Beginning and Intermediate Algebra with Applications & Visualization PDF
Preview Beginning and Intermediate Algebra with Applications & Visualization
Pearson New International Edition Beginning and Intermediate Algebra with Applications and Visualization Gary K. Rockswold Terry A. Krieger Third Edition International_PCL_TP.indd 1 7/29/13 11:23 AM ISBN 10: 1-292-02732-0 ISBN 13: 978-1-292-02732-6 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this book by such owners. ISBN 10: 1-292-02732-0 ISBN 10: 1-269-37450-8 ISBN 13: 978-1-292-02732-6 ISBN 13: 978-1-269-37450-7 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America Copyright_Pg_7_24.indd 1 7/29/13 11:28 AM 123345667899661843265311539351571753 P E A R S O N C U S T O M L I B R AR Y Table of Contents 1. Introduction to Algebra Gary K. Rockswold/Terry A. Krieger 1 2. Linear Equations and Inequalities Gary K. Rockswold/Terry A. Krieger 93 3. Graphing Equations Gary K. Rockswold/Terry A. Krieger 165 4. Systems of Linear Equations in Two Variables Gary K. Rockswold/Terry A. Krieger 263 5. Polynomials and Exponents Gary K. Rockswold/Terry A. Krieger 319 6. Factoring Polynomials and Solving Equations Gary K. Rockswold/Terry A. Krieger 383 7. Rational Expressions Gary K. Rockswold/Terry A. Krieger 445 8. Introduction to Functions Gary K. Rockswold/Terry A. Krieger 531 9. Systems of Linear Equations Gary K. Rockswold/Terry A. Krieger 625 10. Radical Expressions and Functions Gary K. Rockswold/Terry A. Krieger 667 11. Quadratic Functions and Equations Gary K. Rockswold/Terry A. Krieger 751 12. Exponential and Logarithmic Functinos Gary K. Rockswold/Terry A. Krieger 837 13. Conic Sections Gary K. Rockswold/Terry A. Krieger 915 I 111999990000005788991193137 Appendix: Using the Graphing Calculator Gary K. Rockswold/Terry A. Krieger 959 Appendix: Sets Gary K. Rockswold/Terry A. Krieger 971 Appendix: Linear Programming Gary K. Rockswold/Terry A. Krieger 981 Appendix: Synthetic Division Gary K. Rockswold/Terry A. Krieger 989 Glossary Gary K. Rockswold/Terry A. Krieger 993 Bibliography Gary K. Rockswold/Terry A. Krieger 1001 Formulas from Geometry & Algebra Gary K. Rockswold/Terry A. Krieger 1003 Index 1007 II Introduction to Algebra 1 Numbers, Variables, and Expressions 2 Fractions 3 Exponents and Order of Operations 4 Real Numbers and the Number Line 5 Addition and Subtraction of Real Numbers 6 Multiplication and Division of Real Numbers 7 Properties of Real Num- bers 8 Simplifying and Writing Algebraic Expressions NASA/Arena Creative/Shutterstock J Unless you try to do something beyond ust over a century ago only about one in ten workers was in a professional, tech- what you have already nical, or managerial occupation. Today this proportion is nearly one in three, and the mastered, you will study of mathematics is essential for anyone who wants to keep up with the technologi- never grow. cal changes that are occurring in nearly every occupation. Mathematics is the l anguage of technology . —RONALD E. OSBORN I n the information age, mathematics is being used to describe human behavior in areas such as economics, medicine, advertising, social networks, and Internet use. For example, mathematics can help maximize the impact of advertising by analyzing social networks such as Facebook. Today’s business managers need employees who not only understand human behavior but can also describe that behavior using mathematics. I t’s just a matter of time before the m ajority of the workforce will need the analytic skills that are taught in mathematics classes every day. No matter what career path you choose, a solid background in mathematics will provide you with opportunities to reach your full potential in your vocation, income level, and lifestyle. Source: A. Greenspan, “T he Eco- nomic Importance of Improving Math-Science Education. ” From Chapter 1 of Beginning and Intermediate Algebra with Applications and Visualization. Third Edition. Gary K. Rockswold, Terry A. Krieger. Copyright © 2013 by Pearson Education, Inc. All rights reserved. 1 INTRODUCTION TO ALGEBRA 1 Numbers, Variables, and Expressions Natural Numbers and Whole Numbers ● P rime Numbers and Composite Numbers ● Variables, Algebraic Expressions, and Equations ● Translating Words to Expressions ck A LOOK INTO MATH N Numbers are an important concept in every society. A number system once used in south- o st ern Africa consisted of only the numbers from 1 to 20. Numbers larger than 20 were named er utt by counting groups of t wenties. For example, the number 67 was called three twenties h S and seven. This base-20 number system would not work well in today’s technologically n/ ci advanced world. In this section, we introduce two sets of numbers that are used extensively ar M in the modern world: natural numbers and whole numbers. z s Kir Natural Numbers and Whole Numbers NEW VOCABULARY One important set of numbers is the set of n atural numbers . These numbers comprise the counting numbers and may be expressed as follows. n N atural numbers n W hole numbers 1, 2, 3, 4, 5, 6,N n P roduct n F actors B ecause there are infinitely many natural numbers, three dots are used to show that the list n P rime number continues without end. A second set of numbers is called the w hole numbers , and may be n C omposite number expressed as follows. n P rime factorization 0, 1, 2, 3, 4, 5,N n V ariable n A lgebraic expression W hole numbers include the natural numbers and the number 0. n E quation n F ormula N REAL-WORLD CONNECTION Natural numbers and whole numbers can be used when data are not broken into fractional parts. For example, the bar graph in Figure 1 shows the number of CRITICAL THINKING apps on a student’s iPad for the first 5 months after buying the device. Note that both natu- ral numbers and whole numbers are appropriate to describe these data because a fraction of Give an example from every- an app is not possible. day life of natural number or whole number use. iPad Apps 50 45 42 ps 40 37 p A of 30 29 er mb 20 18 u N 10 0 Jan. Feb. Mar. Apr. May Month Figure 1 STUDY TIP Bring your book, notebook, and a pen or pencil to every class. Write down major concepts pre- sented by your instructor. Your notes should also include the meaning of words written in bold type in the text. Be sure that you understand the meaning of these important words. 2 INTRODUCTION TO ALGEBRA Prime Numbers and Composite Numbers When two natural numbers are multiplied, the result is another natural number. For exam- ple, multiplying the natural numbers 3 and 4 results in 12, a natural number. The result 12 is called the product and the numbers 3 and 4 are factors of 1 2 . # 3 4 = 12 factor factor product # NOTE: Products can be expressed in several ways. For example, the product 3 4 = 12 can also be written as 3 * 4 = 12 and 3(4) = 12. A natural number greater than 1 that has o nly itself and 1 as natural number factors is a prime number . The number 7 is prime because the only natural number factors of 7 are 1 and 7. The following is a partial list of prime numbers. There are infinitely many prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 READING CHECK A natural number greater than 1 that is not prime is a c #omposite number . For example, the natural number 15 is a composite number because 3 5 = 15 . In other words, 15 has • How do prime numbers factors other than 1 and itself. differ from composite Every composite number can be written as a product of prime numbers. For example, numbers? we can use a factor tree such as the one shown in Figure 2 to find the prime factors of the composite number 120. Branches of the tree are made by writing each composite number as a product that includes the smallest possible prime factor of the composite number. 120 CRITICAL THINKING 2 60 Suppose that you draw a tree diagram for the prime factor- 2 2 30 ization of a prime number. Describe what the tree will 2 2 2 15 look like. 2 2 2 3 5 Figure 2 Prime Factorization of 120 Figure 2 shows that the prime factorization of 120 is # # # # 120 = 2 2 2 3 5. Every composite number has a unique prime factorization, and it is customary to write the prime factors in order from smallest to largest. MAKING CONNECTIONS Factor Trees and Prime Factorization The prime factors of 120 can also be found using the following tree. Even though this tree is differ- ent than the one used earlier, the prime factors it reveals are the same. 120 10 12 2 5 3 4 2 5 3 2 2 No matter what tree is used, a prime factorization is a lways unique. 3 INTRODUCTION TO ALGEBRA E XAMPLE 1 Classifying numbers as prime or composite Classify each number as prime or composite, if possible. If a number is composite, write it as a product of prime numbers. (a) 31 (b) 1 (c) 35 (d) 200 Solution (a) The only factors of 31 are itself and 1, so the number 31 is prime. (b) The number 1 is neither prime nor composite because prime and composite numbers 200 must be greater than 1. (c) The number 35 is composite b#ecause 5 and 7 are factors. It can be written as a product 2 100 of prime numbers as 35 = 5 7. (d) The number 200 is composite because 10 and 20 are factors. A factor tree can be used 2 2 50 to write 200 as a product of prime numbers#, as# sh#own# in Figure 3 . The factor tree reveals that 200 can be factored as 200 = 2 2 2 5 5 . 2 2 2 25 Now Try Exercises 13, 15, 17, 21 2 2 2 5 5 Figure 3 Prime Factorization of 200 Variables, Algebraic Expressions, and Equations # There #are 12 inches in 1 foot, so 5 feet equal 5 12 = 60 inches. Similarly, in 3 feet there are 3 12 = 36 inches. To convert feet to inches frequently, Table 1 might help. TABLE 1 Converting Feet to Inches Feet 1 2 3 4 5 6 7 Inches 12 24 36 48 60 72 84 However, this table is not helpf#ul in converting 11 feet to inches. We could expand Table 1 into Table 2 to include 11 12 = 132 , but expanding the table to accommodate every possible value for feet would be impossible. TABLE 2 Converting Feet to Inches Feet 1 2 3 4 5 6 7 11 Inches 12 24 36 48 60 72 84 132 READING CHECK Variables are often used in mathematics when tables of numbers are inadequate. A variable is a symbol, typically an italic letter such as x , y , z , or F , used to represent an • What is a variable? unknown quantity. In the preceding example, the number of feet could be represented by the • Give one reason for using variable F , and the corresponding number of inches could be represented# by the variable I . variables in mathematics. The number of inches in F feet is given by the algebraic expression 12 F . That is, to cal- culate the number of inches in F feet, multiply F by 12. The relationship between feet F and inches I is shown using the e quation or formula # I = 12 F. # A dot ( ) is used to indicate multiplication because a multiplication sign ( * ) can be con- fused with the variable x . Many times the multiplication sign is omitted altogether. Thus all three formulas # I = 12 F, I = 12F, and I = 12(F ) 4 INTRODUCTION TO ALGEBRA represent the same relationship between feet and inches. If we wish to find the number of inches in 10 feet, for example, we can replace F in one of these formulas with the number 10. If we let F = 10 in the first formula, # I = 12 10 = 120. That is, there are 120 inches in 10 feet. More formally, #an a lgebraic expression consists of numbers, variables, operation sym- bols such as +, -, , and ,, and grouping symbols such as parentheses. An equation is a mathematical statement that two algebraic expressions are equal. Equations always con- tain an equals sign. A formula is a special type of equation that expresses a relationship between two or more quantities. The formula I = 12F states that to calculate the number of inches in F feet, multiply F by 12. E XAMPLE 2 Evaluating algebraic expressions with one variable Evaluate each algebraic expression for x = 4 . x (a) x + 5 (b) 5x (c) 15 - x (d) (x - 2) Solution (a) Replace x with 4 in the expression x + 5 to obtain 4 + 5 = 9 . # (b) The expression 5x indicates multiplication of 5 and x . Thus 5x = 5 4 = 20 . (c) 15 - x = 15 - 4 = 11 (d) Perform all arithmetic operations inside parentheses first. x 4 4 = = = 2 (x - 2) (4 - 2) 2 Now Try Exercises 45, 47, 51, 53 N REAL-WORLD CONNECTION Some algebraic expressions contain more than one variable. For example, if a car travels 120 miles on 6 gallons of gasoline, then the car’s mileage is 120 = 20 miles per gallon. In general, if a car travels M miles on G gallons of gasoline, 6 then its mileage is given by the expression M . Note that M contains two variables, M and G , G G whereas the expression 12F contains only one variable, F . E XAMPLE 3 Evaluating algebraic expressions with two variables Evaluate each algebraic expression for y = 2 and z = 8 . z (a) 3yz (b) z - y (c) y Solution # # (a) Replace y with 2 and z with 8 to obtain 3yz = 3 2 8 = 48 . (b) z - y = 8 - 2 = 6 z 8 (c) = = 4 y 2 Now Try Exercises 55, 57, 59 5
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