www.FreeEngineeringbooksPdf.com Concept Investigations—Helping you to discover connections Concept Investigations are great learning tools to help you to explore and generalize patterns and relationships such as the graphical and algebraic representations of the functions you’ll study CONCEPT INVESTIGATION This particular Concept Investigation shows you how algebraic concepts can be studied via pattern recognition This may be a review topic for you What direction is that line going in? Use your graphing calculator to examine the following Start by setting up your calculator by doing the following steps ● ● Clear all equations from the Y5 screen (Press Y= , Change the window to a standard window (Press ZOOM , ) (ZStandard).) Now your calculator is ready to graph equations The Y5 screen is where equations will be put into the calculator to graph them or evaluate them at input values Several Explanatory margin boxes simple equations will be graphed to investigate how the graph of an equation for a line like the “What’s That Mean” reacts to changes in the equation (Note that your calculator uses y as the dependent one here appear as needed (output) variable and x as the independent (input) variable.) to ensure you understand key cients concepts terms window Graph the following equations that have positive coef¿ on aorstandard Enter each equation in its own row (Y1, Y2, Y3, ) (Note: To enter an x, you use the X,T,⍜,n button next to the ALPHA button.) a y x c y 5 x Now graph the following equations that have negative coef¿cients b y 22 x c y 25 x 96369_01_ch01_1.1-1.3.indd 39 d y 28 x Coefficient The number in front of a variable expression is the coefficient For example: 27 x Calculator steps and tips are listed here Don’t forget to go to the “Using the Graphing Calculator” Appendix when you need further help In your own words, describe what a negative coef¿cient of x does to the graph 27 is the coefficient for x is the coefficient of x Using Your TI Graphing Calculator In entering fractions in the calculator, it is often best to use parentheses y Θ 1/5 Ι x On many graphing calculators, 11/09/10 parentheses are needed in almost all situations In some calculators, when Graph the following equations with coefficients that are between zero and one c y x d y x 3:01 AM 1/5 x is entered, the calculator will interpret this as a y x b y 5x 5x2 Remember that a variable that is by itself (x) has a coefficient of b y x d y x In your own words, describe what the coef¿cient (number in front) of x does to the graph Remember to read graphs from left to right a y x What’s That Mean? Describing math in your own words will help you understand and remember the concepts _ 5x instead of e y 0.9 x In your own words, describe what a coef¿cient of x between and does to the graph x To be sure the calculator does what you intend, using parentheses is a good idea The TI-83 does not need parentheses in some situations, but in other situations, they are required To keep confusion down, one option is to use parentheses around every fraction Extra parentheses not usually create a problem, but not having them where they are needed can cause miscalculations Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_Endsheet_IFC.indd 26/10/10 4:39 PM BEGINNING ALGEBRA Connecting Concepts through Applications MARK CLARK PALOMAR COLLEGE CYNTHIA ANFINSON PALOMAR COLLEGE Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd i 11/11/10 8:10 PM This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com Beginning Algebra: Connecting Concepts through Applications Mark Clark & Cynthia Anfinson © 2012 Brooks/Cole, Cengage Learning Publisher: Charlie Van Wagner Developmental Editor: Don Gecewicz, Carolyn Crockett, Rita Lombard Assistant Editor: Stefanie Beeck, Carrie Jones ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher Editorial Assistant: Jennifer Cordoba For product information and technology assistance contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 Media Editor: Heleny Wong Marketing Manager: Gordon Lee For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Marketing Coordinator: Shannon Myers Marketing Communications Manager: Darlene Macanan Content Project Manager: Cheryll Linthicum Further permissions questions can be e-mailed to permissionrequest@cengage.com Art Director: Vernon T Boes Print Buyer: Karen Hunt Library of Congress Control Number: 2010929055 Rights Acquisitions Specialist: Roberta Broyer Student Edition: Rights Acquisitions Specialist: Don Schlotman Production Service: MPS Content Services, Jill Traut ISBN-13: 978-0-534-41938-7 ISBN-10: 0-534-41938-0 Text Designer: Diane Beasley Loose-leaf Edition: Art Editor: Leslie Lahr, Lisa Torri ISBN-13: 978-1-111-56929-7 ISBN-10: 1-111-56929-0 Photo Researcher: Bill Smith Group Text Researcher: Isabel Alves Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Copy Editor: Barbara Willette Illustrator: Matrix Art Services, Jade Myers; Lori Heckelman Cover Designer: Lawrence Didona Cover Image: Masterfile Royalty Free Compositor: MPS Limited, a Macmillan Company Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd To learn more about Brooks/Cole, visit www.cengage.com/brookscole Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Printed in the United States of America 15 14 13 12 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd ii 11/11/10 8:10 PM To my wife Christine, our children Will and Rosemary, and my parents for their love and support MC To my husband Fred and son Sean, thank you for your love and support CA Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd iii 11/11/10 8:10 PM About the Authors MARK CLARK graduated from California State University, Long Beach, with Bachelor’s and Master’s degrees in Mathematics He is a full-time Associate Professor at Palomar College and has taught there for the past 13 years He is committed to teaching his students through applications and using technology to help them both to understand the mathematics in context and to communicate their results clearly CYNTHIA (CINDY) ANFINSON graduated from UC San Diego’s Revelle College, with a Bachelor of Arts degree in Mathematics She went to graduate school at Cornell University under the Army Science and Technology Graduate Fellowship and graduated from Cornell in 1989 with a Master of Science degree in Applied Mathematics She is currently an Associate Professor of Mathematics at Palomar College and has been teaching there since 1995 Cindy Anfinson was a finalist for Palomar College’s 2002 Distinguished Faculty Award About The Cover This cover image tells a story Our goal with this cover was to represent how people interact and connect with technology in their daily lives We selected this cover as it illustrates the fundamental idea of the Clark/Anfinson series—connecting concepts to applications and rote mathematics to the real world since the skills and concepts in this series have their foundation in applications from the world around us Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd iv 11/11/10 8:10 PM Brief Contents CHAPTER R REVIEW OF PREALGEBRA CHAPTER BUILDING BLOCKS OF ALGEBRA 61 CHAPTER EQUATIONS AND INEQUALITIES WITH LINEAR ONE VARIABLE 133 CHAPTER LINEAR EQUATIONS WITH TWO VARIABLES 205 CHAPTER SYSTEMS OF LINEAR EQUATIONS 311 CHAPTER EXPONENTS AND POLYNOMIALS 415 CHAPTER FACTORING AND QUADRATIC EQUATIONS 479 CHAPTER RATIONAL EXPRESSIONS AND EQUATIONS 563 CHAPTER RADICAL EXPRESSIONS AND EQUATIONS 647 CHAPTER MODELING DATA 741 APPENDIX A ANSWERS TO PRACTICE PROBLEMS A-1 APPENDIX B ANSWERS TO SELECTED EXERCISES B-1 v Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd v 11/11/10 8:10 PM Contents R Review of Prealgebra R.1 Operations with Integers Natural Numbers, Whole Numbers, and Integers • Number Lines • Relations Between Numbers • Absolute Value • Opposite of a Number • Operations with Integers • Order of Operations R.2 Operations with Fractions 18 Prime Numbers and Prime Factorization • Simplifying Fractions and Equivalent Fractions • Fractions on Number Lines • Addition and Subtraction of Fractions • Multiplication and Division of Fractions • Order of Operations R.3 Operations with Decimals and Percents 30 Place Value • Relationships Between Fractions and Decimals • Graphing Decimals on a Number Line • Rounding Decimals • Addition and Subtraction of Decimals • Multiplication and Division of Decimals • Order of Operations with Decimals • What Is a Percent? • Converting Between Percents, Decimals, and Fractions • Problem Solving with Percents R.4 The Real Number System 43 Rational Numbers • Irrational Numbers • The Real Number System • Exact and Approximate Answers Chapter Chapter Chapter Chapter 1.1 R Summary 51 R Review Exercises R Test 57 R Projects 58 54 Building Blocks of Algebra Exponents, Order of Operations, and Properties of Real Numbers 62 Exponents • Scientific Notation • Order of Operations • Properties of Real Numbers 1.2 Algebra and Working with Variables 75 Constants and Variables • Evaluating Expressions • Unit Conversions • Defining Variables • Translating Sentences into Expressions • Generating Expressions from Input-Output Tables vi Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd vi 11/11/10 8:10 PM Contents vii 1.3 Simplifying Expressions 94 Like Terms • Addition and Subtraction Properties • Multiplication and Distributive Properties • Simplifying Expressions 1.4 Graphs and the Rectangular Coordinate System 105 Data Tables • Bar Graphs • Scatterplots • Rectangular Coordinate System Chapter Chapter Chapter Chapter 1 1 Summary 123 Review Exercises Test 130 Projects 132 127 Linear Equations and Inequalities with One Variable 2.1 Addition and Subtraction Properties of Equality 134 Recognizing Equations and Their Solutions • Addition and Subtraction Properties of Equality • Solving Literal Equations 2.2 Multiplication and Division Properties of Equality 148 Multiplication and Division Properties of Equality • Solving Multiple-Step Equations • Generating Equations from Applications • More on Solving Literal Equations 2.3 Solving Equations with Variables on Both Sides 164 Solving Equations with Variables on Both Sides • Solving Equations That Contain Fractions • Equations That Are Identities or Have No Solution • Translating Sentences into Equations and Solving 2.4 Solving and Graphing Linear Inequalities on a Number Line 175 Introduction to Inequalities • Solving Inequalities • Interval Notation and Number Lines • Compound Inequalities Chapter Summary 191 Chapter Review Exercises 196 Chapter Test 199 Chapter Projects 200 Cumulative Review Chapters 1–2 202 Linear Equations with Two Variables 3.1 Graphing Equations with Two Variables 206 Using Tables to Represent Ordered Pairs and Data • Graphing Equations by Plotting Points • Graphing Nonlinear Equations by Plotting Points • Vertical and Horizontal Lines 3.2 Finding and Interpreting Slope 217 Interpreting Graphs • Determining a Rate of Change • Calculating Slope • Interpreting Slope Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_00_FM_pi-xxiv.indd vii 11/11/10 8:10 PM B-85 Appendix B 23 a Month Average Daily High Temperature (degrees Fahrenheit) 45 55 65 75 83 82 T 27 a The vertical intercept is (0, 0) This means that if pictures are framed, there is $0 profit Average Daily High Temperature for Rapid City, SD b The vertex is _ 6, 108 + This means that the frame shop will make the maximum amount of average profit of $108 when they frame pictures 85 Temperature (degrees Fahrenheit) 80 75 70 c n 2, n 10 The shop must frame either pictures or 10 pictures to make an average profit of $60 per frame 65 60 55 28 a The vertical intercept is (0, 0) This means that if apple trees are sold, there is $0 profit 50 45 40 35 30 b P 58,041.4 According to the model, the median price of a home in Detroit, Michigan, in the year 2010 would be about $58,041 This answer is possible but seems too low to be reasonable This could be an example of model breakdown c P 220,169.4 According to the model, the median price of a home in Detroit, Michigan, in the year 2012 would be about 2$20,169 This answer is not reasonable because a home price cannot be a negative value This is an example of model breakdown Source: www.weather.com b reach a high of approximately $110,493 in the year 2007, then the median home prices will begin to decrease m Month 10 11 12 The display of data follows a parabolic pattern 24 a P b The vertex is _ 4, 64 + This means that the nursery will make the maximum amount of average profit of $64 when they sell trees c n Ϸ 5.87, n Ϸ 2.13 The nursery must sell either trees or trees to make an average profit of about $50 per tree Chapter Test Median Price of a Home in Detroit, MI a Quadratic 115,000 b Linear c Neither Median price (dollars) 110,000 a Let d the dog’s age in years, and let h the human age in years 105,000 100,000 95,000 b 90,000 Conversion of Dog’s Age to Human Age 85,000 h 80,000 60 75,000 56 t Years since 2000 10 52 48 44 Human age (years) b The display of data follows a parabolic pattern 25 a The vertex of the model is approximately _ 9, 85 + This means that the average daily high temperature in Rapid City, South Dakota, will reach a high of approximately 858F in the ninth month of the year (September), then the daily average high temperature will begin to decrease b T 75.11 According to the model, the estimated average high temperature in Rapid City, South Dakota, in December would be about 758F This answer does not seem reasonable for that city and is considerably different from the reported value of 378F This is an example of model breakdown c T 29.11 According to the model, the estimated average high temperature in Rapid City, South Dakota, in February would be about 298F This answer seems reasonable and is not considerably different from the reported value of 388F 26 a The vertex of the model is _ 6.5, 110,493 + This means that the median price of a home in Detroit, Michigan, will 40 36 32 28 24 20 16 12 4 Dog’s age (years) 10 11 d c See scatterplot in part b d h 4.17d 14.83 using the points (1, 19) and (7, 44) e h Ϸ 47.78 According to the model, the age of an 8-yearold dog approximately corresponds to a human age of 48 years This answer seems reasonable f d Ϸ 22.41 According to the model, the age of a 5-yearold human approximately corresponds to a dog’s age of 22 years This answer is not reasonable because Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_12_App B_B1-B86.indd 85 09/11/10 10:01 PM B-86 CHAPTER Modeling Data you cannot have a negative age This is a case of model breakdown a Let t the year, and let p the population in millions b a The vertex is _ 0, 50 + This means that the rock is at its maximum height of 50 feet after seconds from the time it is dropped, or before it has been dropped Population of West Virginia p 1.828 1.825 b The x-intercepts are approximately _ 1.77, + and _ 21.77, + The intercept _ 1.77, + means that 1.77 seconds after being dropped, the rock will be feet above the ground This makes sense in terms of this situation The intercept _ 21.77, + means that the rock would be feet above the ground 1.77 seconds before being dropped, and this does not make sense in terms of this situation 1.822 1.819 Population (millions) c t ഠ 84, t ഠ 249 The model predicts that the population of Mexico City will reach 10 million people in the years 1984 and 1851 1984 may make sense in this situation 1851 is too far in the past for that large a population 1.816 1.813 1.810 1.807 1.804 1.801 1.798 1.795 t 19 94 19 95 19 96 19 97 19 98 19 99 20 00 20 01 20 02 Year Yes, the scatterplot looks linear c See scatterplot in part b d p 20.004t 9.807 e p 1.759 According to the model, the population of West Virginia in the year 2012 would be 1.759 million This answer seems reasonable y –3 –2 –1 –3 x –6 –9 –12 a The graph of the data follows a parabolic pattern b The estimated vertex from the graph is (1, 3) a h 100 90 80 70 60 50 40 30 20 10 –2.5 –2 –1.5 –1 –0.5 –10 0.5 1.5 2.5 t b The graph of the data follows a parabolic pattern c The estimated vertex from the graph is (0, 85) a The vertex is _ 17.75, 1.12 + This means that in the year 1918, Mexico City reached its minimum population of 1.14 million, then the population began to rise b The vertical intercept is (0, 1.75) This means that in the year 1900, the population of Mexico City would have been 1.75 million This may be model breakdown if the actual population in 1900 was not more than the population in 1918 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com 19380_12_App B_B1-B86.indd 86 09/11/10 10:01 PM Index A Absolute value, 6–7 distance as, 6–7, 10–11 graphing an equation containing, 211–212 opposite of a number and, in order-of-operations agreement, 67–68 AC method of factoring, 497–499 Addends, 13, 66 Addition, 13, 66 associative property of, 69–70 commutative property of, 69–70, 96 of decimals, 34–35 distributive property and, 72 of fractions, 23–26 of integers, 8–10 of like terms, 95–96 negative numbers in, 8–10 order-of-operations agreement and, 13–15, 27–28, 37, 65–68, 71–73 phrases used for, 82 of polynomials, 441–444 properties of, 69–70, 96 of radical expressions, 665–668 of rational expressions with common denominators, 584–586 of rational expressions with unlike denominators, 590–596 of signed numbers, 8–10 undoing subtraction, 138, 151 Addition method, for solving systems of equations, 350 Addition property of equality, 138, 140, 142, 154 Addition property of inequalities, 177 Airplane speed, 340–342 Algebra, 75–76 basic operations used in, 66 Algebraic expressions See Expressions Amount of substance in solution, 358–359 in value formula, 343 Angles complementary, 360 supplementary, 360 of triangles, 171 Applications answer as complete sentence, 139, 157 answer that makes sense, 157 defining variables in, 81–82, 84–85 finding equations of lines in, 274–278 functions in, 293–294 inequalities in, 182, 185–186 integer problems, 342–343 percent mixtures, 358–359 percents in, 39–40 proportions in, 619–620, 622–624 of Pythagorean Theorem, 702 quadratic equations in, 716–718 radical equations in, 689–690 rational equations in, 564–565 rounding answer in, 157 simple interest, 356–358, 389–390 speed, 340–342 systems of equations in, 324–326, 340–345, 355–359 systems of inequalities in, 384–386, 388–391 translating into equations, 155–158, 170–172 value mixtures, 343–345 vertex of quadratic equation in, 528–529 Approximate values, 47–49, 662 See also Estimation; Rounding of square roots, 47–48, 650, 662 Approximately equal to (