BEGINNING & I N T E R M E D I AT E A LG E B R A LIAL / HORNSBY / McGINNIS seventh edition EDITION Beginning and Intermediate Algebra Margaret L Lial American River College John Hornsby University of New Orleans Terry McGinnis Vice President, Courseware Portfolio Management: Chris Hoag Director, Courseware Portfolio Management: Michael Hirsch Courseware Portfolio Manager: Karen Montgomery Courseware Portfolio Assistant: Kayla Shearns Managing Producer: Scott Disanno Content Producer: Lauren Morse Producers: Stacey Miller and Noelle Saligumba Managing Producer: Vicki Dreyfus Associate Content Producer, TestGen: Rajinder Singh Content Managers, MathXL: Eric Gregg and Dominick Franck Manager, Courseware QA: Mary Durnwald Senior Product Marketing Manager: Alicia Frankel Product Marketing Assistant: Brooke Imbornone Senior Author Support/Technology Specialist: Joe Vetere Full Service Vendor, Cover Design, Composition: Pearson CSC Full Service Project Management: Pearson CSC (Carol Merrigan) 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third-party content appear on page C-1, which constitutes an extension of this copyright page PEARSON, ALWAYS LEARNING, MyLab™ Math, MathXL, and TestGen are exclusive trademarks in the U.S and/or other countries owned by Pearson Education, Inc or its affiliates Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos 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, 1949- author | McGinnis, Terry, author Title: Beginning and intermediate algebra / Margaret L Lial (American River College), John Hornsby (University of New Orleans), Terry McGinnis Description: 7th edition | Boston : Pearson, [2020] | Includes index Identifiers: LCCN 2019000104 | ISBN 9780134895994 (student edition) | ISBN 0134895991 (student edition) Subjects: LCSH: Algebra Textbooks Classification: LCC QA152.3 L52 2020 | DDC 512.9 dc23 LC record available at https://lccn.loc.gov/2019000104 1 19 ISBN 13: 978-0-13-489599-4 ISBN 10: 0-13-489599-1 CONTENTS Preface viii 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 Prealgebra Review R.1 Fractions 1 R.2 Decimals and Percents 16 The Real Number System 27 1.1 Exponents, Order of Operations, and Inequality 28 1.2 Variables, Expressions, and Equations 36 1.3 Real Numbers and the Number Line 42 1.4 Adding and Subtracting Real Numbers 51 1.5 Multiplying and Dividing Real Numbers 65 SUMMARY EXERCISES Performing Operations with Real Numbers 77 1.6 Properties of Real Numbers 78 1.7 Simplifying Expressions 88 Chapter Summary 94 Chapter Review Exercises 97 Chapter Mixed Review Exercises 100 Chapter Test 100 Chapters R and Cumulative Review Exercises 102 Linear Equations and Inequalities in One Variable 103 2.1 The Addition Property of Equality 104 2.7 Ratio, Proportion, and Percent 157 2.2 The Multiplication Property of Equality 112 2.8 Further Applications of Linear Equations 169 2.3 Solving Linear Equations Using Both Properties of Equality 117 2.4 Clearing Fractions and Decimals When Solving Linear Equations 125 SUMMARY EXERCISES Applying Methods for Solving Linear Equations 131 2.5 Applications of Linear Equations 132 2.6 Formulas and Additional Applications from Geometry 146 2.9 Solving Linear Inequalities 182 Chapter Summary 196 Chapter Review Exercises 200 Chapter Mixed Review Exercises 203 Chapter Test 204 Chapters R–2 Cumulative Review Exercises 205 iii iv Contents Linear Equations in Two Variables 207 3.1 Linear Equations and Rectangular Coordinates 208 SUMMARY EXERCISES Applying Graphing and Equation-Writing Techniques for Lines 261 Chapter Summary 262 Chapter Review Exercises 265 Chapter Mixed Review Exercises 267 Chapter Test 268 Chapters R–3 Cumulative Review Exercises 269 3.2 Graphing Linear Equations in Two Variables 219 3.3 The Slope of a Line 231 3.4 Slope-Intercept Form of a Linear Equation 245 3.5 Point-Slope Form of a Linear Equation and Modeling 253 Exponents and Polynomials 271 4.1 The Product Rule and Power Rules for Exponents 272 4.6 Special Products 317 4.7 Dividing Polynomials 323 4.2 Integer Exponents and the Quotient Rule 280 Chapter Summary 332 Chapter Review Exercises 335 Chapter Mixed Review Exercises 338 Chapter Test 338 Chapters R–4 Cumulative Review Exercises 340 SUMMARY EXERCISES Applying the Rules for Exponents 290 4.3 Scientific Notation 291 4.4 Adding, Subtracting, and Graphing Polynomials 299 4.5 Multiplying Polynomials 310 Factoring and Applications 343 5.1 Greatest Common Factors; Factoring by Grouping 344 5.2 Factoring Trinomials 353 5.3 More on Factoring Trinomials 360 5.4 Special Factoring Techniques 369 SUMMARY EXERCISES Recognizing and Applying Factoring Strategies 379 5.5 Solving Quadratic Equations Using the Zero-Factor Property 382 5.6 Applications of Quadratic Equations 390 Chapter Summary 402 Chapter Review Exercises 405 Chapter Mixed Review Exercises 407 Chapter Test 408 Chapters R–5 Cumulative Review Exercises 409 Contents Rational Expressions and Applications 411 6.1 The Fundamental Property of Rational Expressions 412 6.2 Multiplying and Dividing Rational Expressions 422 6.3 Least Common Denominators 429 6.4 Adding and Subtracting Rational Expressions 436 6.5 Complex Fractions 444 6.6 Solving Equations with Rational Expressions 454 SUMMARY EXERCISES Simplifying Rational Expressions vs Solving Rational Equations 466 6.7 Applications of Rational Expressions 468 Chapter Summary 478 Chapter Review Exercises 483 Chapter Mixed Review Exercises 485 Chapter Test 486 Chapters R–6 Cumulative Review Exercises 487 Linear Equations, Graphs, and Systems 489 7.1 Review of Graphs and Slopes of Lines 490 7.2 Review of Equations of Lines; Linear Models 508 7.3 Solving Systems of Linear Equations by Graphing 522 7.4 Solving Systems of Linear Equations by Substitution 531 7.5 Solving Systems of Linear Equations by Elimination 539 SUMMARY EXERCISES Applying Techniques for Solving 7.6 Systems of Linear Equations in Three Variables 548 7.7 Applications of Systems of Linear Equations 557 Chapter Summary 573 Chapter Review Exercises 578 Chapter Mixed Review Exercises 582 Chapter Test 583 Chapters R–7 Cumulative Review Exercises 584 Systems of Linear Equations 546 Inequalities and Absolute Value 587 8.1 Review of Linear Inequalities in One Variable 588 8.2 Set Operations and Compound Inequalities 596 8.3 Absolute Value Equations and Inequalities 605 SUMMARY EXERCISES Solving Linear and Absolute Value Equations and Inequalities 616 8.4 Linear Inequalities and Systems in Two Variables 617 Chapter Summary 626 Chapter Review Exercises 629 Chapter Mixed Review Exercises 631 Chapter Test 631 Chapters R–8 Cumulative Review Exercises 632 v vi Contents Relations and Functions 635 9.1 Introduction to Relations and Functions 636 9.2 Function Notation and Linear Functions 647 9.3 Polynomial Functions, Graphs, Operations, and Composition 656 9.4 Variation 669 10 Chapter Summary 679 Chapter Review Exercises 681 Chapter Mixed Review Exercises 683 Chapter Test 684 Chapters R–9 Cumulative Review Exercises 685 Roots, Radicals, and Root Functions 687 10.1 Radical Expressions and Graphs 688 10.6 Solving Equations with Radicals 739 10.2 Rational Exponents 700 10.7 Complex Numbers 746 10.3 Simplifying Radicals, the Distance Formula, and Circles 709 10.4 Adding and Subtracting Radical Expressions 723 10.5 Multiplying and Dividing Radical Expressions 728 Chapter 10 Summary 754 Chapter 10 Review Exercises 759 Chapter 10 Mixed Review Exercises 762 Chapter 10 Test 763 Chapters R–10 Cumulative Review Exercises 764 SUMMARY EXERCISES Performing Operations with Radicals and Rational Exponents 738 11 Quadratic Equations, Inequalities, and Functions 767 11.1 Solving Quadratic Equations by the Square Root Property 768 11.2 Solving Quadratic Equations by Completing the Square 774 11.3 Solving Quadratic Equations by the Quadratic Formula 782 11.4 Equations That Lead to Quadratic Methods 789 SUMMARY EXERCISES Applying Methods for Solving Quadratic Equations 800 11.5 Formulas and Further Applications 801 11.6 Graphs of Quadratic Functions 810 11.7 More about Parabolas and Their Applications 819 11.8 Polynomial and Rational Inequalities 830 Chapter 11 Summary 839 Chapter 11 Review Exercises 843 Chapter 11 Mixed Review Exercises 846 Chapter 11 Test 847 Chapters R–11 Cumulative Review Exercises 849 Contents 12 Inverse, Exponential, and Logarithmic Functions 851 12.1 Inverse Functions 852 12.2 Exponential Functions 861 12.3 Logarithmic Functions 870 12.4 Properties of Logarithms 879 12.5 Common and Natural Logarithms 886 12.6 Exponential and Logarithmic Equations; Further Applications 894 13 Chapter 12 Summary 903 Chapter 12 Review Exercises 907 Chapter 12 Mixed Review Exercises 910 Chapter 12 Test 912 Chapters R–12 Cumulative Review Exercises 913 Nonlinear Functions, Conic Sections, and Nonlinear Systems 917 13.1 Additional Graphs of Functions 918 13.2 Circles Revisited and Ellipses 924 13.3 Hyperbolas and Functions Defined by Radicals 934 13.4 Nonlinear Systems of Equations 942 13.5 Second-Degree Inequalities and Systems of Inequalities 948 14 vii Chapter 13 Summary 955 Chapter 13 Review Exercises 958 Chapter 13 Mixed Review Exercises 960 Chapter 13 Test 961 Chapters R–13 Cumulative Review Exercises 962 Further Topics in Algebra 965 14.1 Sequences and Series 966 14.2 Arithmetic Sequences 972 14.3 Geometric Sequences 980 14.4 The Binomial Theorem 990 Chapter 14 Summary 996 Chapter 14 Review Exercises 999 Chapter 14 Mixed Review Exercises 1000 Chapter 14 Test 1001 Chapters R–14 Cumulative Review Exercises 1002 Appendix A Review of Exponents, Polynomials, and Factoring (Transition from Beginning to Intermediate Algebra) 1005 Appendix B Synthetic Division 1013 Answers to Selected Exercises A-1 Photo Credits C-1 Index I-1 PREFACE WELCOME TO THE 7TH EDITION The first edition of Marge Lial’s Beginning and Intermediate Algebra was published in 1996, and now we are pleased to pres ent the 7th edition—with the same successful, well-rounded framework that was established 24 years ago and updated to meet the needs of today’s students and professors The names Lial and Miller, two faculty members from American River College in Sacramento, California, have become synonymous with excellence in Developmental Mathematics, Precalculus, Finite Mathematics, and Applications-Based Calculus With Chuck Miller’s passing, Marge Lial was joined by a team of carefully selected coauthors who partnered with her John Hornsby (University of New Orleans) joined Marge in this capacity in 1992, and in 1999, Terry McGinnis became part of this developmental author team Since Marge’s passing in 2012, John and Terry have dedicated themselves to carrying on the Lial/Miller legacy In the preface to the first edition of Intermediate Algebra, Marge Lial wrote “ . . . the strongest theme . . . is a combination of readability and suitability for the book’s intended audience: students who are not completely selfconfident in mathematics as they come to the course, but who must be self-confident and proficient . . . by the end of the course.” Today’s Lial author team upholds these same standards With the publication of the 7th edition of Beginning and Intermediate Algebra, we proudly present a complete course program for students who need developmental algebra Revisions to the core text, working in concert with such innovations in the MyLab Math course as Skill Builder and Learning Catalytics, combine to provide superior learning opportunities appropriate for all types of courses (traditional, hybrid, online) We hope you enjoy using it as much as we have enjoyed writing it We welcome any feedback that you have as you review and use this text WHAT’S NEW IN THIS EDITION? We are pleased to offer the following new features and resources in the text and MyLab IMPROVED STUDY SKILLS These special activities are now grouped together at the front of the text, prior to Chapter R Study Skills Reminders that refer students to specific Study Skills are found liberally throughout the text Many Study Skills viii now include a Now Try This section to help students implement the specific skill REVISED EXPOSITION With each edition of the text, we con tinue to polish and improve discussions and presentations of topics to increase readability and student understanding This edition is no exception NEW FIGURES AND DIAGRAMS For visual learners, we have included more than 50 new mathematical figures, graphs, and diagrams, including several new “hand drawn” style graphs These are meant to suggest what a student who is graphing with paper and pencil should obtain We use this style when introducing a particular type of graph for the first time ENHANCED USE OF PEDAGOGICAL COLOR We have thor oughly reviewed the use of pedagogical color in discussions and examples and have increased its use whenever doing so would enhance concept development, emphasize important steps, or highlight key procedures INCREASED Concept Check AND WHAT WENT WRONG? EXERCISES The number of Concept Check exercises, which facilitate students’ mathematical thinking and conceptual understanding, and which begin each exercise set, has been increased We have also more than doubled the number of WHAT WENT WRONG? exercises that highlight common student errors INCREASED RELATING CONCEPTS EXERCISES We have doubled the number of these flexible groups of exercises, which are located at the end of many exercise sets These sets of problems were specifically written to help students tie concepts together, compare and contrast ideas, identify and describe patterns, and extend concepts to new situations They may be used by individual students or by pairs or small groups working collaboratively All answers to these exer cises appear in the student answer section ENHANCED MYLAB MATH RESOURCES MyLab exercise cov 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 RELATING CONCEPTS exercise sets (both even- and oddnumbered problems) are now assignable in MyLab Math SKILL BUILDER These exercises offer just-in-time additional adaptive practice in MyLab Math The adaptive engine tracks student performance and delivers, to each individual, questions that adapt to his or her level of understanding This new feature enables instructors to assign fewer questions for C-2 Photo Credits CHAPTER p 587 Leonid Shcheglov/Shutterstock; p 601 Jeayesy/123RF; p 603 Michaeljung/Shutterstock; p 604 Chad McDermott/Shutterstock; p 611 Leonid Shcheglov/Shutterstock; p 615 Katherine Martin/123RF; p 625 Derek Meijer/Alamy Stock Photo; p 634 National Atomic Museum Foundation CHAPTER p 635 Michelangeloop/Shutterstock; p 653 Rido/Shutterstock; p 655 Neelsky/Shutterstock; p 662 Gorbelabda/Shutterstock; p 666 Anton Starikov/Shutterstock; p 668 Terry McGinnis; p 674 Dana Bartekoske/123RF; p 677 Dmitry Deshevykh/Alamy Stock Photo; p 678 Cathy Yeulet/123RF CHAPTER 10 p 687 Giulio Meinardi/Fotolia; p 699 Stoupa/Shutterstock; p 708 Maria Moroz/Fotolia; p 721 (dog) Anneka/Shutterstock, (screen) Cobalt/Fotolia; p 722 Kevin Kipper/Alamy Stock Photo; p 745 Johnson Space Center/NASA; p 765 Scol22/Fotolia CHAPTER 11 p 767 Sarot Chamnankit/123RF; p 770 Portrait of Galileo (ca 1853), Charles Knight Steel engraving Library of Congress Prints and Photographs Division [LC-DIG-pga-38085]; p 791 Phovoir/Shutterstock; p 804 Micro10x/Shutterstock; p 809 Look Die Bildagentur der Fotografen GmbH/Alamy Stock Photo; p 810 Steven J Everts/123RF; p 814 Zurijeta/Shutterstock; p 818 Andriy Popov/123RF; p 819 Alexwhite/Shutterstock; p 828 Kaband/Shutterstock; p 829 Bonniej/E+/Getty Images; p 839 Hartrockets/iStock/Getty Images CHAPTER 12 p 851 Chien321/Shutterstock; p 853 Frontpage/Shutterstock; p 860 Mark Higgins/Shutterstock; p 869 Bernhard Staehli/Shutterstock; p 870 Maksym Bondarchuk/123RF; p 875 Per Tillmann/Fotolia; p 878 Prisma by Dukas Presseagentur GmbH/Alamy Stock Photo; p 887 Romrodphoto/Shutterstock; p 889 Georgios Kollidas/Fotolia; p 892 (top) Jim West/Alamy Stock Photo, (bottom) Silvano Rebai/Fotolia; p 893 WaterFrame/Alamy Stock Photo; p 901 Sergey Nivens/Shutterstock; p 902 Armadillo Stock/Shutterstock; p 908 Shutterstock; p 910 Syda Productions/Shutterstock; p 911 IndustryAndTravel/Alamy Stock Photo CHAPTER 13 p 917 Terry McGinnis; p 921 Samuel Acosta/Shutterstock; p 934 Petr Student/Shutterstock; p 941 Faiz Azizan/Shutterstock CHAPTER 14 p 965 Callie Daniels; p 966 Madeinitaly4k/Shutterstock; p 970 Frank11/Shutterstock; p 979 Blend Images/Shutterstock; p 980 Creativa Images/Shutterstock; p 988 Paul Hakimata Photography/Shutterstock; p 991 Atlaspix/Alamy Stock Photo; p 1001 Jtbaskinphoto/Shutterstock; p 1003 Baloncici/123RF APPENDIX A Absolute value distance definition of, 606–607 evaluating, 48 explanation of, 47, 605 simplifying square roots using, 694 solving equations involving, 605–611 solving inequalities involving, 605–611 Absolute value equations, 605–611 Absolute value functions, 918 Absolute value inequalities, 605–611 Addition associative property of, 79 commutative property of, 78 of complex numbers, 749 of decimals, 17–18 of fractions, 7–9, 10 with grouping symbols, 29, 56–57, 77 identity element for, 80 identity property of, 80–81 inverse for, 81 of multivariable polynomials, 304 of negative numbers, 52 on a number line, 51–53 in order of operations, 29, 77 of polynomial functions, 659–660 of polynomials, 302–303, 304, 1006 of radical expressions, 723–725 of rational expressions, 436–439 of real numbers, 51–54, 56–58 of signed numbers, 51–54 summary of properties of, 84 of terms, 310 word phrases for, 57–58 Addition property of equality, 104–109 of inequality, 183–185, 589 Additive identity element, 80 Additive inverse explanation of, 47, 81 finding for real numbers, 46–47, 55 symbol for, 56 Agreement on domain, 642 Algebraic expressions distinguishing from equations, 39–40 evaluating, 36–37 explanation of, 36 from word phrases, 37–38, 91, 123 simplifying, 82, 88–91 Angles complementary, 139–140 measure of, 139–140, 148–149 right, 139 straight, 139, 148–149 INDEX supplementary, 139–140, 568 vertical, 148–149, 568 Annuity explanation of, 984 ordinary, 984 terms of, 984 Apogee, 673, 933 Approximately equal symbol, 695 Area of a circle, 152 of a rectangle, 146 rules for exponents for, 277–278 of a trapezoid, 146, 152 of a triangle, 152 Area problem, 803 Arithmetic mean, 969–970 Arithmetic progression, 972 Arithmetic sequence application of, 974 common difference of, 972–973 explanation of, 972 general term of, 973–974 specified term of, 974–975 sum of terms of, 976–978 Associative properties distinguishing from commutative, 79–80 explanation of, 79, 84 Asymptotes of a hyperbola, 934–935 of reciprocal function, 918 Average, 76, 969–970 Average rate of change, 501–502 Axis of a coordinate system, 212, 490 of a hyperbola, 934 of a parabola, 305, 810, 813 of symmetry, 305 B Base of an exponential expression, 28, 272 in a percentage discussion, 162 Binomial(s) conjugates of, 732–734 explanation of, 301 factoring, 370 finding greater powers of, 320 finding product of the sum and difference of two terms, 318–319 multiplication by FOIL method, 312–313 multiplication of, 728–729 raising to a power, 990–994 squares of, 317–318 steps to multiply by FOIL method, 312 Binomial coefficient formula, 992 Binomial expansion general, 993 specified term of, 994–995 Binomial theorem, 993–994 Boundary line, 617 Braces explanation of, 29 as a grouping symbol, 29 for set notation, 38 Brackets as grouping symbol, 29–31 Brin, Sergey, 296 C Calculator graphing method See Graphing calculators Cartesian coordinate system explanation of, 212, 490 plotting points on, 213–214 Center of circle, 716, 924–927 of ellipse, 927 Center-radius form of a circle, 717, 925, 927 Change-of-base rule, 890–891 Circle(s) area of, 152 center of, 716, 924–927 center-radius form of, 717, 925, 927 circumference of, 152 equation of, 556, 716–717, 925–927 explanation of, 924 graph, 11 graphing, 716–717 graphs of, 716, 924–926, 937 radius of, 924–927 Circumference of a circle, 152 Closed interval, 588 Coefficient(s), 88, 299, 992 Combinations, explanation of, 992 Combined variation, 674 Common denominators, 7–8 Common difference of an arithmetic sequence, 972–973 Common factors, 3, 344 Common logarithms applications of, 887–888 evaluating, 886–887 explanation of, 886 Common ratio of a geometric sequence, 980–981 Commutative properties distinguishing from associative, 79–80 explanation of, 78, 84 Complementary angles, 139–140 I-1 I-2 Index Completing the square method for solving quadratic equations, 774–779, 800 Complex conjugates, 750 Complex fractions explanation of, 444 simplifying, 445–450 simplifying with negative exponents, 450–451 steps to simplify, 445, 447 Complex numbers addition of, 749 conjugates of, 750 division of, 750–751 explanation of, 748 imaginary part of, 748 multiplication of, 749–750 nonreal, 748 real part of, 748 standard form of, 748 subtraction of, 749 Components, 490 Composite function explanation of, 662 finding, 662–664 Composite numbers, Composition of functions, 662–664 Compound inequalities with and, 596–598 explanation of, 596 with or, 598–601 Compound interest continuous, 899 formula for, 280, 808, 898–899 Concours d’Elegance, 444 Conditional equation, 121–122 Conic sections examples of, 924 explanation of, 924 summary of, 937 Conjugate(s) of a binomial, 732–734 of a complex number, 750 explanation of, 320 Consecutive integers even, 138, 392 explanation of, 137, 391 odd, 138–139, 392 solving problems involving, 137–139 Consistent system, 525–526 Constant(s), 36 Constant function, 650–651 Constant of variation, 669 Consumer Price Index (CPI), 804 Continuous compounding explanation of, 899 formula for, 899 Contradiction, 122 Coordinate(s) of a point, 44, 212 of points in a plane, 490 Coordinate system Cartesian, 212, 490 explanation of, 212 origin of, 212 quadrants of, 212 rectangular, 212, 490 Cost, unit, 158 Cost-benefit equation, 893 Counting numbers See Natural numbers Cross products of a proportion, 159–160 Cube(s) difference of, 374–375, 376 of a number, 28 perfect, 374 sum of, 375–376 Cube root(s) application of, 691 explanation of, 691 of negative numbers, 691 symbol for, 691 Cube root function explanation of, 693 graph of, 693 Cubing function, 657 D Data, interpreting, 48, 59 Data modeling, 514–517 Data sets, 256–257 Decay applications of, 866–867 exponential, 866–867, 900 Decibel, 888 Decimal(s) adding, 17–18 dividing, 19–20 equivalents, 21 explanation of, 16 linear equations with, 128–129 multiplying, 18, 20 place value in, 16 repeating, 21, 43 rounding of, 19–20 solving linear systems with, 536–537 subtracting, 17–18 terminating, 21, 43 writing as percents, 21–22, 162 writing fractions as, 20–21 writing percents as, 21–22, 162 written as fractions, 17 Decimal places, 18 Degree explanation of (angle measure unit), 139 of a polynomial, 301 of a term, 301 Denominator(s) adding fractions with different, 8–9 adding fractions with same, common, 7–8 explanation of, 1, 3–4 least common, 8, 429–431 in rational expressions, 412, 431–432 rationalizing, 730–734 Dependent equations explanation of, 525–526 substitution method for solving, 534–535 Dependent variable, 638 Depreciation, 230, 911 Descartes, René, 212, 490 Descending powers, 300 Difference of cubes, 374–375, 376 explanation of, 9, 54 of squares, 369–371, 376 Direct variation explanation of, 669 as a power, 671 solving problems involving, 669–671 Discriminant, 785–787, 822–823 Disjoint interval, 588 Distance, rate, and time relationship, 174–176, 469–470, 561–563, 790 Distance formulas distance between points, 714–715 distance to the horizon, 699, 722 explanation of, 174 for falling objects, 382, 390 Distributive properties addition property of equality and, 109 explanation of, 82–84 Diversity, index of, 911 Dividend, 6, 19, 68 Divisibility tests for numbers, 76, 344 Division of complex numbers, 750–751 of decimals, 19–20 explanation of, 68 of fractions, 6–7 long, 324–329 multiplication property of equality extended to, 113 in order of operations, 29, 77 of polynomial functions, 659, 661 of polynomials, 323–329 by powers of ten, 20 of rational expressions, 424–426 of real numbers, 67–73 with scientific notation, 293 of signed numbers, 68 synthetic, 1013–1016 word phrases for, 72 by zero, 69 Divisor(s) explanation of, 6, 19, 68 factors as, 344 Domain agreement on, 642 of a function, 639–640 of a relation, 639–640 Double negative rule, 47 Index Double solution, 386–387 Doubling time, 893, 898 Downward opening parabola, 813 E e, 889 Earthquake, intensity of, 298 Elements of a set, 38 Elimination method explanation of, 539 for solving equivalent equations, 544 for solving linear systems, 539–544, 546 for solving nonlinear systems, 944–945 steps to solve by, 540–541 Elimination number, 156 Ellipse(s) center of, 927 equation of, 928 explanation of, 924 foci of, 927 graphs of, 927–930, 937, 939 horizontal shift of, 929 intercepts of, 927–928 vertical shift of, 929 Empty set explanation of, 122 symbols for, 122 Equality addition property of, 104–109 multiplication property of, 112–116 symbol for, 31, 38 Equation(s) absolute value, 605–611 of a circle, 556, 716–717, 925–927 conditional, 121–122 dependent, 525–526 distinguishing from expressions, 39–40, 73, 454–455 of an ellipse, 928 equivalent, 104, 544 explanation of, 38, 104 exponential, 864–865, 894–895, 898–899 first-degree, 491 from sentences, 72–73 graphs of, 491, 936 of a horizontal line, 224–225, 493, 512, 514 of hyperbola, 934 independent, 525–526 of an inverse function, 854–855 linear in one variable See Linear equations in one variable linear in three variables, 548–553 linear in two variables See Linear equations in two variables linear system of, 548 of lines, 508–517 literal, 149–151 logarithmic, 872, 894–897 nonlinear, 942 nonlinear systems of, 942–946 power rule for, 739–740 quadratic See Quadratic equations quadratic in form, 789, 793–796 radical, 739–743 rational, 789 second-degree, 936, 944–945 simplifying before solving, 109, 115–116 slope of, 236–237 solution set of, 104, 196 solutions of, 38, 104 solving with rational expressions, 455–462 square root property of, 768–772 translating sentences into, 72–73 of a vertical line, 224–225, 493, 512, 514 working, 549 Equilibrium demand, 531 Equilibrium supply, 531 Equivalent equations, 104, 544 Equivalent forms for a rational expression, 417–418 Equivalent inequalities, 589 Euler, Leonhard, 889 Even consecutive integers, 138, 392 Exponent(s) negative, 1005 power rules for, 1005 product rule for, 1005 quotient rule for, 1005 zero, 1005 Exponential decay, 866–867, 900 Exponential equations applications of, 898–899 explanation of, 864 general method for solving, 895 properties for solving, 864, 894 steps to solve, 864 Exponential expressions base of, 28, 272 evaluating, 28 explanation of, 28, 272 Exponential functions applications of, 866–867 characteristics of graph of, 863 converting to logarithmic form, 871 explanation of, 861 graphs of, 861–863 Exponential growth, 866, 899–900 Exponential notation, 700–703 Exponents explanation of, 28, 272 integer, 280–287 negative, 281–284, 286, 450–451 in order of operations, 29, 77 positive, 284, 286 power rules for, 274–277, 286 product rule for, 272–274, 276–277, 286 quotient rule for, 284–286 rational, 700–706 and scientific notation, 291–294 I-3 summary of rules for, 276, 286 zero, 281, 286 Expressions algebraic See Algebraic expressions distinguishing from equations, 39–40, 73, 454–455 equality and inequality symbols in, 33 exponential, 28, 272 from word phrases, 91 quadratic, 382 radical See Radical expressions rational See Rational expressions simplifying, 82, 88–91 terms of, 88–89, 299 Extraneous solutions, 739 Extremes of a proportion, 158 F Factor(s) common, 3, 344 distinguishing between terms and, 89 explanation of, 2, 344 greatest common, 3–4, 344–348 of integers, 67 of numbers, 2, 344 prime, Factor tree, Factorial notation, 991 Factoring binomials, 370 difference of cubes, 374–375, 376 difference of squares, 369–371 explanation of, 344 with four terms by grouping, 349–350 greatest common factor, 346–348 by grouping, 348–350 guidelines for, 356 perfect square trinomials, 371–373, 376 polynomials, 1008–1009 solving quadratic equations, 768 special techniques, 369–376 sum of cubes, 375–376 summary for polynomials, 379 summary of special techniques, 376 trinomials, 353–358, 360–366 zero-factor property, 382–388, 768 Farads, 695 Fibonacci, 688 Fibonacci sequence, 965 Finite sequence, 966 First-degree equation See Linear equations First-degree equations See also Linear equations explanation of, 491 graph of, 491 Fixed cost, 253 Foci (sing: Focus) of ellipse, 927 of hyperbola, 934 Focus variable, 549 I-4 Index FOIL method, 728–729, 750 explanation of, 312 factoring binomials, 370 factoring trinomials, 354, 363–366 inner product of, 312 outer product of, 312 Formula(s) binomial coefficient, 992 for compound interest, 898–899 distance See Distance formulas explanation of, 146 Galileo’s, 770, 843 Heron’s, 700 midpoint, 494 perimeter, 558 of the Pythagorean theorem, 393, 713–714, 802 quadratic, 782–787, 800 resonant frequency, 695 for simple interest, 153, 172, 898 slope, 495 solving for a specified variable of, 801–802 with square roots, 801 vertex, 821 Fourth root, 691 Fraction(s) adding, 7–9, 10 applications of, 10 complex, 444–451 denominator of, dividing, 6–7 equivalents, 21 explanation of, improper, 1, least common denominator of, 8, 429–431 linear equations with, 125–127 linear inequalities with, 592 linear systems with, 535–536 lowest terms of, mixed numbers, 4, 8, 10 multiplying, 5–6 numerator of, operations on, 5–10 proper, reciprocals of, 6, 67, 81 simplifying, subtracting, 9–10 writing as percents, 23 writing decimals as, 17 writing percents as, 22–23 written as decimals, 20–21 Fraction bar explanation of, 1, 68 as grouping symbol, 29–31 Froude, William, 809 Froude number, 809 Function(s) absolute value, 918 coding information using, 860 composite, 662–664 composition of, 662–664 constant, 650–651 cube root, 693 cubing, 657 definitions, variations of, 643 domain of, 639–640 equation of the inverse of, 854–855 evaluating, 647–649 explanation of, 636–637 exponential, 861–867, 871 greatest integer, 920–921 identity, 657 as Input-output machine, 639 inverse of, 852–857 linear, 508, 650–651, 657 logarithmic, 874–875 one-to-one, 852–854 operations on, 659–661 polynomial, 656–664 quadratic See Quadratic functions range of, 639–640 reciprocal, 918 square root, 692, 918, 938–939 squaring, 657 step, 920 translation of, 919–920 vertical line test for, 640–641 Function notation, 647–650 Fundamental property of rational expressions, 414 Fundamental rectangle of hyperbola, 935 Future value of an ordinary annuity, 984 F(x) notation, 647–650 G Galileo Galilei, 382, 390, 770 Galileo’s formula, 770, 843 Garfield, James A., 722 General binomial expansion, 993 General term of an arithmetic sequence, 973–974 of a geometric sequence, 981 of a sequence, 966–967 Generalized square root function, 938–939 Geometric progression, 980 Geometric sequence common ratio of, 980–981 explanation of, 980 general term of, 981 specified term of, 982 sum of terms of, 982–984 Geometry formulas, explanation of, 146 See also specific formulas Googol, 296 Grade, 495 Graph(s) of absolute value function, 918 of boundary passing through origin, 619 circle, 11 of circles, 716, 924–926, 937 of cube root functions, 693 of elementary functions, 918–920 of ellipses, 927–930, 937 of equations, 491, 936 explanation of, 219 of exponential functions, 861–863 of first-degree equations, 491 of greatest integer functions, 921 of horizontal lines, 493 of hyperbolas, 934–937 of intervals on number line, 588 of inverses, 856–857 line, 208–209, 246–247 of linear equations, 219–226, 491 of linear inequalities, 617–621 of linear systems, 548–549 of lines passing through origin, 493 of logarithmic functions, 874–875 of ordered pairs of numbers, 490 of parabolas, 305, 810, 825–826, 937 pie charts, 11 of polynomial functions, 657–658 of quadratic functions, 810–816, 821–822 of quadratic inequalities, 831 of radical expressions, 692–693 of reciprocal functions, 918 of second-degree inequalities, 948–952 of semicircles, 938 of square root functions, 692, 918, 938–939 of systems of inequalities, 950–952 of vertical lines, 493 Graphing explanation of, 219 horizontal lines, 224–225 inequalities, 182–183 linear equations in two variables, 219–226, 522–527 linear inequalities in one variable, 182–183 numbers, 44 ordered pairs, 212–213 parabolas, 305, 658 polynomials, 305 rational numbers, 44 vertical lines, 224–225 Graphing calculators for approximation of roots, 695–696 for displaying binomial coefficients, 993 for generating quadratic models, 816 for generating sequences, 967 for graphing circles or ellipses, 930 square viewing window in, 930 standard viewing window in, 495 Greater powers of binomials, 320 “(is) Greater than” explanation of, 46 ordering of real numbers, 46 symbol for, 31, 182, 588 “(is) Greater than or equal to,” 31, 182, 588 Greatest common factor explanation of, 3, 344 factoring out, 346–348 finding, 344–346 of numbers, 344 Index steps to find, 345 for variable terms, 345–346 Greatest integer functions graphs of, 921 method for applying, 920–921 Grouping factoring by, 348–350 factoring by four terms, 349–350 factoring trinomials by, 360–362 Grouping symbols addition with, 56–57 explanation of, 29 subtraction with, 56–57 Growth applications of, 866 exponential, 866, 899–900 H Half-closed interval, 588 Half-life, 900 Half-open interval, 588 Henrys, 695 Heron’s formula, 700 Horizontal hyperbola, 935 Horizontal line(s) equation of, 224–225, 493, 512, 514 explanation of, 224 graph of, 493 graphing, 224–225 slope of, 235–236, 498 Horizontal line test for a one-to-one function, 853–854 Horizontal parabola, 825–826 Horizontal shift of an ellipse, 929 method for applying, 919–920 of a parabola, 812–813 Hyperbola(s) asymptotes of, 934–935 equations of, 934 explanation of, 924 foci of, 934 fundamental rectangle of, 935 graphs of, 934–937 intercepts of, 934 transverse axis of, 934 Hypotenuse of a right triangle, 393, 713 I i explanation of, 746 powers of, 751 Identity, 122 Identity element for addition, 80 for multiplication, 80 Identity function, 657 Identity properties, 80–81, 84 Imaginary part of a complex number, 748 Imaginary unit explanation of, 746 powers of, 751 Improper fractions converting between mixed numbers and, explanation of, Incidence rate, 421 Inconsistent system explanation of, 525–526 solving, 553 substitution method for solving, 534 Independent equations, 525–526 Independent variable, 638 Index of a radical, 691 Index of diversity, 911 Index of summation, 968 Inequality(ies) absolute value, 605–611 addition property of, 183–185, 589 applied problems using, 189–190 compound, 596–601 equivalent, 589 explanation of, 31, 182, 588 graphing, 182–183 linear in one variable See Linear inequalities in one variable linear in two variables See Linear inequalities in two variables multiplication property of, 185–187, 590–592 polynomial, 834–835 quadratic, 830–834 rational, 835–837 second-degree, 948–952 solving linear, 182–192 symbols of, 31–33, 45–46, 182, 588 systems of, 950–952 three-part, 190–192, 593–594 Infinite geometric sequence explanation of, 985 sum of terms of, 985–987 Infinite interval, 588 Infinite sequence explanation of, 966 terms of, 966 Infinity symbol, 182, 588 Inner product, 312 Input-output machine, 639 Integers consecutive, 137–139, 391 consecutive even, 138, 392 consecutive odd, 138–139, 392 explanation of, 42–43 as exponents, 280–287 factors of, 67 Intensity of an earthquake, 298 Intercepts of ellipse, 927–928 graphing, 221–223 of hyperbola, 934 of a linear equation, 221–223, 245–249 of a parabola, 821–823 point-slope form, 253–254 slope-intercept form, 245–249 x-, 491–492 y-, 491–492, 509 Interest compound, 280, 808, 898 formula for, 170 simple, 153, 172–173, 898 Intersection of linear inequalities, 620 of sets, 596 Interval notation, 182–183, 192, 588 Interval on a number line, 182–183 Inverse additive, 46–47, 55–56, 81 multiplicative, 67, 81 Inverse of a function equation of, 854–855 explanation of, 852 graph of, 856–857 steps to find the equation of, 854 symbol for, 852 Inverse properties, 81, 84 Inverse variation explanation of, 671–672 as a power, 672 Irrational numbers, 44, 690 J Joint variation, 673–674 L Leading term, 299 Least common denominators (LCDs) explanation of, 8, 429 finding, 8, 429–431 rational expressions and, 455 steps to find, 429 Legs of a right triangle, 393, 713 Leonardo of Pisa, 688 “(is) Less than” explanation of, 46 ordering of real numbers, 46 symbol for, 31, 182, 588 “(is) Less than or equal to,” 31, 182, 588 Light-year, 339 Like terms combining, 89, 299–300 explanation of, 89, 299 Limit notation, 985 Line(s) equations of, 508–517 graphing, 208–209, 246–247 horizontal See Horizontal line(s) number See Number line parallel, 238–240 perpendicular, 238–240 slope of, 231–240, 495–498 of symmetry, 305 vertical See Vertical line(s) I-5 I-6 Index Line graphs explanation of, 208 interpreting, 208–209 Line segment, midpoint of, 494 Linear equations in one variable applications of, 132–140, 169–176 with decimal coefficients, 128–129 explanation of, 104 with fractions, 125–127 geometric applications of, 146–148 with infinitely many solutions, 121–122 with no solutions, 122 solution set of, 594 solving, 117–123, 125–129 steps to solve, 117 Linear equations in three variables explanation of, 548 graphs of, 548–549 system of, 548–553 Linear equations in two variables explanation of, 209, 491 graph of, 491 graphing, 219–226, 522–527 intercepts of, 221–223 point-slope form of, 247–249, 253–255, 510–511, 514 slope-intercept form of, 245–249, 255, 508, 514 slope of, 236–237 solution of, 209 standard form of, 209, 253, 254–255, 491, 511, 514 summary of forms of, 255, 514 systems of See Systems of linear equations use to model, 225–226, 256–257 x-intercept of, 491–492 y-intercept of, 491–492, 509 Linear functions, 508, 650–651, 657 Linear inequalities in one variable explanation of, 589 with fractions, 592 graph of, 182–183 solution of, 187–188 solution sets of, 184, 594 solving using addition property, 589 solving using multiplication property, 590–592 steps to solve, 187, 591 three-part, 593–594 Linear inequalities in two variables boundary line of graph, 617 explanation of, 617 graphs of, 617–621 intersection of, 620 region of solution, 617 system of See Systems of inequalities Linear models, creating, 514–517 Linear programming, 625 Linear system of equations See System of linear equations Linear systems See Systems of linear equations Literal equations, 149–151 Lithotripter, 932 Logarithm(s) alternative forms of, 882–883 change-of-base rule for, 890–891 common, 886–888 evaluating, 871–872, 886–887 explanation of, 870–871 exponential form of, 871 natural, 889–890 power rule for, 881–882 product rule for, 879–880, 882 properties of, 873, 879–884, 882 quotient rule for, 880–881, 882 Logarithmic equations explanation of, 872 properties for solving, 894 solving, 895–897 steps to solve, 897 Logarithmic functions applications of, 875 with base a, 874–875 characteristics of graph of, 875 converting to exponential form, 871 graphs of, 874–875 Long division, 324–329 Lowest terms of a fraction, of a rational expression, 414–417 writing radical quotients in, 734 M Magic number, 156 Mapping of sets, 638 Mathematical model, 42 Maximum value of a quadratic function, 824–825 Mean, arithmetic, 969–970 Means of a proportion, 158 Measure of an angle, 139–140, 148–149 Midpoint of a line segment formula, 494 Minimum value of a quadratic function, 824–825 Minuend, 54 Mixed numbers applications with, 10 converting between improper fractions and, explanation of, 4, 43 Mixture problems formula for, 170 gasoline-oil, 136 steps to solve, 170–172, 560–561 Model(s) data, 514–517 mathematical, 42 quadratic, 394–395 quadratic functions as, 804–805, 814–816 using a linear equation to, 225–226, 256–257 Money denomination problems, 173–174 Money problems, 559 Monomials dividing polynomials by, 323–324 explanation of, 301 multiplying, 310 Motion problems, 174–176, 469–470, 561–563, 790 Multiplication associative property of, 79 of binomials, 312–313, 728–729 commutative property of, 78 of complex numbers, 749–750 of decimals, 18, 20 distributive property, 82–84, 109 FOIL method of, 312–313, 728–729 of fractions, 5–6 identity element for, 80 identity property of, 80–81 inverse property for, 81 of a monomial, 310 in order of operations, 29, 77 of a polynomial, 310–312 of polynomial functions, 659, 661 of polynomials, 1007 by powers of ten, 20 of radical expressions, 728–729 of radicals, 709 of radicals with different indexes, 713 of rational expressions, 422–424, 426 of real numbers, 65–67, 70–72 with scientific notation, 293 of signed numbers, 65–67 of sum and difference of two terms, 318–319 summary of properties of, 84 of terms, 310 using logarithms, 879–880 word phrases for, 71–72 by zero, 65 Multiplication property of equality, 112–116 of inequality, 185–187, 590–592 of zero, 65 Multiplicative identity element, 80 Multiplicative inverse, 67, 81 Multivariable polynomials adding, 304 subtracting, 304 N Nanometer, 294 Natural logarithms applications of, 890 evaluating, 889–890 explanation of, 889 Natural numbers explanation of, 1, 42 negative of, 42–43 opposite of, 42–43 Negative exponents changing to positive, 284, 286 Index explanation of, 282, 286, 1005 in rational expressions, 450–451 scientific notation and, 292 Negative infinity symbol, 182 Negative numbers adding, 51–52 cube roots of, 691 dividing, 68 explanation of, 42–43 as exponents, 281–284, 286 multiplying, 66–67 square roots of, 746–747 subtracting, 55–56 symbol for, 56 Negative reciprocals, 500 Negative slope, 235, 499 Negative square roots, 688 Newtons, 670 n factorial, 991 Noncollinear points, 556 Nonlinear equations, 942 Nonlinear systems of equations explanation of, 942 solving, 942–946 Nonreal complex numbers, 748 “(is) Not equal to,” 31 Notation exponential, 700–703 factorial, 991 function, 647–650 interval, 182–183, 192, 588 limit, 985 scientific, 291–294 set-builder, 43, 192, 525 sigma, 968 standard, 293 subscript, 233, 494 summation, 968–969 nth root(s) explanation of, 691 exponential notation for, 700–703 finding for nth powers, 694 Null set explanation of, 122 symbols for, 122 Number(s) absolute value of, 47–48, 694 complex, 748–751 composite, cube of, 28 divisibility tests for, 76, 344 factors of, 2, 344 fractions, graph of, 44 greatest common factor of, 344–348 imaginary, 748 integers, 42–43 irrational, 44, 690 mixed, 4, 8, 10, 43 natural, 1, 42 negative See Negative numbers nonreal complex, 748 opposite of See Opposite(s) ordered pair of, 490 ordering of, 46 perfect square, 690 positive, 43 prime, 2, 345 prime factors of, pure imaginary, 748 rational, 43–44 real See Real numbers reciprocal of, 6, 67 signed See Signed numbers solving problems about, 468 square of, 28, 688 square roots of See Square root(s) whole, 1, 42 Number line(s) addition on, 51–53 explanation of, 42 graphing a number on, 44 graphing intervals on, 182–183, 588 subtraction on, 54 Numerator(s) explanation of, 1, 3–4 in rational expressions, 412 Numerical coefficient, 88, 299 Numerical expressions evaluating, 70–72 from word phrases, 57–58, 71–72 O Odd consecutive integers, 138–139, 392 Ohm’s law, 754 One, properties of, One-to-one function explanation of, 852 horizontal line test for, 853–854 inverse of, 852–853 Open interval, 588 Operations on functions, 659–661 on polynomials, 1006–1007 on sets, 596, 598 Opposite(s) as additive inverse, 46–47, 55, 81 as negative number, 42–43, 48 quotient of, 417 Order of a radical, 691 Order of operations, 29, 56–57, 77 Ordered pairs completing, 210–211 explanation of, 209, 490 graph of, 490 graphing, 212–213 plotting, 212–213 as solutions, 210 table of, 491 table of values, 211–212 writing solutions as, 209 Ordered triple, 548 Ordering of real numbers, 46 I-7 Ordinary annuity explanation of, 984 future value of, 984 payment period of, 984 Origin explanation of, 212, 490 graphing line passing through, 493 line passing through, 223 Outer product, 312 P Pairs, ordered See Ordered pairs Parabola(s) applications of, 824–825 axis of, 305, 810, 813, 825 axis of symmetry of, 305 explanation of, 305, 810, 924 graph of, 305, 810, 825 graphing, 658 graphs of, 937 horizontal, 825–826 horizontal shift of, 812–813 intercepts of, 822–823 line of symmetry of, 305 summary of graphs of, 826 symmetry of, 810 vertex formula for, 821 vertex of, 305, 810, 813, 819–821, 825 vertical, 811–813 vertical shift of, 811–813 Parallel lines explanation of, 238 slope of, 239, 499–500, 512–513 Parentheses, 29, 588 Pascal, Blaise, 991 Pascal’s triangle, 991 Payment period of an ordinary annuity, 984 Percents and percentages applied problem involving, 24 applied problems involving, 163 equivalents, 21 explanation of, 21, 162 writing as decimals, 21–22, 162 writing as fractions, 22 writing decimals as, 22, 162 writing fractions as, 23 Perfect cubes, 374, 691 Perfect fourth powers, 691 Perfect square trinomials explanation of, 371 factoring of, 371–373 Perfect squares, 370, 690 Perigee, 673, 933 Perimeter explanation of, 15, 147 of a rectangle, 147, 152 of a triangle, 147–148, 152 Perimeter formula, 558 Perpendicular lines explanation of, 238 slopes of, 239, 499–500, 512–513 I-8 Index pH application of, 887–888 explanation of, 887 Pi (π ) approximation of, 152 explanation of, 45, 690 Pie charts, 11 Pisa, Leonardo of, 688 Place value in decimals, 16 Plane coordinates of points in, 490 explanation of, 212 orientation of a line in the, 499 plotting points in, 490 in three-dimensional graphing, 548 Plotting ordered pairs, 212–213 Plotting points, 212–213 Plus or minus symbol, 769 Point(s) coordinates in a plane, 490 noncollinear, 556 Point-slope form, 253–255, 510–511, 514 Polynomial(s) addition of, 302–303, 304, 1006 binomials See Binomial(s) classifying, 301 coefficients of, 299 degree of, 301 of degree two, 305 in descending powers, 300 division of, 323–329 evaluating, 302 factoring, 1008–1009 factoring summary, 379 graphing, 305 graphing equations defined by, 305 leading term of, 299 monomials See Monomials multiplication of, 310–312, 1007 multivariable, 304 numerical coefficients of, 299 operations on, 302–304, 310–312, 323–329 prime, 356 special factoring techniques, 369–376 subtraction of, 303–304, 1006 terms of, 299 trinomials See Trinomials in x, 300 Polynomial function(s) addition of, 659–660 cubing, 657 of degree n, 656 division of, 659, 661 domain of, 657 evaluating, 656 explanation of, 656 graphs of, 657–658 identity, 657 multiplication of, 659, 661 range of, 657 squaring, 657 subtraction of, 659–660 Polynomial inequality solving, 834–835 third-degree, 834–835 Positive exponents changing from negative to, 284, 286 scientific notation and, 292 Positive infinity symbol, 182 Positive numbers, 43 Positive slope, 235, 499 Positive square roots, 688 Power(s) descending, 300 explanation of, 28, 272 Power rule(s) for exponents, 274–277, 286, 1005 for logarithms, 881–882 for radical equations, 739–740 Powers of i explanation of, 751 simplifying, 751 Powers of ten division by, 20 explanation of, 16 multiplication by, 20 Price per unit, 158 Pricing problem, 565–566 Prime factors of a number, Prime numbers, 2, 345 Prime polynomials, 356 Principal square root, 688 Product explanation of, 2, 65 of the sum and difference of two terms, 318–319 translating words and phrases, 71 Product rule for exponents, 272–274, 276–277, 286, 1005 for logarithms, 879–880, 882 for radicals, 709 Progression arithmetic, 972 geometric, 980 Proper fractions, Properties of equality, 104–109, 112–116 of inequality, 183–187, 589, 590 of one, of real numbers, 65, 78–84 Proportional, 669 Proportions applications of, 161 cross products of, 159–160 explanation of, 158 extremes of, 158 means of, 158 solving, 158–160 terms of, 158 Proposed solution, 739 Pure imaginary number, 748 Pyramids, volume of, 153 Pythagorean theorem explanation of, 393, 713–714 proof of, 722 solving applied problems using, 802 Q Quadrants, 212, 490 Quadratic equations applications of, 390–395, 789–792, 802–805 completing the square method for solving, 774–779, 800, 819–820 with complex solutions, 772 discriminant of, 785–787, 822–823 explanation of, 382, 768 with nonreal complex solutions, 772, 778–779 quadratic formula for solving, 783–785, 800 solving radical equations that lead to, 792 square root property for solving, 768–772, 800 standard form of, 382, 768 steps to solve, 384 steps to solve an applied problem, 390 steps to solve by completing the square, 776 substitution method for solving, 793–796 summary of methods for solving, 800 types of solutions, 785–787 zero-factor property for solving, 382–388, 768, 800 Quadratic expressions, 382 Quadratic formula derivation of, 782 explanation of, 783 solving quadratic equations using, 783–785, 800 Quadratic functions applications using, 804–805, 814–816, 824–825 explanation of, 811 general characteristics of, 814 graphs of, 810–816, 821–822 maximum value of, 824–825 minimum value of, 824–825 solving applied problems using, 804–805 steps to graph, 821 Quadratic in form equations, 789, 793–796 Quadratic inequalities explanation of, 830 graphs of, 831 steps to solve, 833 Quadratic models, 394–395 Quotient(s) explanation of, 6, 19, 68 of opposites, 417 translating words and phrases, 72 Index Quotient rule for exponents, 284–286, 1005 for logarithms, 880–881, 882 for radicals, 710 R Radical(s) conditions for simplified form, 710, 738 converting between rational exponents and, 703–704 equations with, 739–743 explanation of, 688 index of, 691 multiplication of, 709 order of, 691 product rule for, 709 quotient rule for, 710 simplifying, 710–713 Radical equations explanation of, 739 extraneous solutions of, 739 power rule for solving, 739–740 solving with additional steps, 740–742 solving with indexes greater than 2, 743 steps for solving, 740 Radical expressions addition of, 723–725 explanation of, 688, 692 graphs of, 692–693 multiplication of, 728–729 rationalizing the denominator of, 730–734 squaring of, 689 subtraction of, 723–725 Radicand, 688 Radius, 924–927 Radius of a circle, 716–717 Range of a function, 639–640 of a relation, 639–640 Rate in motion problems, 561–563 Rate of change, average, 501–502 Rate of work, 470–473 Ratio explanation of, 157 from word phrases, 157–158 Rational equations, solving in quadratic form, 789 Rational exponents converting between radicals and, 703–704 evaluating terms with, 700–703 explanation of, 701 radical form of, 703 rules for, 704–706 Rational expressions adding, 436–439 applications of, 468–473 with denominator zero, 413 dividing, 424–426 equivalent forms for, 417–418 evaluating, 412 explanation of, 412 fundamental property of, 414 in lowest terms, 414–417 multiplying, 422–424, 426 with numerator zero, 412 operations on, 422–426, 436–441 simplifying with negative exponents, 450–451 solving equations with, 455–462 with specified denominator, 431–432 steps for division of, 426 steps for multiplication of, 426 subtracting, 439–441 summary of operations on, 466–467 undefined values for, 413 Rational functions, inverse variation and, 672 Rational inequality explanation of, 835 steps to solve, 835 Rational numbers explanation of, 43 as exponents, 701 graphing, 44 Rationalizing a binomial denominator, 732–734 the denominator, 730–734 the numerator, 737 Real numbers See also Number(s) absolute value of, 47–48 adding, 51–54, 56–58 additive inverse of, 46–47 dividing, 67–73 explanation of, 45 inequality symbols used with, 45–46 multiplying, 65–67, 70–72 opposites of, 46–47 order of operations of, 56–57 ordering of, 46 properties of, 65, 78–84 reciprocals of, 67 sets of, 45 subtracting, 54–59 summary of operations on, 29, 77 Real part of a complex number, 748 Reciprocal(s) explanation of, of fractions, 6, 67, 81 negative, 500 Reciprocal function, 918 Rectangle, area of, 146 Rectangles, perimeter of, 147, 152, 558 Rectangular box, volume of, 152 Rectangular coordinate system explanation of, 212, 490 plotting points in, 490 quadrants of, 490 Regions in the real number plane, 617 Relation definitions of, 636–639 domain of, 639–640 range of, 639–640 Relative error, 611 Remainder theorem, 1015–1016 Repeating decimals, 21, 43 Resonant frequency formula, 695 Richter, Charles F., 298 Richter scale, 298 Right angles, 139 Right triangle(s) hypotenuse of, 393, 713 legs of, 393, 713 Pythagorean theorem for, 393, 713 Rise, 232, 495 Roots calculator approximation of, 695–696 cube, 691 fourth, 691 nth, 691 principal, 688 square, 688–690 Rounding, of decimals, 19–20 Run, 232, 495 S Scale, 495 Scatter diagrams, 214, 515, 815 Scientific notation application of, 291, 294 in calculations, 293–294 dividing with, 293 explanation of, 291 and exponents, 291, 292 multiplying with, 293 steps to write a number in, 292 Scrap value, 911 Second-degree equation See Quadratic equations Second-degree equations, 936, 944–945 Second-degree inequalities explanation of, 948 graphs of, 949–952 Semicircles, 938 Semiperimeter, 700 Sentences equality and inequality symbols in, 33 translating into equations, 72–73 Sequence(s) applications of, 967 arithmetic, 972–978 explanation of, 966 Fibonacci, 965 finite, 966 general term of, 966–967 geometric, 980–987 infinite, 966 terms of, 966 Series explanation of, 968 finite, 968 infinite, 968 I-9 I-10 Index Set(s) elements of, 38 empty, 122 explanation of, 38 intersection of, 596 mapping of, 638 null, 122 operations on, 596, 598 of real numbers, 45 solution See Solution set(s) union of, 596, 598 Set braces, 38 Set-builder notation, 43, 192, 525 Set operations, 596, 598 Sigma notation, 968 Signed numbers adding, 51–54 dividing, 68 explanation of, 43 interpreting data with, 59 multiplying, 65–67 subtracting, 55–56 Similar triangles, 167 Simple interest formula for, 153, 172, 898 solving problems, 173 Simplified form of a radical, 710, 738 Simplifying algebraic expressions, 82, 88–91 equations before solving, 109, 115–116 fractions, Six-step method for solving applied problems, 132 Slope(s) explanation of, 232, 495 formula for, 232, 233, 495 from an equation, 236–237 of a horizontal line, 498 of horizontal lines, 235–236 of a line, 495–498 negative, 235, 499 of parallel lines, 238–239, 499–500, 512–513 of perpendicular lines, 238–239, 499–500, 512–513 positive, 235, 499 undefined, 235–236, 498, 499 of a vertical line, 498 of vertical lines, 235–236 Slope-intercept form, 245–249, 508, 514 Solution(s) double, 386–387 of equations, 38, 104 writing as ordered pairs, 209 Solution set(s) of an equation, 104, 196 of equations and inequalities, 594 of system of inequalities, 950–952 of a system of linear equations, 522 Solving for a specified variable with power rule, 743 with square roots, 801 Speed in motion problems, 561–563 Spheres, volume of, 153 Square(s) of binomials, 317–318 difference of, 369–371 of a number, 28, 688 perfect, 370, 690 sum of, 370 Square bracket in interval notation, 588 Square root(s) of a, 689 explanation of, 688 finding, 689 identifying types of, 690 negative, 688 of a negative real number, 746–747 of a number, 688 positive, 688 principal, 688 simplifying, 694 solving formulas involving, 801–802 symbol for, 688 Square root function explanation of, 692 generalized, 938–939 graph of, 692 graphs of, 918, 938–939 Square root property for solving quadratic equations, 768–772, 800 Square viewing window, 930 Squaring function, 657 Squaring of radical expressions, 689 Standard form of a complex number, 748 of a linear equation, 209, 253, 254–255, 491, 511, 514 of a quadratic equation, 382, 768 Standard notation, 293 Standard viewing window, 495 Step functions, 920 Straight angles, 139, 148–149 Study skills analyzing test results, S-9 completing homework, S-4 managing time, S-6 preparing for math final exam, S-10 reading math textbook, S-2 reviewing a chapter, S-7 taking lecture notes, S-3 taking math tests, S-8 using math textbook, S-1 using study cards, S-5 Subscript notation, 233, 494 Substitution method explanation of, 531 to solve nonlinear systems, 942–944 for solving dependent equations, 534–535 for solving inconsistent systems, 534 for solving linear systems, 531–537, 546 for solving quadratic equations, 793–796 steps to solve by, 533 Subtraction addition property of equality extended to, 106 of complex numbers, 749 of decimals, 17–18 explanation of, 55 of fractions, 9–10 with grouping symbols, 56–57 of a multivariable polynomial, 304 on a number line, 54 in order of operations, 29, 77 of polynomial functions, 659–660 of polynomials, 303–304, 1006 of radical expressions, 723–725 of rational expressions, 439–441 of real numbers, 54–59 of signed numbers, 55–56 symbol for, 56 word phrases for, 57–58 Subtrahend, 54 Sum of cubes, 375–376 explanation of, of measures of angles of a triangle, 568 of squares, 370 of terms of a geometric sequence, 982–984, 985–987 of terms of an arithmetic sequence, 976–978 Summation notation, 968–969 Supplementary angles, 139–140, 568 Supply and demand, 531 Symbol(s) for additive inverse, 56 for cube root, 691 for empty sets, 122 for equality, 31, 33, 38 grouping, 29, 56–57 for inequality, 31–33, 45–46, 182, 588 for the inverse of a function, 852 for negative infinity, 182 for negative numbers, 56 for null set, 122 plus or minus, 769 for positive infinity, 182 for square roots, 688 for subtraction, 56 word conversions to, 32, 57–58 Symmetry about an axis, 810 Symmetry axis of a parabola, 305 Synthetic division, 1013–1016 System of linear equations in three variables applications of, 564–566 explanation of, 548 geometry of, 548–549, 564–565 graphs of, 548–549 inconsistent, 553 special cases of, 552–553 steps to solve, 549 Index System of linear equations in two variables applications of, 557–566 steps to solve applications of, 557 Systems of inequalities explanation of, 950 graphs of, 950–952 solution set of, 950–952 Systems of linear equations alternative method for solving, 542–543 choosing a method to solve, 546–547 consistent, 525–526 with decimals, 536–537 elimination method for solving, 539–544, 546 explanation of, 522 with fractions, 535–536 graphing method for solving, 522–527 inconsistent, 525–526 with no solution, 525–526 solution of, 522 solution set of, 522 solving by graphing, 522–527 steps for solving by elimination, 540 steps for solving by substitution, 533 steps to solve by graphing, 524 substitution method for solving, 531–537 summary of outcomes, 526 Systems of nonlinear equations, 942–946 T Table of data, interpreting, 48 Table of ordered pairs, 491 Table of values, 211–212 Term(s) adding, 310 of an annuity, 984 of a binomial expansion, 994–995 combining, 89, 299–300 degree of, 301 distinguishing between factors and, 89 explanation of, 88–89 of an expression, 88–89, 299 like, 89, 299 multiplying, 310 numerical coefficient of, 88, 299 of a polynomial, 299 of a proportion, 158 of a sequence, 966 unlike, 89, 299–300 Terminating decimals, 21, 43 Tests for divisibility, 76, 344 Third-degree polynomial inequalities, 834–835 Three-part inequalities explanation of, 593 solving, 190–192, 593–594 Threshold sound, 888 Threshold weight, 708 Time in motion problems, 561–563 Tolerance, 611 Traffic intensity, 421 Translating sentences into equations, 72–73 word phrases into algebraic expressions, 37–38, 91, 123 Translations of functions, 918–920 Transverse axis of hyperbola, 934 Trapezoids, area of, 146, 152 Triangle(s) area of, 152 Pascal’s, 991 perimeter of, 147–148, 152 right, 393, 713 similar, 167 sum of angles of, 568 Trinomials explanation of, 301 factoring of, 353–358, 360–366 perfect square, 371–373, 376 Triple, ordered, 548 U Undefined rational expressions, 413 Undefined slope, 236, 498, 499 Union of sets, 596, 598 of solution sets, 196 Unit cost, 158 Unit pricing, 158 Unlike terms, 89, 299–300 V Variable(s) dependent, 638 explanation of, 36 formulas to evaluate, 146–148 independent, 638 solving for specified, 149–151, 461–462 Variable cost, 253 Variation combined, 674 constant of, 669 direct, 669–671 explanation of, 669 inverse, 671–673 joint, 673–674 steps to solve problems with, 671 Vertex of a parabola explanation of, 305, 810, 813, 825 finding, 819–821 formula for, 821 Vertical angles, 148–149, 568 Vertical hyperbola, 936 Vertical line(s) equation of, 224–225, 493, 512, 514 explanation of, 224 graph of, 493 I-11 graphing, 224–225 slope of, 235–236, 498 Vertical line test for a function, 640–641 Vertical parabola graphing, 811–813 vertex of, 811, 819–821 x-intercepts of, 822–823 Vertical shift of ellipse, 929 method for applying, 919–920 of a parabola, 811–813 Volume explanation of, 152 of a pyramid, 153 of a rectangular box, 152 of a sphere, 153 W Whole numbers, 1, 42 Windchill factor, 708 Word phrases for addition, 57–58 to algebraic expressions, 37–38, 91, 123 for division, 72 to expressions, 91 for multiplication, 71–72 to numerical expressions, 57–58, 71–72 to ratios, 157–158 for subtraction, 57–58 Word statements to equations, 72–73 Words to symbols conversions, 32, 57–58 Work problems, 791–792 Work rate problems, 470–473 Working equation, 549 X x-axis, 212, 490 x-intercept explanation of, 221, 491 of a line, 491–492 of a parabola, 822–823 Y y-axis, 212, 490 y-intercept explanation of, 221, 491 of a line, 491–492, 509 slope-intercept form and, 245–248 Z Zero division by, 69 multiplication by, 65 Zero denominator in a rational expression, 413 Zero exponent, 281, 286, 1005 Zero-factor property, 382–388, 768, 800 Triangles and Angles Right Triangle Triangle has one 90° (right) angle Right Angle Measure is 90° c a 90° b Pythagorean Theorem (for right triangles) a2 + b2 = c2 Isosceles Triangle Two sides are equal Straight Angle Measure is 180° B 180° AB = BC A C Equilateral Triangle All sides are equal Complementary Angles The sum of the measures of two complementary angles is 90° B AB = BC = CA A C Supplementary Angles The sum of the measures of two supplementary angles is 180° B A AB AC BC = = DE DF EF Angles and are supplementary C Similar Triangles Corresponding angles are equal Corresponding sides are proportional A = D, B = E, C = F Angles and are complementary Sum of the Angles of Any Triangle A + B + C = 180° Vertical Angles Vertical angles have equal measures E B C A F Angle = Angle D Angle = Angle Geometry Formulas Figure Formulas Illustration Square Perimeter: P = 4s s Area: 𝒜 = s2 s s s Rectangle Perimeter: P = 2L + 2W Area: 𝒜 = LW W L Triangle Perimeter: P = a + b + c a Area: 𝒜 = bh Parallelogram c h b Perimeter: P = 2a + 2b b Area: 𝒜 = bh a h a b Trapezoid Perimeter: P = a + b + c + B Area: 𝒜 = h1b + B2 b a h c B Circle Diameter: d = 2r Circumference: C = 2pr C = pd Area: 𝒜 = pr Chord r d Geometry Formulas Figure Formulas Illustration Cube Volume: V = e3 Surface area: S = 6e2 e e e Rectangular Solid Volume: V = LWH Surface area: S = 2HW + 2LW + 2LH Right Circular Cylinder H W L Volume: V = pr 2h Surface area: S = 2prh + 2pr (Includes both circular bases) h r Cone Volume: V = pr h Surface area: S = pr 2r + h2 + pr (Includes circular base) Right Pyramid Volume: V = Volume: V = r Bh B = area of the base Sphere h h pr Surface area: S = 4pr r Other Formulas Distance: d = rt 1r = rate or speed, t = time2 Percent: p = br 1p = percentage, b = base, r = rate2 Temperature: F = C + 32 C = 1F - 322 Simple Interest: I = prt 1p = principal or amount invested, r = rate or percent, t = time in years2 ... 2, 3, 5, 7, 1 1, 1 3, 1 7, 1 9, 2 3, 2 9, 3 1, 37 First dozen prime numbers A natural number greater than that is not prime is a composite number 4, 6, 8, 9, 1 0, 1 2, 1 4, 1 5, 1 6, 1 8, 2 0, 21 First dozen... Hornsby Terry McGinnis DEDICATION To BK and Vangie E .J. H To Andrew and Tyler Mom Resources for Success Get the Most Out of MyLab Math for Beginning and Intermediate Algebra, Seventh Edition by Lial, ... fractions Interpret data in a circle graph 1, 2, 3, 4, c, The three dots, or ellipsis points, indicate that each list of numbers continues in the same way indefinitely 0, 1, 2, 3, 4, c, and