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A variety of options are available to personalize learning in MyLab Math: ❍❍ With Personalized Homework, students take a quiz or test and receive a subsequent homework assignment that is personalized based on their performance. This way, students can focus on just the topics they have not yet mastered. ❍❍ Skill Builder offers adaptive practice that is designed to increase students’ ability to complete their assignments. By monitoring student performance on their homework, Skill Builder adapts to each student’s needs and provides just-in-time, in-assignment practice to help them improve their proficiency of key learning objectives. Available for select MyLab™ courses. Visit pearson.com/mylab/math and click Get Trained to make sure you’re getting the most out of MyLab Math. Intermediate EDITION 13 Algebra Margaret L. Lial American River College John Hornsby University of New Orleans Terry McGinnis Vice President, Courseware Portfolio Management: Chris Hoag NOTICE: Director, Courseware Portfolio Management: Michael Hirsch This work is Courseware Portfolio Manager: Karen Montgomery solely for the use Courseware Portfolio Assistant: Kayla Shearns of instructors and Managing Producer: Scott Disanno administrators for Content Producer: Lauren Morse the purpose of Producers: Stacey Miller and Noelle Saligumba teaching courses Managing Producer: Vicki Dreyfus and assessing Associate Content Producer, TestGen: Rajinder Singh Content Managers, MathXL: Eric Gregg and Dominick Franck student learning. 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Such references are not in- tended 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, 1949- author. | McGinnis, Terry, author. Title: Intermediate algebra / Margaret L. Lial (American River College), John Hornsby (University of New Orleans), Terry McGinnis. Description: 13th edition. | Boston : Pearson, [2020] | Includes index. Identifiers: LCCN 2018048199 | ISBN 9780134895987 (student edition) | ISBN 0134895983 (student edition) | ISBN 9780134996752 (annotated instructor edition) | ISBN 0134996755 (annotated instructor edition) Subjects: LCSH: Algebra--Textbooks. Classification: LCC QA152.3 .L534 2020 | DDC 512.9--dc23 LC record available at https://lccn.loc.gov/2018048199 1 18 ISBN 13: 978-0-13-489598-7 ISBN 10: 0-13-489598-3 CONTENTS Preface vii Study Skills S-1 STUDY SKILL 1 Using Your Math Text S-1 STUDY SKILL 6 Managing Your Time S-6 STUDY SKILL 2 Reading Your Math Text S-2 STUDY SKILL 7 Reviewing a Chapter S-7 STUDY SKILL 3 Taking Lecture Notes S-3 STUDY SKILL 8 Taking Math Tests S-8 STUDY SKILL 4 Completing Your Homework S-4 STUDY SKILL 9 Analyzing Your Test Results S-9 STUDY SKILL 5 Using Study Cards S-5 STUDY SKILL 10 Preparing for Your Math Final Exam S-10 R Review of the Real Number System 1 R.1 Fractions, Decimals, and Percents 1 R.5 Properties of Real Numbers 45 R.2 Basic Concepts from Algebra 14 Chapter R Summary 52 R.3 Operations on Real Numbers 26 Chapter R Test 54 R.4 Exponents, Roots, and Order of Operations 36 1 Linear Equations, Inequalities, and Applications 55 1.1 Linear Equations in One Variable 56 1.7 Absolute Value Equations and Inequalities 125 1.2 Formulas and Percent 65 SUMMARY EXERCISES Solving Linear and Absolute Value 1.3 Applications of Linear Equations 78 Equations and Inequalities 136 1.4 Further Applications of Linear Equations 92 Chapter 1 Summary 137 Chapter 1 Review Exercises 142 SUMMARY EXERCISES Applying Problem-Solving Chapter 1 Mixed Review Exercises 145 Techniques 101 Chapter 1 Test 146 1.5 Linear Inequalities in One Variable 103 Chapters R and 1 Cumulative Review 1.6 Set Operations and Compound Inequalities 116 Exercises 148 2 Linear Equations, Graphs, and Functions 149 2.1 Linear Equations in Two Variables 150 2.5 Introduction to Relations and Functions 199 2.2 The Slope of a Line 161 2.6 Function Notation and Linear Functions 210 2.3 Writing Equations of Lines 176 Chapter 2 Summary 219 SUMMARY EXERCISES Finding Slopes and Equations Chapter 2 Review Exercises 222 of Lines 191 Chapter 2 Mixed Review Exercises 224 Chapter 2 Test 225 2.4 Linear Inequalities in Two Variables 192 Chapters R–2 Cumulative Review Exercises 227 iii iv Contents 3 Systems of Linear Equations 229 3.1 Systems of Linear Equations in Two Chapter 3 Summary 270 Variables 230 Chapter 3 Review Exercises 273 3.2 Systems of Linear Equations in Three Chapter 3 Mixed Review Exercises 275 Variables 245 Chapter 3 Test 276 Chapters R–3 Cumulative Review 3.3 Applications of Systems of Linear Exercises 277 Equations 254 4 Exponents, Polynomials, and Polynomial Functions 279 4.1 Integer Exponents 280 4.6 Dividing Polynomials 324 4.2 Scientific Notation 290 Chapter 4 Summary 331 4.3 Adding and Subtracting Polynomials 296 Chapter 4 Review Exercises 334 Chapter 4 Mixed Review Exercises 337 4.4 Polynomial Functions, Graphs, and Chapter 4 Test 337 Composition 302 Chapters R–4 Cumulative Review 4.5 Multiplying Polynomials 315 Exercises 338 5 Factoring 341 5.1 Greatest Common Factors and Factoring Chapter 5 Summary 376 by Grouping 342 Chapter 5 Review Exercises 379 5.2 Factoring Trinomials 348 Chapter 5 Mixed Review Exercises 381 Chapter 5 Test 382 5.3 Special Factoring 356 Chapters R–5 Cumulative Review 5.4 A General Approach to Factoring 363 Exercises 382 5.5 Solving Quadratic Equations Using the Zero-Factor Property 367 6 Rational Expressions and Functions 385 6.1 Rational Expressions and Functions; Multiplying 6.5 Applications of Rational Expressions 421 and Dividing 386 6.6 Variation 433 6.2 Adding and Subtracting Rational Chapter 6 Summary 443 Expressions 396 Chapter 6 Review Exercises 447 6.3 Complex Fractions 405 Chapter 6 Mixed Review Exercises 449 6.4 Equations with Rational Expressions Chapter 6 Test 450 and Graphs 411 Chapters R–6 Cumulative Review Exercises 452 SUMMARY EXERCISES S implifying Rational Expressions vs. Solving Rational Equations 419 Contents v 7 Roots, Radicals, and Root Functions 455 7.1 Radical Expressions and SUMMARY EXERCISES Performing Operations with Radicals Graphs 456 and Rational Exponents 502 7.2 Rational Exponents 464 7.6 Solving Equations with Radicals 503 7.3 Simplifying Radicals, the Distance 7.7 Complex Numbers 510 Formula, and Circles 473 Chapter 7 Summary 518 7.4 Adding and Subtracting Radical Chapter 7 Review Exercises 523 Expressions 487 Chapter 7 Mixed Review Exercises 526 7.5 Multiplying and Dividing Radical Chapter 7 Test 527 Expressions 492 Chapters R–7 Cumulative Review Exercises 528 8 Quadratic Equations, Inequalities, and Functions 531 8.1 The Square Root Property and 8.6 More about Parabolas and Their Completing the Square 532 Applications 579 8.2 The Quadratic Formula 541 8.7 Polynomial and Rational Inequalities 590 8.3 Equations That Lead to Quadratic Methods 549 Chapter 8 Summary 599 SUMMARY EXERCISES Applying Methods for Solving Chapter 8 Review Exercises 603 Quadratic Equations 560 Chapter 8 Mixed Review Exercises 606 Chapter 8 Test 607 8.4 Formulas and Further Applications 561 Chapters R–8 Cumulative Review 8.5 Graphs of Quadratic Functions 570 Exercises 609 9 Inverse, Exponential, and Logarithmic Functions 611 9.1 Inverse Functions 612 Chapter 9 Summary 663 9.2 Exponential Functions 621 Chapter 9 Review Exercises 667 Chapter 9 Mixed Review Exercises 670 9.3 Logarithmic Functions 630 Chapter 9 Test 672 9.4 Properties of Logarithms 639 Chapters R–9 Cumulative Review 9.5 Common and Natural Logarithms 646 Exercises 673 9.6 Exponential and Logarithmic Equations; Further Applications 654 10 Nonlinear Functions, Conic Sections, and Nonlinear Systems 677 10.1 Additional Graphs of Functions 678 Chapter 10 Summary 715 10.2 Circles Revisited and Ellipses 684 Chapter 10 Review Exercises 718 Chapter 10 Mixed Review Exercises 720 10.3 Hyperbolas and Functions Defined Chapter 10 Test 721 by Radicals 694 Chapters R–10 Cumulative Review 10.4 Nonlinear Systems of Equations 702 Exercises 722 10.5 Second-Degree Inequalities and Systems of Inequalities 708 vi Contents 11 Further Topics in Algebra 725 11.1 Sequences and Series 726 Chapter 11 Summary 756 11.2 Arithmetic Sequences 732 Chapter 11 Review Exercises 759 Chapter 11 Mixed Review Exercises 760 11.3 Geometric Sequences 740 Chapter 11 Test 761 11.4 The Binomial Theorem 750 Chapters R–11 Cumulative Review Exercises 762 Appendix Synthetic Division 765 Answers to Selected Exercises A-1 Photo Credits C-1 Index I-1 PREFACE now include a Now Try This section to help students imple- WELCOME TO THE 13TH EDITION ment the specific skill. The first edition of Marge Lial’s Intermediate Algebra was published in 1972, and now we are pleased to present REVISED EXPOSITION With each edition of the text, we con- tinue to polish and improve discussions and presentations the 13th edition—with the same successful, well-rounded of topics to increase readability and student understanding. framework that was established 50 years ago and updated This edition is no exception. to meet the needs of today’s students and professors. The names Lial and Miller, two faculty members from American NEW FIGURES AND DIAGRAMS For visual learners, we have River College in Sacramento, California, have become syn- included more than 50 new mathematical figures, graphs, and onymous with excellence in Developmental Mathematics, diagrams, including several new “hand drawn” style graphs. Precalculus, Finite Mathematics, and Applications-Based These are meant to suggest what a student who is graphing Calculus. with paper and pencil should obtain. We use this style when With Chuck Miller’s passing in 1986, Marge Lial was introducing a particular type of graph for the first time. joined by a team of carefully selected coauthors who part- nered with her. John Hornsby (University of New Orleans) ENHANCED USE OF PEDAGOGICAL COLOR We have thor- joined Marge in this capacity in 1992, and in 1999, Terry oughly reviewed the use of pedagogical color in discussions McGinnis became part of this developmental author team. and examples and have increased its use whenever doing so Since Marge’s passing in 2012, John and Terry have dedi- would enhance concept development, emphasize important cated themselves to carrying on the Lial/Miller legacy. steps, or highlight key procedures. In the preface to the first edition of Intermediate Algebra, Marge Lial wrote INCREASED Concept Check AND WHAT WENT WRONG? EXERCISES The number of Concept Check exercises, which “ . . . the strongest theme . . . is a combination of facilitate students’ mathematical thinking and conceptual readability and suitability for the book’s intended understanding, and which begin each exercise set, has been audience: students who are not completely self- increased. We have also more than doubled the number confident in mathematics as they come to the course, of  WHAT WENT WRONG? exercises that highlight com- but who must be self-confident and proficient . . . by mon student errors. the end of the course.” INCREASED RELATING CONCEPTS EXERCISES We have Today’s Lial author team upholds these same standards. With doubled the number of these flexible groups of exercises, the publication of the 13th edition of Intermediate Algebra, which are located at the end of many exercise sets. These we proudly present a complete course program for students sets of problems were specifically written to help students who need developmental algebra. Revisions to the core text, tie concepts together, compare and contrast ideas, identify working in concert with such innovations in the MyLab Math and describe patterns, and extend concepts to new situations. course as Skill Builder and Learning Catalytics, combine to They may be used by individual students or by pairs or small provide superior learning opportunities appropriate for all groups working collaboratively. All answers to these exer- types of courses (traditional, hybrid, online). cises appear in the student answer section. We hope you enjoy using it as much as we have enjoyed writing it. We welcome any feedback that you have as you ENHANCED MYLAB MATH RESOURCES MyLab exercise cov- review and use this text. erage in the revision has been expanded, and video coverage has also been expanded and updated to a modern format for today’s students. WHAT WENT WRONG? problems and all WHAT’S NEW IN THIS EDITION? RELATING CONCEPTS exercise sets (both even- and odd- We are pleased to offer the following new features and numbered problems) are now assignable in MyLab Math. resources in the text and MyLab. SKILL BUILDER These exercises offer just-in-time additional IMPROVED STUDY SKILLS These special activities are now adaptive practice in MyLab Math. The adaptive engine grouped together at the front of the text, prior to Chapter R. tracks student performance and delivers, to each individual, Study Skills Reminders that refer students to specific Study questions that adapt to his or her level of understanding. This Skills are found liberally throughout the text. Many Study Skills new feature enables instructors to assign fewer questions for vii viii Preface homework, allowing students to complete as many or as few Identifying functions and domains from equations questions as they need. (Section 2.6) LEARNING CATALYTICS This new student response tool uses Determining strategies for factoring polynomials students’ own devices to engage them in the learning process. (Section 5.4) Problems that draw on prerequisite skills are included at the Solving quadratic equations with double solutions beginning of each section to gauge student readiness for the (Section 5.5) section. Accessible through MyLab Math and customizable to instructors’ specific needs, these problems can be used to Solving rational equations (Section 6.4) generate class discussion, promote peer-to-peer learning, and Concepts and relationships among real numbers, provide real-time feedback to instructors. More information nonreal complex numbers, and imaginary numbers; can be found via the Learning Catalytics link in MyLab Math. simplifying powers of i (Section 7.7) Specific exercises notated in the text can be found by search- ing LialIntermediate# where the # is the chapter number. Solving quadratic equations using the quadratic for- mula (Section 8.2) Solving exponential and logarithmic equations CONTENT CHANGES (Sections 9.2, 9.3) Specific content changes include the following: ● Exercise sets have been scrutinized and updated with a LIAL DEVELOPMENTAL HALLMARK FEATURES renewed focus on conceptual understanding and skill de- velopment. Even and odd pairing of the exercises, an im- We have enhanced the following popular features, each of portant feature of the text, has been carefully reviewed. which is designed to increase ease of use by students and/ or instructors. ● Real world data in all examples and exercises and their accompanying graphs has been updated. ● Emphasis on Problem-Solving We introduce our six- step problem-solving method in Chapter 2 and integrate ● An increased emphasis on fractions, decimals, and per- it throughout the text. The six steps, Read, Assign a cents appears throughout the text. Chapter R begins with Variable, Write an Equation, Solve, State the Answer, a new section that thoroughly reviews these topics. And and Check, are emphasized in boldface type and we have included an all-new set of Cumulative Review repeated in examples and exercises to reinforce the Exercises, many of which focus on fractions, decimals, problem-solving process for students. We also provide and percents, at the end of Chapter 1. Sets of Cumulative students with PROBLEM-SOLVING HINT boxes that feature Review Exercises in subsequent chapters now begin with helpful problem-solving tips and strategies. new exercises that review skills related to these topics. ● Helpful Learning Objectives We begin each section ● Solution sets of linear inequalities in Sections 1.5–1.7 with clearly stated, numbered objectives, and the in- are now graphed first before writing them using interval cluded material is directly keyed to these objectives so notation. that students and instructors know exactly what is cov- ● Scientific notation is covered in a separate section in ered in each section. Chapter 4. ● Cautions and Notes One of the most popular features of ● Chapters 10 and 11 on Nonlinear Functions, Conic previous editions is our inclusion of information marked Sections, Nonlinear Systems, and Further Topics in ! CAUTION and NOTE to warn students about common Algebra, previously available online in MyLab Math, errors and to emphasize important ideas throughout the are now included in the text. The material has been fully exposition. The updated text design makes them easy revised and updated. to spot. ● Presentations of the following topics have been en- ● Comprehensive Examples The new edition features hanced and expanded, often including new examples a multitude of step-by-step, worked-out examples and exercises: that include pedagogical color, helpful side com- ments, and special pointers. We give special attention Evaluating exponential expressions (Section R.4) to checking example solutions—more checks, desig- Geometric interpretation of slope as rise/run (Sec- nated using a special CHECK tag and ✓, are included tion 2.2) than in past editions. Solving systems of linear equations in three variables ● More Pointers There are more pointers in examples and (Section 3.2) discussions throughout this edition of the text. They