during all phases of the revision process. Carol Merrigan provided excellent production work. Special thanks go out to Paul Lorczak, Hal Whipple, and Sarah Sponholz for their excellent accuracy-checking. We thank Lucie Haskins, who provided an accurate index. We appreciate the valuable suggestions for Chapter 5 from Mary Hill of College of Dupage and the detailed suggestions from Kyle Linden of St. Charles Community College. As an author team, we are committed to providing the best possible college algebra and trigonometry course to help instructors teach and students succeed. As we continue to work toward this goal, we welcome any comments or suggestions you might send, via e-mail, to math@pearson.com. Margaret L. Lial John Hornsby David I. Schneider Callie J. Daniels
R Review of Basic Concepts Positive and negative numbers, used to represent gains and losses on a board such as this one, are examples of real numbers encountered in applications of mathematics. 1 R.1 Fractions, Decimals, and Percents R.2 Sets and Real Numbers R.3 Real Number Operations and Properties R.4 Integer and Rational Exponents R.5 Polynomials R.6 Factoring Polynomials R.7 Rational Expressions R.8 Radical Expressions
2 CHAPTER R Review of Basic Concepts The shaded region represents of the circle. 3 8 Figure 1 Writing a Fraction in LowestTerms Step 1 Write the numerator and denominator in factored form. Step 2 Replace each pair of factors common to the numerator and denominator with 1. Step 3 Multiply the remaining factors in the numerator and in the denominator. (This procedure is sometimes called “simplifying the fraction.”) EXAMPLE 1 Writing Fractions in LowestTerms Write each fraction in lowest terms. (a) 10 15 (b) 15 45 (c) 150 200 SOLUTION (a) 10 15 = 2 # 5 3 # 5 = 2 3 # 5 5 = 2 3 # 1 = 2 3 5 is the greatest common factor of 10 and 15. (b) 15 45 = 1 # 15 3 # 15 = 1 3 # 1 = 1 3 (c) 150 200 = 3 # 50 4 # 50 = 3 4 # 1 = 3 4 50 is the greatest common factor of 150 and 200. Another strategy is to choose any common factor and work in stages. 150 200 = 15 # 10 20 # 10 = 3 # 5 # 10 4 # 5 # 10 = 3 4 # 1 # 1 = 3 4 The same answer results. S Now Try Exercises 7 and 15. Remember to write 1 in the numerator. Recall that fractions are a way to represent parts of a whole. See Figure 1. In a fraction, the numerator gives the number of parts being represented. The denominator gives the total number of equal parts in the whole. The fraction bar represents division Qa b = a , bR. Fraction bar 3 8 Numerator Denominator A fraction is classified as either a proper fraction or an improper fraction. Proper fractions 1 5 , 2 7 , 9 10 , 23 25 Numerator is less than denominator. Value is less than 1. Improper fractions 3 2 , 5 5 , 11 7 , 28 4 Numerator is greater than or equal to denominator. Value is greater than or equal to 1. R.1 Fractions, Decimals, and Percents ■ LowestTerms of a Fraction ■ Improper Fractions and Mixed Numbers ■ Operations with Fractions ■ Decimals as Fractions ■ Operations with Decimals ■ Fractions as Decimals ■ Percents as Decimals and Decimals as Percents ■ Percents as Fractions and Fractions as Percents Lowest Terms of a Fraction A fraction is in lowest terms when the numerator and denominator have no factors in common (other than 1).
3 R.1 Fractions, Decimals, and Percents EXAMPLE 2 Converting an Improper Fraction to a Mixed Number Write 59 8 as a mixed number. SOLUTION Because the fraction bar represents division Qa b = a , b, or b)aR, divide the numerator of the improper fraction by the denominator. Denominator of fraction 7 8)59 56 3 Quotient Numerator of fraction Remainder 59 8 = 7 3 8 S Now Try Exercise 17. EXAMPLE 3 Converting a Mixed Number to an Improper Fraction Write 6 4 7 as an improper fraction. SOLUTION Multiply the denominator of the fraction by the natural number, and then add the numerator to obtain the numerator of the improper fraction. 7 # 6 = 42 and 42 + 4 = 46 The denominator of the improper fraction is the same as the denominator in the mixed number, which is 7 here. 6 4 7 = 7 # 6 + 4 7 = 46 7 S Now Try Exercise 21. Improper Fractions and Mixed Numbers A mixed number is a single number that represents the sum of a natural (counting) number and a proper fraction. Mixed number 2 3 4 = 2 + 3 4 Operations with Fractions Figure 2 illustrates multiplying fractions. of is equivalent to ∙ , which equals of the circle. 3 8 3 4 1 2 3 4 1 2 1 2 Figure 2 Multiplying Fractions If a b and c d are fractions 1b≠0, d≠02, then a b # c d = a # c b # d . That is, to multiply two fractions, multiply their numerators and then multiply their denominators.
4 CHAPTER R Review of Basic Concepts Two numbers are reciprocals of each other if their product is 1. For example, 3 4 # 4 3 = 12 12 , or 1. Division is the inverse or opposite of multiplication, and as a result, we use reciprocals to divide fractions. Figure 3 illustrates dividing fractions. EXAMPLE 4 Multiplying Fractions Multiply 3 8 # 4 9. Write the answer in lowest terms. SOLUTION 3 8 # 4 9 = 3 # 4 8 # 9 Multiply numerators. Multiply denominators. = 12 72 Multiply. = 1 # 12 6 # 12 The greatest common factor of 12 and 72 is 12. = 1 6 1 # 12 6 # 12 = 1 6 # 1 = 1 6 S Now Try Exercise 27. Make sure the product is in lowest terms. 4 4 is equivalent to ∙ , which equals of the circle. 1 8 1 2 1 2 1 4 1 2 Figure 3 Dividing Fractions If a b and c d are fractions 1b≠0, d≠0, c ≠02, then a b ÷ c d = a b # d c . That is, to divide by a fraction, multiply by its reciprocal. (a) 3 4 , 8 5 = 3 4 # 5 8 Multiply by 5 8 , the reciprocal of 8 5. = 3 # 5 4 # 8 Multiply numerators. Multiply denominators. = 15 32 (b) 5 8 , 10 = 5 8 # 1 10 Multiply by 1 10 , the reciprocal of 10. = 5 # 1 8 # 2 # 5 Multiply and factor. = 1 16 Make sure the answer is in lowest terms. Remember to write 1 in the numerator. Think of 10 as 10 1 here. EXAMPLE 5 Dividing Fractions Divide. Write answers in lowest terms as needed. (a) 3 4 , 8 5 (b) 5 8 , 10 (c) 1 2 3 , 4 1 2 SOLUTION
5 R.1 Fractions, Decimals, and Percents (c) 1 2 3 , 4 1 2 = 5 3 , 9 2 Write each mixed number as an improper fraction. = 5 3 # 2 9 Multiply by 2 9, the reciprocal of 9 2. = 10 27 Multiply. The quotient is in lowest terms. S Now Try Exercises 37, 43, and 49. Adding and Subtracting Fractions If a b and c b are fractions 1b≠02, then add or subtract as follows. a b + c b = a +c b and a b − c b = a −c b That is, to add or subtract two fractions having the same denominator, add or subtract the numerators and keep the same denominator. If the denominators are different, first find the least common denominator (LCD). Write each fraction as an equivalent fraction with this denominator. Then add or subtract as above. EXAMPLE 6 Adding and Subtracting Fractions Add or subtract as indicated. Write answers in lowest terms as needed. (a) 2 10 + 3 10 (b) 4 15 + 5 9 (c) 15 6 - 4 9 (d) 4 1 2 - 1 3 4 SOLUTION (a) 2 10 + 3 10 = 2 + 3 10 Add numerators. Keep the same denominator. = 5 10 = 1 2 Write in lowest terms. 15 = 3 # 5 and 9 = 3 # 3, so the LCD is 3 # 3 # 5 = 45. Write equivalent fractions with the common denominator. Add numerators. Keep the same denominator. (b) 4 15 + 5 9 = 4 15 # 3 3 + 5 9 # 5 5 = 12 45 + 25 45 = 37 45 1 8 3 8 1 4 8 5 1 2 5 Figure 4 Adding Fractions 3 8 1 8 2 2 8 5 1 4 5 Figure 5 Subtracting Fractions Figures 4 and 5 illustrate adding and subtracting fractions.
6 CHAPTER R Review of Basic Concepts (c) 15 16 - 4 9 = 15 16 # 9 9 - 4 9 # 16 16 Because 16 and 9 have no common factors except 1, the LCD is 16 # 9 = 144. = 135 144 - 64 144 Write equivalent fractions with the common denominator. = 71 144 Subtract numerators. Keep the common denominator. (d) Method 1 4 1 2 - 1 3 4 = 9 2 - 7 4 Write each mixed number as an improper fraction. = 18 4 - 7 4 Find a common denominator. The LCD is 4. = 11 4 , or 2 3 4 Subtract. Write as a mixed number. Method 2 4 1 2 = 4 2 4 = 3 6 4 The LCD is 4. 4 2 4 = 3 + 1 + 2 4 = 3 + 4 4 + 2 4 = 3 6 4 -1 3 4 = -1 3 4 = -1 3 4 2 3 4 , or 11 4 The same answer results. S Now Try Exercises 55, 57, 71, and 75. Fractions are one way to represent parts of a whole. Another way is with a decimal fraction or decimal, a number written with a decimal point. 9.25, 14.001, 0.3 Decimal numbers See Figure 6. Each digit in a decimal number has a place value, as shown below. 4, 8 9 6, 3 2 8 millions ones or units tens hundreds thousands ten thousands hundred thousands 9 7 2 1 tenths hundredths ten-thousandths thousandths Whole number part Decimal point read “and” Fractional part • Each successive place value is ten times greater than the place value to its right and one-tenth as great as the place value to its left. Think: 9 2 # 2 2 = 18 4 3 parts of the whole 10 are shaded. As a fraction, 3 10 of the figure is shaded. As a decimal, 0.3 is shaded. Both of these numbers are read “three-tenths.” FIGURE 6
7 R.1 Fractions, Decimals, and Percents Decimals as Fractions Converting a Decimal to a Fraction Read the decimal using the correct place value. Write it in fractional form just as it is read. • The numerator will be the digits to the right of the decimal point. • The denominator will be a power of 10—that is, 10 for tenths, 100 for hundredths, and so on. EXAMPLE 7 Writing Decimals as Fractions Write each decimal as a fraction. (Do not write in lowest terms.) (a) 0.95 (b) 0.056 (c) 4.2095 SOLUTION (a) We read 0.95 as “ninety-five hundredths.” 0.95 = 95 100 For hundredths (b) We read 0.056 as “fifty-six thousandths.” 0.056 = 56 1000 For thousandths (c) We read 4.2095, which is greater than 1, as “Four and two thousand ninety-five ten-thousandths.” 4 .2095 = 4 2095 10,000 Write the decimal number as a mixed number. = 42,095 10,000 Write the mixed number as an improper fraction. S Now Try Exercises 85, 89, and 91. Do not confuse 0.056 with 0.56, read “fifty-six hundredths,” which is the fraction 56 100. Think: 10,000 # 4 + 2095 Operations with Decimals EXAMPLE 8 Adding and Subtracting Decimals Add or subtract as indicated. (a) 6.92 + 14.8 + 3.217 (b) 47.6 - 32.509 SOLUTION (a) Place the digits of the decimal numbers in columns by place value. Attach zeros as placeholders. 6.92 6.920 14.8 becomes 14.800 + 3.217 + 3.217 24.937 Attach 0s. (b) 47.6 47.600 - 32.509 becomes - 32.509 15.091 Write the decimal numbers in columns, attaching 0s to 47.6. S Now Try Exercises 93 and 97. Be sure to line up decimal points. 6.92 is equivalent to 6.920. 14.8 is equivalent to 14.800.
8 CHAPTER R Review of Basic Concepts Multiplying and Dividing by Powers of 10 (Shortcuts) • To multiply by a power of 10, move the decimal point to the right as many places as the number of zeros. • To divide by a power of 10, move the decimal point to the left as many places as the number of zeros. In both cases, insert 0s as placeholders if necessary. EXAMPLE 9 Multiplying and Dividing Decimals Multiply or divide as indicated. In part (c), round the answer to two decimal places. (a) 29.3 * 4.52 (b) 0.05 * 0.3 (c) 8.949 , 1.25 SOLUTION (a) Multiply normally. Place the decimal point in the answer as shown. 29.3 * 4.52 586 1465 1172 132.436 (b) Here 5 * 3 = 15. Be careful placing the decimal point. 2 decimal places 1 decimal place 0.05 * 0.3 = 0.015 1 decimal place 2 decimal places 1 + 2 = 3 3 decimal places 2 + 1 = 3 decimal places Attach 0 as a placeholder in the tenths place. Do not write 0.150. (c) 1.25 )8.949 Move each decimal point two places to the right. 7.159 125)894.900 875 199 125 740 625 1150 1125 25 Move the decimal point straight up, and divide as with whole numbers. Attach 0s as placeholders. We carried out the division to three decimal places so that we could round the answer to two decimal places, obtaining 7.16. S Now Try Exercises 105 and 113. NOTE To round 7.159 in Example 9(c) to two decimal places (that is, to the nearest hundredth), we look at the digit to the right of the hundredths place. • If this digit is 5 or greater, we round up. • If this digit is less than 5, we drop the digit(s) beyond the desired place. Hundredths place 7.159 9, the digit to the right of the hundredths place, is 5 or greater. ≈ 7.16 Round 5 up to 6. ≈ means is approximately equal to.
9 R.1 Fractions, Decimals, and Percents S Now Try Exercises 117 and 125. EXAMPLE 10 Multiplying and Dividing by Powers of 10 Multiply or divide as indicated. (a) 48.731 * 10 (b) 48.731 * 1000 (c) 48.731 , 10 (d) 48.731 , 1000 SOLUTION (a) 48.731 * 10 = 48.731 = 487.31 (b) 48.731 * 1000 = 48.731 = 48,731 Move the decimal point one place to the right. Move the decimal point three places to the right. (c) 48.731 , 10 = 48.731 = 4.8731 Move the decimal point one place to the left. (d) 48.731 , 1000 = 048.731 = 0.048731 Move the decimal point three places to the left. Converting a Fraction to a Decimal Because a fraction bar indicates division, write a fraction as a decimal by dividing the numerator by the denominator. EXAMPLE 11 Writing Fractions as Decimals Write each fraction as a decimal. (a) 19 8 (b) 2 3 SOLUTION (a) 19 8 2.375 8)19.000 16 30 24 60 56 40 40 0 Divide 19 by 8. Add a decimal point and as many 0s as necessary to 19. 19 8 = 2.375 Terminating decimal (b) 2 3 0.6666c 3)2.0000c 18 20 18 20 18 20 18 2 2 3 = 0.6666c Repeating decimal = 0.6 A bar is written over the repeating digit(s). ≈0.667 Nearest thousandth S Now Try Exercises 141 and 147. Fractions as Decimals
10 CHAPTER R Review of Basic Concepts Percent, Fraction, and Decimal Equivalents 1%= 1 100 =0.01, 10%= 10 100 =0.10, 100%= 100 100 =1 For example, 73% means “73 per one hundred.” 73%= 73 100 = 0.73 Essentially, we are dropping the % symbol from 73% and dividing 73 by 100. Doing this moves the decimal point, which is understood to be after the 3, two places to the left. Writing 0.73 as a percent is the opposite process. 0.73 = 0.73 # 100%= 73% 100%= 1 Moving the decimal point two places to the right and attaching a % symbol gives the same result. EXAMPLE 12 Converting Percents and Decimals by Moving the Decimal Point Convert each percent to a decimal and each decimal to a percent. (a) 45% (b) 250% (c) 9% (d) 0.57 (e) 1.5 (f) 0.007 SOLUTION (a) 45%= 0.45 (b) 250%= 2.50, or 2.5 (c) 9%= 09%= 0.09 (d) 0.57 = 57% (e) 1.5 = 1.50 = 150% (f) 0.007 = 0.7% S Now Try Exercises 151 and 161. Converting Percents and Decimals (Shortcuts) • To convert a percent to a decimal, move the decimal point two places to the left and drop the % symbol. • To convert a decimal to a percent, move the decimal point two places to the right and attach a % symbol. Drop %. Divide by 100. (Move decimal point two places left.) Decimal Percent 0.85 85% Multiply by 100. Attach %. (Move decimal point two places right.) Percents as Decimals and Decimals as Percents The word percent means “per 100.” Percent is written with the symbol %. “One percent” means “one per one hundred,” or “one one-hundredth.” See Figure 7. Figure 7 35 of the 100 squares are shaded. That is, 35 100, or 35%, of the figure is shaded.
RkJQdWJsaXNoZXIy NjM5ODQ=