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1. Irrational Rational numbers numbers 5 V3 7 0 Integers Noninteger wy 3 2 1 fractions O i 2 3h woz positive and negative Q Al T73 0 5 Negative Whole integers numbers a 3 2 1 OMI Natural Zero numbers Ue Pes hares Figure 1 1 Subsets of Real Numbers Section 1 1 The Real Number System 3 Even with the set of integers there are still many quantities in everyday life that you cannot describe accurately The costs of many items are not in whole dollar amounts but in parts of dollars such as 1 19 and 39 98 You might work 85 hours or you might miss the first half of a movie To describe such quantities you can expand the set of integers to include fractions The expanded set is called the set of rational numbers Formally a real number is rational if it can be written as the ratio p q of two integers where q 0 the symbol means does not equal Here are some examples of rational numbers 2 1 1 125 2 P 3 0 333 8 0 125 and ll 1 126126 The decimal representation of a rational number is either terminating or repeating For instance the decimal representation of 4 0 25 is terminating and the decimal representation of 4 0 363636 0 3 11 is repeating The overbar symbol over 36 indicates which digits repeat A real number that cannot be written as a ratio of two integers is
2. _e 1 0 1 2 4 5 gt 6 lt 3 22 4 6 5 5 7 6 6 9 8 7 14 5 93 105 114 60 12 3 5 4 2 15 4 16 2 a Distributive Property b Additive Inverse Property a Associative Property of Addition b Multiplicative Identity Property Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 1 5 9 11 15 17 19 21 23 31 37 41 45 49 55 61 67 73 Answers to Odd Numbered Exercises Quizzes and Tests All 19 1068 20 20 8640 9 l u 2 13 8 N 15 4c 10 21 a The sum of the parts of the circle graph is equal to 1 2 6 5 Section 1 4 page 37 17 030L 19 21 n 8 23 3x2 4 Wnts 102 pe Oates Bele 25 A number decreased by 2 e 8 gt ae Jema me La 27 A number increased by 50 Me a e A ae I g a 29 Two decreased by three times a number dl A 13 4x 3 55 a m9 31 The ratio of a number and 2 Commutative Property of Addition 33 Four fifths of a number i a 35 Eight times the difference of a number and 5 Bias Pp 37 The sum of a number and 10 divided by 3 o pn a o ET PERI 39 The square of a number decreased by 3 41 0 25n 43 0 10m 45 5m 10n 8y 33 8x 18y 35 6x 2x 30 274 422 52 8 39 x 3xy y 47 55t 49 Fa 51 0 45y 8x 4x 12 43 189 3y 6 53 0 01251 55 L 0 20L 0 80L T3 2x4 47 5x 2x 57 8 25 0 6
3. j 2 SE 1 12 17 22 Differences 5 5 5 5 5 The differences are constant a 79 aand c Using a specific case may make it easier to see the form of the expression for the general case page 52 a 52 Jo b 4 0 52 V9 3 4 0 52 3 V9 V2 1 2 3 4 5 6 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part 7 17 25 33 41 43 53 61 65 67 69 75 77 79 87 91 97 99 101 Answers to Odd Numbered Exercises Quizzes and Tests a 21 2n 2n 2 4n 2 ile i te 22 Perimeter 21 2 0 6 3 21 144 b E Area I 0 6 0 612 1215 6 5 4 3 2 1 0 1 c 3 ot ooo 1 0 1 2 3 d 2 4 4 3 2 1 0 lt 9 lt 14L14 13 73 15 5 72 19 11 21 230 23 41 8 4o 2 4 2 8 98 4200 35 4 37 14 39 2 SIEI EEE ays a 4a 8 8 45 67 47 1296 49 16 S51 20 55 98 57 1 165 469 01 59 800 Additive Inverse Property 63 Distributive Property Associative Property of Addition Commutative Property of Multiplication Distributive Property 71 u 3v 73 8a 3a 17 17 3 2 W ae i 1 12 7 52 1 251 52 This page contains answers for this chapter only 9x 81 5v 83 5x 10 85 5x y 15a 18b 89 a 0 b 3 a 19 b 8 93 12 2n 95 y 49 The sum of two times a number and 7 The d
4. 12 1 23 4 52 1 52 5 13 12 12 13 a3 6 4 10 8 4 10 8 a 7 3 12 9 3 12 9 i 8 5 10 8 8 5 10 26 27 8 510 8 10 5 10 10 11 HO 15 7 9 7 15 10 2x 10 2 x 28 show a counterexample Skills 15 3 6 5 y Distributive Property 5 6 z Distributive Property 8 y 4 25 x Additive Inverse Property 13x 13x x 8 1 Additive Identity Property 8x 0 Are subtraction and division associative If not In Exercises 21 28 complete the statement using the specified property of real numbers See Example 2 CA 21 Commutative Property of Multiplication Associative Property of Addition Commutative Property of Addition Multiplicative Identity Property In Exercises 29 40 give a the additive inverse and b 12 1 9k 9k the multiplicative inverse of the quantity 13 10x 1 29 10 30 18 10x 31 19 32 37 14 0 4x 4x i 33 5 34 3 15 2x 2x 0 16 4 3B x 4 3 x 35 3 36 5 37 6z z 0 38 2y y 0 17 32 x 3 2 3x 18 3 6 b 3 64 3 5b 19 x 1 1 0 39 x 2 x 2 20 6x 3 6 x 6 3 a yal Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Z 61 a a b b Licensed to iChapters User In Exercises 41 44 rewrite the expression using the Associative
5. Disbursements 0 322t 3 75t 27 6 9 lt t lt 15 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 40 103 104 107 108 109 Chapter 1 F A 50 pens Fundamentals of Algebra 304 20 g E P 9 10 11 12 B 14 5 Year 9 lt 1999 Figure for 103 and 104 Annual total in billions of dollars Graphically approximate the total amount of FFEL disbursements in the year 2000 Then use the model to confirm your estimate algebraically Use the model and a calculator to complete the table showing the total yearly disbursements in bil lions of dollars from 1999 to 2005 Round each amount to the nearest tenth Year 1999 2000 2001 2002 Amount Year 2003 2004 2005 Amount 105 A Geometry The roof shown in the figure is made up of two trapezoids and two triangles Find the total area of the roof For a trapezoid area 5h b b where b and b are the lengths of the bases and h is the height lt 40 ft 106 Exploration a A convex polygon with n sides has n n 3 2 diagonals Verify the formula for a square a pentagon and a hexagon nea b Explain why the formula in part a will always yield a natural number Explaining Concepts Isit possible to evaluate the expression r2 y y 3 when x 5 and y 3 Expl
6. 0 4 In Exercises 73 82 plot the number and its opposite on the real number line Determine the distance of each from 0 73 7 74 4 75 5 76 6 77 3 78 79 gt 80 j 81 4 25 82 3 5 In Exercises 83 90 write the statement using inequality notation 83 x is negative 84 y is more than 25 85 uis at least 16 86 x is nonnegative 87 A bicycle racer s speed s is at least 16 miles per hour and at most 28 miles per hour 88 The tire pressure p is at least 30 pounds per square inch and no more than 35 pounds per square inch 89 The price p is less than 225 90 The average a will exceed 5000 In Exercises 91 94 find two possible values of a 91 Jal 4 92 a 7 93 The distance between a and 3 is 5 94 The distance between a and is 6 Explaining Concepts 95 Every real number is either rational or irrational 96 The distance between a number b and its opposite is equal to the distance between O and twice the number b 97 amp Describe the difference between the rational numbers 0 15 and 0 15 98 amp Is there a difference between saying that a real number is positive and saying that a real number is nonnegative Explain your answer Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 2 Operations with Real Numbers 11 Jupiterlmages Pixland Alamy
7. 5 7 9 veu y niu lt 25 gt 37 35 39 3 47 16 49 3 55 gt 57 gt 63 34 34 65 71 4 7 4 7 45 85 53 7 61 gt 69 2 Jo ll a Bln uiw uw vin uju 83 89 95 97 4 25 eae 4 25 85 u 16 91 4 4 87 16 lt s lt 28 93 2 8 True If a number can be written as the ratio of two integers it is rational If not the number is irrational x lt 0 p lt 225 0 15 is a terminating rational number and 0 15 is a repeating rational number Section 1 2 1 11 19 29 39 47 57 67 77 87 97 105 4 113 121 129 135 137 139 143 145 147 149 151 page 19 45 3 4 5 27 20 13 15 2 22 2 23 4 2 5 2 4 S 31 2 33 60 35 45 95 37 4 9 41 6 4 43 4 2 45 30 48 49 40 51 36 53 5 55 59 61 63 2 65 3 7 6 6 71 3 73 9 75 3 2 79 4 gH 83 7 85 4 89 32 91 16 93 64 95 ot 99 0 027 101 0 32 103 0 107 22 109 6 111 12 27 115 135 117 6 119 6 1 123 30 125 161 127 14 425 171 36 131 133 2533 56 a 7 7 9 25 9 15 24 17 3 8 wie Day Daily gain or loss Tuesday 5 Wednesday 8 S 16 Thursday Friday b The stock gained 24 in value during the week Find the difference between the first bar Monday and the last bar Friday a 10 800 b 15 832 22 c 5032
8. O N gt Nn o g Q lo 2 Q Licensed to iChapters User o BROOKS COLE a CENGAGE Learning Intermediate Algebra Fifth Edition Ron Larson Publisher Charlie Van Wagner Associate Development Editor Laura Localio Assistant Editor Shaun Williams Editorial Assistant Rebecca Dashiell Senior Media Editor Maureen Ross Executive Marketing Manager Joe Rogove Marketing Coordinator Angela Kim Marketing Communications Manager Katherine Malatesta Project Manager Editorial Production Carol Merrigan Art amp Design Manager Jill Haber Senior Manufacturing Coordinator Diane Gibbons Text Designer Jerilyn Bockorick Photo Researcher Sue McDermott Barlow Copy Editor Craig Kirkpatrick Cover Designer Irene Morris Cover Image Frank Schwere Stone Getty Images Compositor Larson Texts Inc TI is a registered trademark of Texas Instruments Inc Chapter Opener and Contents Photo Credits p 1 Janine Wiedel Photolibrary Alamy p 57 Stock Connection Blue Alamy p 125 John Kelly Getty Images p 217 Richard G Bingham II Alamy p 351 Kris Timken Blend Images Jupiterlmages p 368 Antonio Scorza AFP Getty Images p 369 Digital Vision Alamy p 448 Purestock Getty Images p 449 James Marshall The Image Works p 511 Jupiterlmages Banana Stock Alamy p 576 Scott T Baxter Photodisc Punchstock p 577 Ariel Skelley Blend Images Getty Images p 655 Purestock Punchsto
9. 5 4 c 8 3 3042 d 75 2 1 42 Error Analysis In Exercises 154 156 describe and correct the error 154 a a Be Bt 155 12 3 156 3 12 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 3 Properties of Real Numbers 23 1 3 Properties of Real Numbers What You Should Learn 1 gt Identify and use the properties of real numbers 2 gt Develop additional properties of real numbers Basic Properties of Real Numbers The following list summarizes the basic properties of addition and multiplication Although the examples involve real numbers these properties can also be applied Why You Should Learn It l to algebraic expressions Understanding the properties of real numbers will help you to understand and use the properties of algebra z Properties of Real Numbers 1 gt Identify and use the properties of real Let a b and c represent real numbers variables or algebraic expressions numbers Property Example Commutative Property of Addition a b b a 3 t3 e 3 Commutative Property of Multiplication ab ba Bo Tf TED Associative Property of Addition a b c a b 0c 4 2 3 4 2 3 Associative Property of Multiplication ab c a bc o So 7 2e S gt 7 Distributive Property a b c ab ac A Tete eee 3 b c ac 2 53 2 3 5 Study Tip a b c ac bc 5 3 3 sp So 3 alb
10. 7 W ma tad II Mental Math In Exercises 67 72 use the Distributive Property to perform the arithmetic mentally For example you work in an industry in which the wage is 14 per hour with time and a half for overtime So your hourly wage for overtime is 1 14 1 5 14 1 1 14 7 21 67 16 1 75 16 2 4 68 15 13 15 2 3 69 7 62 7 60 2 70 5 51 5 50 1 71 9 6 98 9 7 0 02 72 12 19 95 12 20 0 05 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 30 Chapter 1 Fundamentals of Algebra Solving Problems 73 A Geometry The figure shows two adjoining rectangles Demonstrate the Distributive Property by filling in the blanks to write the total area of the two rectangles in two ways is a 74 A Geometry The figure shows two adjoining rectangles Demonstrate the subtraction version of the Distributive Property by filling in the blanks to write the area of the left rectangle in two ways J lt b gt lt c b c gt 9 A Geometry In Exercises 75 and 76 write the expres sion for the perimeter of the triangle shown in the figure Use the properties of real numbers to simplify the expression 13 4 x 5 3x 2 76 2x 3x 4 2x 4 A Geometry In Exercises 77 and 78 write and simp
11. 9 6 a 2 or x 2 b 2 79 20 2 20 6 47 x 2 6 2 or 2x 12 6 2y 6 5 or 12y 30 81 74 2 x 9 53 37 5 F M 3x 15 57 2x 16 Answers will vary 61 Answers will vary M cD 3 L Original equation x 5 5 3 5 Addition Property of Equality 3 x 5 5 2 Associative Property of Addition x 0 2 Additive Inverse Property x 2 Additive Identity Property 8 13 17 18 Answers to Odd Numbered Exercises Quizzes and Tests 2x 5 6 Original equation Q2x 5 5 6 5 Addition Property of Equality 2x 5 5 11 Associative Property of Addition 2x 0 11 Additive Inverse Property 2x 11 Additive Identity Property 3 2x 3 11 Multiplication Property of Equality Gast Associative Property of Multiplication 1l x 4 Multiplicative Inverse Property 1 x 45 Multiplicative Identity Property 28 69 434 71 62 82 alb c ab ac 4 x 5 3x 2 4x 11 a 2 x 6 2 2x 6x 12 b x 6 2x 2x 12x The additive inverse of a real number is its opposite The sum of a number and its additive inverse is the additive identity zero For example 3 2 3 2 0 Given two real numbers a and b the sum a plus b is the same as the sum b plus a Sample answer 4 O 7 15 18 704 30 407 21 427 304 07 id Chapter Quiz page 31 6 4 5 2 3 3 e aaa 4 2 7 6 5 4
12. Model Labels Rate 12 miles per hour Time hours Expression 12t miles V CHECKPOINT Now try Exercise 47 Using unit analysis you can see that the expression in Example 7 has miles as its unit of measure miles MP oat t hours When translating verbal phrases involving percents be sure you write the percent in decimal form Percent Decimal Form 4 0 04 62 0 62 140 1 40 25 0 25 Remember that when you find a percent of a number you multiply For instance 25 of 78 is given by 0 25 78 19 5 25 of 78 AMPLE 8 Constructing a Mathematical Model A person adds k liters of fluid containing 55 antifreeze to a car radiator Write an algebraic expression that indicates how much antifreeze was added Solution Verbal Beret Number of liters Model antifreeze Labels Percent of antifreeze 0 55 in decimal form Number of liters k liters Expression 0 55k liters Note that the algebraic expression uses the decimal form of 55 That is you compute with 0 55 rather than 55 J CHECKPOINT Now try Exercise 51 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Study Tip You can check that your algebraic expressions are correct for even odd or consecutive integers by substituting an integer for n For instance by letting n 5 you can see that 2n 2 5 10 is an ev
13. Why You Should Learn It Real numbers can be used to represent many real life quantities such as the net profits for Columbia Sportswear Company see Exercise 136 on page 21 1 gt Add subtract multiply and divide real numbers What You Should Learn 1 gt Add subtract multiply and divide real numbers 2 gt Write repeated multiplication in exponential form and evaluate exponential expressions 2 Use order of operations to evaluate expressions 4 gt Evaluate expressions using a calculator and order of operations Operations with Real Numbers There are four basic operations of arithmetic addition subtraction multiplication and division The result of adding two real numbers is the sum of the two numbers and the two real numbers are the terms of the sum The rules for adding real numbers are as follows Addition of Two Real Numbers 1 To add two real numbers with like signs add their absolute values and attach the common sign to the result 2 To add two real numbers with unlike signs subtract the smaller absolute value from the greater absolute value and attach the sign of the number with the greater absolute value Adding Integers a 84 14 84 14 Use negative sign 70 Subtract absolute values b 138 62 138 62 Use common sign 200 Add absolute values a 26 41 0 53 26 41 0 53 Use common sign 26 94 A
14. c ab ac 6 5 3 6 5 6 3 The operations of subtraction and division are not listed at the right be ae be G DA TAE because they do not have many of Additive Identity Property the properties of real numbers For iy D instance subtraction and division GUS e 2 NS 2 are not commutative or associative Multiplicative Identity Property To see this consider the following oleaicasa 5 1 1 5 5 4 3 73 4 Additive Inverse Property 15 5 45415 a a 0 oo 3 0 Multiplicative Inverse Property 1 1 gos az Sos Il a 8 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 24 Chapter 1 Fundamentals of Algebra 4 MPLE 1 Identifying Properties of Real Numbers Identify the property of real numbers illustrated by each statement a 44a 3 4 at4 3 b 6 7 1 ce 3 2 5b 34 2 5 d b 8 0 b 8 Solution a This statement illustrates the Distributive Property b This statement illustrates the Multiplicative Inverse Property c This statement illustrates the Associative Property of Addition d This statement illustrates the Additive Identity Property where b 8 is an algebraic expression iv CHECKPOINT Now try Exercise 5 The properties of real numbers make up the third component of what is called a mathematical system These three components are a set of numbers Section 1 1 op
15. on the real number line b The point representing the real number 2 3 lies between 2 and 3 but closer to 2 on the real number line c The point representing the real number 2 2 25 lies between 2 and 3 but closer to 2 on the real number line Note that the point representing 2 lies slightly to the left of the point representing 2 3 The point representing the real number 0 3 lies between 1 and 0 but closer to 0 on the real number line v CHECKPOINT Now try Exercise 13 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User oo 4 3 2 i 0 Figure 1 6 t t Figure 1 7 a 0 Figure 1 8 a at LF t eo gt 1 0 Figure 1 9 Section 1 1 The Real Number System 5 The real number line provides a way of comparing any two real numbers For instance if you choose any two different numbers on the real number line one of the numbers must be to the left of the other You can describe this by saying that the number to the left is less than the number to the right or that the number to the right is greater than the number to the left as shown in Figure 1 5 i po a b a lt b Figure 1 5 ais to the left of b Order on the Real Number Line If the real number a lies to the left of the real number b on the real number line then a is less than b which is written as a lt b This relationshi
16. x 2xy 2x7 xy y 3a 5ab 9a 4ab a Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 38 Chapter 1 Fundamentals of Algebra In Exercises 41 52 use the Distributive Property to simplify the expression 41 4 2x x 3 42 43 3 6y y 2 44 45 3x 2x 4 46 47 x 5x 2 48 49 3x 17 4x 50 51 5x 7 21 52 In Exercises 53 72 simplify Examples 4 and 5 Z 53 10x 3 2x 5 54 55 x 5x 9 56 57 5a 4a 3 58 59 3 3y 1 2 y 5 60 5 a 6 4 2a 1 61 3 y 2 yy 3 62 x x 5 4 4 x 63 x x 3 3 x 4 64 5 x 1 x 2x 6 amp 65 9a 7 5 7a 3 66 12b 9 7 5b 6 67 3 2x 4 x 8 68 4 5 3 x2 10 69 8x 3x 10 4 3 x 70 5y y 9 6 y 2 71 2 3 b 5 b2 b 3 72 5 3 z 2 z2 z 2 8 z3 4z 2 SSS 2y 1 5f 8t 10 y y 10 Sy 2y 1 6x 9x 4 the expression See 3 x 1 x 6 y By 1 7x 2x 5 In Exercises 73 90 evaluate the expression for the specified values of the variable the reason See Examples 6 8 s If not possible state Expression Values 73 5 3x a x 3 b x 5 74 75 76 77 78 79 80 81 82 83
17. 10 9 4 Evaluate powers and multiply within symbols of grouping 40 5 8 Subtract within symbols of grouping then divide CHECKPOINT Nov try Exercise 109 a Polite pressions tines Calculators and Order of Operations calculator and order of operations When using your own calculator be sure that you are familiar with the use of each of the keys Two possible keystroke sequences are given in Example 14 one for a standard scientific calculator and one for a graphing calculator Technology Tip MPLE 14 Evaluating Expressions on a Calculator Be sure you see the difference between the change sign key a To evaluate the expression 7 5 3 use the following keystrokes and the subtraction key ona Keystrokes Display scientific calculator Also notice the 710050306 8 Selenis difference between the negation key and the subtraction key 7 O 5 amp 30 ENTER 8 Graphing on a graphing calculator b To evaluate the expression 3 4 use the following keystrokes Keystrokes Display 7 x2 isan Technology Discovery 3 O46 i a 30 amp amp 4 ENTER 13 Graphing To discover if your calculator performs the established order of c To evaluate the expression 5 4 3 2 use the following keystrokes operations evaluate 7 5 Keystrokes Display Jta j 3 2 4 exadly as it appears 5001030200 0 5 Seieniitie If your calculator performs the establis
18. 143 Find the volume of a bale of hay in cubic feet if 1728 cubic inches equals cubic foot 144 Approximate the number of bales in a ton of hay Then approximate the volume of a stack of baled hay in cubic feet that weighs 12 tons 2000 lb 1 ton Explaining Concepts True or False n Exercises 145 149 determine whether the statement is true or false Justify your answer 145 The reciprocal of every nonzero rational number is a rational number 146 The product of two fractions is the product of the numerators over the LCD 147 If a negative real number is raised to the 12th power the result will be positive 148 If a negative real number is raised to the 11th power the result will be positive 149 a b b a 150 Are the expressions 27 and 22 equal Explain 151 amp In your own words describe the rules for determining the sign of the product or the quotient of two real numbers 152 In your own words describe the established order of operations for addition and subtraction Without these priorities explain why the expression 6 5 2 would be ambiguous 153 amp Decide which expressions are equal to 27 when you follow the standard order of operations For the expressions that are not equal to 27 see if you can discover a way to insert symbols of grouping that make the expression equal to 27 Discuss the value of symbols of grouping in mathematical communication a 40 10 3 b 52
19. 22 15 square meters 6 125 cubic feet 141 20 square inches a True A nonzero rational number is of the form pP where a and b are integers and a 0 b 0 The reciprocal will b La be which is also rational a True When a negative number is raised to an even power the result is positive False 6 3 2 5 376 If the numbers have like signs the product or quotient is positive If the numbers have unlike signs the product or quotient is negative Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User A10 153 a 40 10 3 27 b 5 3 4 27 c 8 3 30 2 27 d 75 2 1 2 27 65 155 Only common factors not terms of the numerator and denominator can be divided out Section 1 3 page 28 1 3 5 7 9 11 13 15 19 21 25 29 33 37 39 41 45 49 51 55 59 63 Additive Inverse Property Multiplicative Inverse Property Commutative Property of Addition Associative Property of Addition Distributive Property Associative Property of Multiplication Multiplicative Inverse Property Additive Inverse Property 17 Distributive Property Additive Inverse Property 3 15 23 5 6 5 z x 25 27 8S a 10 6 G31 a 19 b 5 67 i 2 35 a b 8 73 1 a 6z b a 75 77 1 32 4 y 43
20. 3 5 56 75 3 23 3 1 245 2 57 3 6 58 36 25 3 3 26 3 59 3 37 3 60 3 3 8 Geass 28 ij 3 5 61 3 2 62 2 3 12 29 34 43 30 53 73 31 102 6 32 85 43 In Exercises 63 68 find the reciprocal 33 85 25 34 36 8 63 6 64 4 35 11 325 34 625 65 3 66 2 36 16 25 54 78 67 2 68 2 37 6 st 38 152 12 In Exercises 69 82 evaluate the expression See In Exercises 39 44 write the expression as a multiplica Example 10 tion problem P 69 70 n 39 9 9 9 9 gt 15 71 48 16 2 72 P Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 20 Chapter 1 Fundamentals of Algebra 73 63 7 74 27 5 105 24 5 22 106 18 32 12 75 5 76 p A 107 28 4 3 5 108 6 7 6 4 m 1 45 m e Z 109 14 2 8 4 110 21 5 7 5 2 e r n gi 4l 4 82 262 B 105 113 5 2 9 18 8 114 8 3 4 12 3 In Exercises 83 88 write the expression using 115 5 14 4 116 2 25 7 exponential notation 6 8 3 9 6 2 83 7 7 7 117 Fa 118 334 84 CAAA ae a 1 1 1 z 85 3 4 4 5 119 ii 7 120 5 27 86 3 3 g G ig L w 2D 87 7 7 7 3243 52 8 2 88 5
21. 5x 4 2x 18 Original equation shown at the right are less formal 5x 4 4 2x 18 4 Subtract 4 from each side than those shown in Examples 5 and 6 on page 27 The importance of the properties is that they can be 5x 2x 2x 2x 14 Add 2x to each side used to justify the steps of a solution 5x 2x 14 Simplify They do not always need to be ao Simplify listed for every step of the solution Ix 14 S Divide each side by 7 J 7 x 2 Simplify Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 26 Chapter 1 Fundamentals of Algebra Each of the additional properties in the list on the preceding page can be proved by using the basic properties of real numbers MPLE 3 Proof of the Cancellation Property of Addition Prove thatifa c b c thena b Use the Addition Property of Equality Solution Notice how each step is justified from the preceding step by means of a property of real numbers at c bt e Write original equation a c c b c oc Addition Property of Equality a e t c b e o Associative Property of Addition a 0 b 0 Additive Inverse Property a b Additive Identity Property wv CHECKPOINT Nov try Exercise 59 AMPLE 4 Proof of a Property of Negation Prove that 1 a a You may use any of the properties of equality and prop erties of zero Sol
22. 84 amp 85 86 87 88 89 90 Expression 3 ax 2 10 4x 3y 10 yy yt5 2x7 5x 3 3x 2y 6x Sy x xy y y xy x yY wees x xy ly xl x y Distance traveled rt Simple interest Prt Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Values a x 6 b x 3 a x 1 b x 3 a y 2 b y 3 a y 2 b y 2 a x 2 b x 3 a x 0 b x 3 a x 0 b x 6 x 1 y b x 6 y 9 a x 2 y 3 b x 1 a x 2 y l1 b x 3 y 2 a x 5 y 2 b x 3 y 3 a x 4 y 2 b x 3 y 3 a x 0 y 10 b x 4 a x 2 y 5 b x 2 y 2 a x 0 y 2 b x 3 y 15 a r 40 t 54 b r 35 t 4 a P 7000 r 0 065 y 4 t 10 b P 4200 r 0 07 t 9 Licensed to iChapters User Section 1 4 Algebraic Expressions 39 Solving Problems A Geometry In Exercises 91 94 find the volume of the rectangular solid by evaluating the expression wh for the dimensions given in the figure 91 92 7 ft 4 in 6 ft 5 in 6 ft 4 in 93 94 18 in 18 cm 27 in 42 cm In Exercises 95 98 evaluate the expression 0 01p 0 05n 0 10d 0 25q to find the value of the given number of pennies p nickels n dimes d and quarters q 95 11 pennies 7 nickels 3 quarters 96 8 pennies 13 nickels 6 dimes 97 43 penn
23. 9 13 3 6 8 6 4 83 24 24 24 24 Note that an additional step is MPLE 6 Adding and Subtracting Fractions needed to simplify the fraction after the numerators have been added 5 9 5 9 i a 17 17 17 Add numerators Simplift 7 implify b 8 12 8 3 12 2 Least common denominator is 24 eee Simpli 24 24 PRE cai Subtract ti 24 ubtract numerators Simplif 24 implify V CHECKPOINT Now try Exercise 21 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 14 Chapter 1 Fundamentals of Algebra Study Tip MPLE 7 Adding Mixed Numbers A quick way to convert the mixed Find the sum of 12 and 1 number 14 into the fraction 2 is to multiply the whole number by the Solution denominator of the fraction and add it ue 11 _9 n 11 ATE MT the result to the numerator as 5 7 5S7 REA follows T i 11 5 e ene east Common denominator 1S 7 4 160 t4_9 5 7 75 5 5 5 63 55 E 35 35 Simplify 2 63 55 118 idi icsi numerators and simputy 35 35 CHECKPOINT Now try Exercise 29 Multiplication of two real numbers can be described as repeated addition For instance 7x3 can be described as 3 3 3 3 3 3 3 Multiplication is denoted in a variety of ways For instance 7 x 3 7 3 7 3 and 7 3 all denote the product 7 times 3 The result of multiplying two real numbers is their product a
24. 92 evaluate the algebraic expression for the specified values of the variable s If not possi ble state the reason Expression Values 89 x 2x 3 a x 3 b x 0 ba ETT a x 0 y 3 b x 5 y 2 91 y 2y 4x a x 4 y 1 b x 2 y 2 Values 6 y 3 4 y 5 Expression 92 2x a7 2y a x b x 1 5 Constructing Algebraic Expressions 1 gt Translate verbal phrases into algebraic expressions and vice versa In Exercises 93 96 translate the verbal phrase into an algebraic expression 93 Twelve decreased by twice the number n 94 One hundred increased by the product of 15 and a number x 95 The sum of the square of a number y and 49 96 Three times the absolute value of the difference of a number n and 3 all divided by 5 In Exercises 97 100 write a verbal description of the algebraic expression without using the variable 97 2y 7 98 5u 3 x 5 4 100 4 a 1 99 2 Construct algebraic expressions with hidden products In Exercises 101 104 write an algebraic expression that represents the quantity in the verbal statement and simplify if possible 101 The amount of income tax on a taxable income of J dollars when the tax rate is 18 102 The distance traveled when you travel 8 hours at the average speed of r miles per hour 103 The area of a rectangle whose length is units and whose width is 5 units less than the length 104 The sum of t
25. Focused During class When you sit down at your desk get all other issues out of your mind by reviewing your notes from the last class and focusing just on math e Repeat in your mind what you are writing in your notes e When the math is particularly difficult ask your instructor for another example e Before doing homework review the concept completing boxes and examples Talk through the examples homework out loud Complete homework as though you were also preparing for a quiz Memorize the different types of problems formulas rules and so on Between Review the concept boxes and check your memory using the checkpoint exercises Concept Check exercises and the What Did You Learn section Preparing Review all your notes that pertain to the for a test upcoming test Review examples of each type of problem that could appear on the test Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Answers to Odd Numbered Exercises Quizzes and Tests A9 Answers to Odd Numbered Exercises Quizzes and Tests CHAPTER 1 Section 1 1 1 5 0 2 13 15 23 50 35 8 in 225 3 page 9 a 1 2 6 b 6 0 1 2 6 c 6 4 0 3 1 2 6 d V6 V2 a V4 4 0 c 4 2 V4 4 0 4 5 5 5 543 a V11 7 2 72 5 4 3 2 1 0 1 2 3 1 a b
26. Property of Addition or the Associative Property of Multiplication 41 32 4 y 42 15 3 x 43 9 6m 44 11 4n In Exercises 45 50 rewrite the expression using the Distributive Property 45 20 2 5 46 3 4 8 47 x 6 2 48 z 10 12 49 6 2y 5 50 4 10 b In Exercises 51 54 use the Distributive Property to simplify the expression 51 7x 2x 52 8x 6x 7x 5x 3x 2 a 29 8 8 at 5 5 In Exercises 55 58 the right side of the statement does not equal the left side Change the right side so that it does equal the left side 55 3 x 5 3x4 5 56 4 x 2 4x 2 57 2 x 8 27 16 58 9 x 4 9x 36 In Exercises 59 62 use the basic properties of real numbers to prove the statement See Examples 3 and 4 59 If ac be and c 0 then a b 60 Ifa c b c thena b 62 a a 0 In Exercises 63 66 identify the property of real numbers that justifies each step See Examples 5 and 6 X 65 X 63 x 5 3 x 5 5 3 5 x 5 5 2 x 0 2 x 2 Section 1 3 Properties of Real Numbers 29 64 x 8 20 x 8 8 20 8 x 8 8 28 x 0 28 x 28 2x 5 6 2x 5 5 6 5 2x 5 5 11 2x 0 11 2x 11 32x 3 11 NI N II x ll VIZ VIZ n 66 3x 4 10 3x 4 4 10 4 3x 4 4 6 3x 0 6 3x 3 3x lon gt 6 w
27. dollars per quarter Number of coins x quarters Expression 0 25x dollars wv CHECKPOINT Now try Exercise 41 AMPLE 6 A cash register contains n nickels and d dimes Write an algebraic expression for this amount of money in cents Constructing a Mathematical Model Solution Verbal Value Number Value Number Model of nickel of nickels of dime of dimes Labels Value of nickel 5 cents per nickel Number of nickels n nickels Value of dime 10 cents per dime Number of dimes d dimes Expression 5n 10d cents V CHECKPOINT Now try Exercise 45 In Example 6 the final expression 5n 10d is measured in cents This makes sense as described below 5 cents niekel eee 10 cents z d dimes Note that the nickels and dimes divide out leaving cents as the unit of measure for each term This technique is called unit analysis and it can be very helpful in determining the final unit of measure Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 44 Chapter 1 Fundamentals of Algebra MPLE 7 Constructing a Mathematical Model A person riding a bicycle travels at a constant rate of 12 miles per hour Write an algebraic expression showing how far the person can ride in t hours Solution For this problem use the formula Distance Rate Time Verbal Rate Time
28. even or odd Explain Developing Skills In Exercises 1 24 translate the verbal phrase into an algebraic expression See Examples 1 3 GY 1 2 O lN DANAU 10 11 12 13 14 15 16 17 18 X 19 20 21 The sum of 23 and a number n Twelve more than a number n The sum of 12 and twice a number n The total of 25 and three times a number n Six less than a number n Fifteen decreased by three times a number n Four times a number n minus 10 The product of a number y and 10 is decreased by 35 Half of a number n Seven fifths of a number n The quotient of a number x and 6 The ratio of y and 3 Eight times the ratio of N and 5 Fifteen times the ratio of x and 32 The number c is quadrupled and the product is increased by 10 The number u is tripled and the product is increased by 250 Thirty percent of the list price L Twenty five percent of the bill B The sum of a number and 5 divided by 10 The sum of 7 and twice a number x all divided by 8 The absolute value of the difference between a num ber and 8 22 23 24 The absolute value of the quotient of a number and 4 The product of 3 and the square of a number is decreased by 4 The sum of 10 and one fourth the square of a number In Exercises 25 40 write a verbal description of the algebraic expression without using the variable See Example 4 oY 25 26 27 28 29 30 31 32 33 34
29. of a 2 a a 3 a a 0 Determine the absolute value of a real number as a lt b This relationship can also be described by saying j ms The double negative of a is a that b is greater than a and writing b gt a a and a are additive inverses Use the real number line to find the distance between two real numbers If a is areal number then the absolute value of a is a ifa 20 lal ifa lt 0 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part If a and b are two real numbers such that a lt b then the distance between a and b is given by b a Licensed to iChapters User 1 2 Operations with Real Numbers Assignment What Did You Learn 51 Due date Perform operations on real numbers 1 To add with like signs Add the absolute values and attach the common sign 2 To add with unlike signs Subtract the smaller absolute value from the greater absolute value and attach the sign of the number with the greater absolute value 3 To subtract b from a Add b to a 4 To multiply two real numbers With like signs the product is positive With unlike signs the product is negative The product of zero and any other real number is zero 5 To divide a by b Multiply a by the reciprocal of b 6 To add fractions Write the fractions so that they have the same denominator Add the nu
30. 0q 59 n 5n 6n 122 51x 51 107 35t 53 12x 35 61 On 1 4 Qn 3 On4 5 o 49 4y 9 bP ea 59 Ty 7 6 6323 12 65 44a 22 3 merta Sarea MRN 6x 96 69 12x 2x 71 2b 4b 36 67 b 0 75b 0 375b a 3 b 10 75 a 6 b 9 69 Perimeter 2 2w 2w 6w Area 2w w 2w a 7 b Il 71 Perimeter 6 2x 3 x 3 x 4x 12 77 79 81 85 87 91 97 101 103 107 109 111 113 a Not possible undefined b z a 13 b 36 83 a 7 b 7 a Not possible undefined b 5 a 3 b 0 89 a 210 b 140 252 ft 93 3888 in 95 1 21 7 23 99 4b b 90 23 500 million 23 775 million 22 billion 22 3 billion 105 1440 square feet No When y 3 the expression is undefined To remove a set of parentheses preceded by a minus sign distribute 1 to each term inside the parentheses For example 13 10 5 13 10 5 18 A factor can consist of a sum of terms the term x is part of the sum x y which is a factor of x y z No There are an infinite number of values of x and y that would satisfy 8y 5x 14 For example x 10 and y 8 would be a solution and so would x 2 and y 3 Section 1 5 page 47 1 23 n 3 12 2n Sn 6 7 4n 10 73 75 77 81 Review Exercises 1 3 Area 3x 6x 9x or 6 2x 3 x 9x b b 50 b 50b square meters n 0 1 2 3 4 5
31. 32 Fundamentals of Algebra William Thomas Cain Getty Images n d Why You Should Learn It Many real life quantities can be determined by evaluating algebraic expressions For instance in Example 9 on page 36 you will evaluate an algebraic expression to find yearly revenues of gambling industries 1 gt Identify the terms and coefficients of algebraic expressions Study Tip It is important to understand the difference between a term and a factor Terms are separated by addition whereas factors are separated by multiplication For instance the expression 4x x 2 has three factors 4 x and x 2 What You Should Learn 1 gt Identify the terms and coefficients of algebraic expressions 2 gt Simplify algebraic expressions by combining like terms and removing symbols of grouping Evaluate algebraic expressions by substituting values for the variables Algebraic Expressions One of the basic characteristics of algebra is the use of letters or combinations of letters to represent numbers The letters used to represent the numbers are called variables and combinations of letters and numbers are called algebraic expressions Here are some examples x 2 3x 2x 3y 2x3 y L Algebraic Expression A collection of letters called variables and real numbers called constants combined using the operations of addition subtraction multiplication or division is called an al
32. 35 36 37 38 A 2 D y 50 2y 3 2 3x Ty 4 HG MNu GOI NIN 4 Nn Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 48 39 40 Chapter 1 Fundamentals of Algebra ya xw 2 In Exercises 41 64 write an algebraic expression that represents the specified quantity in the verbal statement and simplify if possible See Examples 5 10 oY 41 42 43 44 45 46 o 47 48 49 50 The amount of money in dollars represented by n quarters The amount of money in dollars represented by x nickels The amount of money in dollars represented by m dimes The amount of money in dollars represented by y pennies The amount of money in cents represented by m nickels and n dimes The amount of money in cents represented by m dimes and n quarters The distance traveled in t hours at an average speed of 55 miles per hour The distance traveled in 5 hours at an average speed of r miles per hour The time required to travel 320 miles at an average speed of r miles per hour The average rate of speed when traveling 320 miles in hours The amount of antifreeze in a cooling system con taining y gallons of coolant that is 45 antifreeze 52 53 54 55 56 amp 57 58 X 59 60 61 62 63 64 The amount of water in g quarts o
33. 4 interest compounded monthly the total amount in the account after 18 years will be 12 et 0 04 216 sof 1 204 iff 0 04 Use a calculator to determine this amount How much of the amount in part b is earnings from interest 138 Savings Plan A Geometr the figure a b c You save 60 per month for 30 years How much money has been set aside during the 30 years If the money in part a is deposited in a savings account earning 3 interest compounded monthly the total amount in the account after 30 years will be 12 ae 0 03 360 af x 283 Ja Use a calculator to determine this amount How much of the amount in part b is earnings from interest In Exercises 139 142 find the area of The area A of a rectangle is given by A length width and the area A of a triangle is given by A i 139 base height 140 lt 14 cm 142 m 10 ft gt Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 22 Chapter 1 Fundamentals of Algebra Volume In Exercises 143 and 144 use the following information A bale of hay is a rectangular solid weighing approximately 50 pounds It has a length of 42 inches a width of 18 inches and a height of 14 inches The volume V of a rectangular solid is given by V length width height
34. 5 22 5 4 1 V CHECKPOINT Now try Exercise 73 MPLE 7 Evaluating Algebraic Expressions Evaluate each algebraic expression when x 2 and y 1 a y xl b x 2xy y Solution a When x 2 and y 1 the expression y x has a value of 1 3 3 b When x 2 and y 1 the expression x 2xy y has a value of 22 2 2 1 1 2 44 441 9 V CHECKPOINT Now try Exercise 81 Evaluating an Algebraic Expression Evaluate a when x 4 andy 3 X Solution When x 4 and y 3 the expression 2xy x 1 has a value of 2 4 3 24 4 1 3 8 wo CHECKPOINT Now try Exercise 85 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 36 Chapter 1 Fundamentals of Algebra AMPLE 9 Using a Mathematical Model Q The yearly revenues in billions of dollars for gambling industries in the United States for the years 1999 to 2005 can be modeled by Revenue 0 157 1 19 11 0 9 lt t lt 15 where f represents the year with t 9 corresponding to 1999 Create a table that shows the revenue for each of these years Source 2005 Service Annual Survey Solution To create a table of values that shows the revenues in billions of dollars for the gambling industries for the years 1999 to 2005 evaluate the expr
35. 5 5 5 5 5 343 6 2 123 124 A In Exercises 89 102 evaluate the exponential expres i 4 sion See Example 11 8 9 89 25 90 53 amp 91 2 4 92 3 3 93 43 94 64 95 5 96 5 97 98 3 99 0 3 3 100 0 2 4 101 5 0 4 3 102 3 0 8 amp 125 5 6 13 2 5 6 3 In Exercises 125 130 evaluate the expression using a calculator Round your answer to two decimal places See Example 14 126 6 9 6 1 4 2 16 127 5 3 400 128 300 1 09 156 24 In Exercises 103 124 evaluate Examples 12 and 13 103 16 6 10 the expression See 104 18 12 4 500 129 7 955 20 Solving Problems Circle Graphs In Exercises 131 and 132 find the unknown fractional part of the circle graph 133 Account Balance During one month you made the following transactions in your non interest bearing checking account Find the balance at the end of the month 130 5 100 3 64 4 1 BALANCE NUMERO oare maana Ramer y leel ancunr 8261868 3 1 Pay 1236 45 2154 3 3 Magazine 25 62 2155 3 6 Insurance 455 00 3 12 Withdrawal 125 00 2156 3 15 Mortgage 715 95 Figure for 133 134 Profit The midyear financial statement of a cloth ing company showed a profit of 1 345 298 55 At the close of the year the financial statement showed a profit for the yea
36. 5 Add two numbers at a time 5 Add V CHECKPOINT Now try Exercise 19 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 2 Operations with Real Numbers 13 To add or subtract fractions it is useful to recognize the equivalent forms of fractions as illustrated below a a a a All o are positive b b b b p a a a a All i are negative b b gt b b Study Tip Addition and Subtraction of Fractions Like Denominators The sum and difference of two fractions with like denominators c 0 are Here is an alternative method for adding and subtracting fractions with unlike denominators b 0 andd 0 a b_atb E C C ad bc bd ad bc Unlike Denominators To add or subtract two fractions with unlike denominators first rewrite the fractions so that they have the same denominator and then apply the first rule To find the least common denominator LCD for two or more fractions find the least common multiple LCM of their denominators For instance the LCM of 6 and 8 is 24 To see this consider all multiples of 6 6 12 18 24 30 36 42 48 and all multiples of 8 8 16 24 32 40 48 The numbers 24 and 48 are common multiples and the number 24 is the smallest of the common multiples To add and 2 proceed as follows 26 48 1 3 14 33 4 9 4
37. ANNUAL PICNIC TOMORROW tw ft lt 6w ft gt Explaining Concepts 79 Which are equivalent to 4x 81 amp When a statement is translated into an algebraic expression explain why it may be helpful to use a a x multiplied by 4 specific case before writing the expression b x increased by 4 c the product of x and 4 d the ratio of 4 and x 82 amp When each phrase is translated into an algebraic 80 amp Ifnis an integer how are the integers 2n 1 expression is order important Explain and 2n 1 related Explain a y multiplied by 5 b 5 decreased by y c y divided by 5 d the sum of 5 and y Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 50 Chapter 1 Fundamentals of Algebra Use these two pages to help prepare for a test on this chapter Check off the key terms and key concepts you know You can also use this section to record your assignments Plan for Test Success Date of test Things to review LI Key Terms p 50 LI Key Concepts pp 50 51 L Your class notes C Your assignments Key Terms L set p 2 C subset p 2 C real numbers p 2 _ natural numbers p 2 positive integers p 2 L whole numbers p 2 L negative integers p 2 L integers p 2 L fractions p 3 _ rational numbers p 3 L irrational numbers p 3 L real number line p 4
38. C origin p 4 _ nonnegative real number p 4 C inequality symbols p 5 Key Concepts 1 1 The Real Number System Assignment Study dates and times JaM eM Ei Jame C Study Tips pp 2 6 7 13 14 15 17 23 25 32 33 34 42 43 45 L Technology Tips pp 3 18 36 C Mid Chapter Quiz p 31 L opposites p 7 additive inverses p 7 C absolute value p 7 C sum p 11 L difference p 12 _ least common denominator p 13 _ least common multiple p 13 L product p 14 LI factor p 14 L reciprocal p 15 L quotient p 15 LI dividend p 15 C divisor p 15 L numerator p 15 C denominator p 15 L Review Exercises pp 52 54 L Chapter Test p 55 L Video Explanations Online CI Tutorial Online L exponential form p 16 _ base p 16 L exponent p 16 L order of operations p 16 _ variables p 32 _ algebraic expressions p 32 _ variable terms p 32 L constant term p 32 LI coefficient p 32 L like terms p 33 L simplify p 33 L evaluate p 35 L consecutive integers p 45 Due date If the real number a lies to the left of the real number b on the real number line then a is less than b which is written Use the real number line to order real numbers Use properties of opposites and additive inverses Let a be a real number 1 a is the opposite
39. Exercise 59 Sometimes an expression may be written directly from a diagram using a common geometric formula as shown in the next example Constructing a Mathematical Model Write expressions for the perimeter and area of the rectangle shown in Figure 1 16 2w in m w 12 in gt Figure 1 16 Solution For the perimeter of the rectangle use the formula Perimeter 2 Length 2 Width Verbal Length Width Mea ea 2 Labels Length w 12 inches Width 2w inches Expression 2 w 12 2 2w 2w 24 4w 6w 24 inches For the area of the rectangle use the formula Area Length Width Verbal Length Width Model Labels Length w 12 inches Width 2w inches Expression w 12 2w 2w 24w square inches v CHECKPOINT Now try Exercise 69 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 5 Constructing Algebraic Expressions Concept Check The phrase reduced by implies what operation The word ratio indicates what operation A car travels at a constant rate of 45 miles per hour for t hours The algebraic expression for the distance traveled is 45t What is the unit of measure of the algebraic expression Go to pages 50 51 to record your assignments 4 Let n represent an integer Is the expression n n 1
40. ain Is it possible to evaluate the expression 3x 5y 18z when x 10 and y 8 Explain amp State the procedure for simplifying an algebraic expression by removing a set of parentheses preceded by a minus sign such as the parentheses in a b c Then give an example 110 amp How can a factor be part of a term in an algebraic expression Explain and give an example 111 amp How can an algebraic term be part of a factor in an algebraic expression Give an example 112 amp You know that the expression 180 10x has a value of 100 Is it possible to determine the value of x with this information Explain and find the value if possible 113 amp You know that the expression 8y 5x has a value of 14 Is it possible to determine the values of x and y with this information Explain and find the values if possible Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 5 Constructing Algebraic Expressions 41 Expressions What You Should Learn 1 gt Translate verbal phrases into algebraic expressions and vice versa 2 gt Construct algebraic expressions with hidden products Translating Phrases Clive Brunskill Getty Images In this section you will study ways to construct algebraic expressions When you Why You Should Learn It translate a verbal sentence or phrase into an algebraic ex
41. ate algebraic expressions To evaluate an algebraic expression substitute numerical values for each of the variables in the expression and simplify Due date Assignment Translate verbal phrases into algebraic expressions and vice versa Addition sum plus greater than increased by more than exceeds total of Subtraction difference minus less than decreased by subtracted from reduced by the remainder Multiplication product multiplied by twice times percent of Division quotient divided by ratio per Write labels for integers The following expressions are useful ways to denote integers 1 2n denotes an even integer for n 1 2 3 2 2n 1 and 2n 1 denote odd integers for n 1 2 3era 3 n n 1 n 2 denotes a set of consecutive integers Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 52 Chapter 1 Fundamentals of Algebra 1 1 The Real Number System 1 gt Understand the set of real numbers and the subsets of real numbers In Exercises 1 and 2 which of the real numbers in the set are a natural numbers b integers c rational numbers and d irrational numbers 1 3 4 0 V2 52 4 Vo 2 98 141 2 3 99 12 34 In Exercises 3 and 4 list all members of the set 3 The natural numbers between 2 3 and 6 1 4 T
42. bers 1 4 5 6 2 2 3 In Exercises 3 and 4 find the distance between the two real numbers 3 15 and 7 4 8 75 and 2 25 In Exercises 5 and 6 evaluate the expression 5 7 6 6 9 8 In Exercises 7 16 evaluate the expression Write fractions in simplest form 7 32 18 8 12 17 93244 10 5 4 11 3 2 10 12 33 13 3 14 3 18 2 3 4 62 12 2 10 15 3 27 25 5 16 In Exercises 17 and 18 identify the property of real numbers illustrated by each statement 17 a Bu 5 8 u 8 5 b 10x 10x 0 18 a 7 y z 7 y z b 2x 1 2x 19 During one month you made the following transactions in your non interest bearing checking account Find the balance at the end of the month BALANCE NUMBER OR PAYMENT CODE DATE TRANSACTION DESCRIPTION cure ae ee ON 1406 98 2103 1 5 Car Payment 375 03 2104 1 7 Phone 59 20 1 8 Withdrawal 225 OO 12 Deposit 320 45 20 You deposit 45 in a retirement account twice each month How much will you deposit in the account in 8 years 21 Determine the unknown fractional part of the circle graph at the left Explain how you were able to make this determination Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Chapter 1
43. cause 2 2 and 2 is greater than 1 a b 4 4 because 4 4 and 4 4 c 12 lt 15 because 12 12 15 15 and 12 is less than 15 d 3 gt 3 because 3 3 and 3 is greater than 3 e 2 gt 2 because 2 2 and 2 is greater than 2 f 3 3 because 3 3 and 3 is equal to 3 V CHECKPOINT Now try Exercise 55 When the distance between the two real numbers a and b was defined as b a the definition included the restriction a lt b Using absolute value you can generalize this definition That is if a and b are any two real numbers then the distance between a and b is given by Distance between a and b b al a b For instance the distance between 2 and 1 is given by 2 1 3 3 Distance between 2 and 1 Go to page xxii for ways to Create a Positive Study Environment You could also find the distance between 2 and 1 as follows 1 2 3 3 Distance between 2 and 1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 1 The Real Number System 9 Concept Check 1 Two real numbers are plotted on the real number 3 Is the number 7 a rational number Explain why or line How can you tell which number is greater why not 4 The distance betwee
44. ck p 706 SuperStock Alamy p 707 UpperCut Images Alamy Printed in the United States of America 1234567 12 11 10 09 08 2010 Brooks Cole Cengage Learning 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 For product information and technology assistance contact us at Cengage Learning Customer amp Sales Support 1 800 354 9706 For permission to use material from this text or product submit all requests online at www cengage com permissions Further permissions questions can be e mailed to permissionrequest cengage com Library of Congress Control Number 2008931148 Student Edition ISBN 13 978 0 547 10217 7 ISBN 10 0 547 10217 8 Annotated Instructor s Edition ISBN 13 978 0 547 10220 7 ISBN 10 0 547 10220 8 Brooks Cole 10 Davis Drive Belmont CA 94002 3098 USA 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 yo
45. dd absolute values b 3 2 0 4 3 2 0 4 Use positive sign 2 8 Subtract absolute values V CHECKPOINT Now try Exercise 9 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 12 Chapter 1 Fundamentals of Algebra The result of subtracting two real numbers is the difference of the two num bers Subtraction of two real numbers is defined in terms of addition as follows Subtraction of Two Real Numbers To subtract the real number b from the real number a add the opposite of b toa That is a b a D AMPLE 3 Subtracting Integers Find each difference a 9 21 D 15 Solution a 9 21 9 21 Add opposite of 21 E 21 9 2 Use negative sign and subtract absolute values b 15 8 15 8 Add opposite of 8 15 4 8 23 Use common sign and add absolute values Find each difference a 2 5 2 7 b 7 02 13 8 Solution a 2 5 2 7 2 5 2 7 Add opposite of 2 7 _ _ Use positive sign and 2 7 2 5 0 2 subtract absolute values b 7 02 13 8 7 02 13 8 Add opposite of 13 8 Use common sign and add absolute values 7 02 13 8 20 82 Evaluate 13 7 11 4 Solution 13 7 11 4 13 7 114 4 Add opposites 20 1
46. en integer 2n 1 2 5 1 9 is an odd integer and 2n 1 2 5 1 11 is an odd integer Section 1 5 Constructing Algebraic Expressions 45 When assigning labels to two unknown quantities hidden operations are often involved For example two numbers add up to 18 and one of the numbers is assigned the variable x What expression can you use to represent the second number Let s try a specific case first then apply it to a general case Specific Case If the first number is 7 the second number is 18 7 11 General Case If the first number is x the second number is 18 x The strategy of using a specific case to help determine the general case is often helpful in applications Observe the use of this strategy in the next example Using Specific Cases to Model General Cases a A person s weekly salary is d dollars Write an expression for the person s annual salary b A person s annual salary is y dollars Write an expression for the person s monthly salary Solution a Specific Case If the weekly salary is 300 the annual salary is 52 300 dollars General Case If the weekly salary is d dollars the annual salary is 52 d or 52d dollars b Specific Case If the annual salary is 24 000 the monthly salary is 24 000 12 dollars General Case If the annual salary is y dollars the monthly salary is y 12 or y 12 dollars V CHECKPOINT Now try Exercise 57 In mathematics it is usefu
47. erations with the set of numbers Section 1 2 and properties of the operations with the numbers Section 1 3 Set of gt Operations with Properties of Numbers the Numbers gt the Operations Note that the properties of real numbers can be applied to variables and algebra ic expressions as well as to real numbers AMPLE 2 Using the Properties of Real Numbers Complete each statement using the specified property of real numbers a Multiplicative Identity Property 4a 1 b Associative Property of Addition b 8 3 c Additive Inverse Property 0 5c d Distributive Property 7 b 7 5 Solution a By the Multiplicative Identity Property 4a 1 4a b By the Associative Property of Addition b 8 3 b 8 3 c By the Additive Inverse Property 0 5c 5c d By the Distributive Property 7 b 7 5 7 b 5 V CHECKPOINT Now try Exercise 21 To help you understand each property of real numbers try stating the prop erties in your own words For instance the Associative Property of Addition can be stated as follows When three real numbers are added it makes no difference which two are added first Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 3 Properties of Real Numbers 25 2 Develop additional properties ofreal_ Additional Properties of Real Numbers numbe
48. ers on the real 25 3 26 3 3 number line See Example 2 2 _ 10 5 3 27 3 3 28 3 5 13 a 3 O O 52 14 a 8 b f 6 75 d 3 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 10 Chapter 1 Fundamentals of Algebra In Exercises 29 40 find the distance between the pair of real numbers See Example 4 X 29 31 33 35 37 39 4 and 10 12 and 7 8 and 0 0 and 35 18 and 32 6 and 9 30 32 34 36 38 40 In Exercises 41 54 evaluate Example 5 41 10 43 225 45 85 47 16 a9 3 51 3 5 53 a 75 and 20 54 and 32 14 and 6 0 and 125 35 and 0 12 and 7 the expression See 42 62 44 14 46 36 5 48 25 50 52 1 4 54 7 In Exercises 55 62 place the correct symbol lt gt or between the pair of real numbers See Example 6 55 6 1471 s Ars i 74 2l 27 1 8 _ 4 5 56 58 60 62 2 12l 150 310 12 5 25 3 JEE In Exercises 63 72 find the opposite and the absolute value of the number 63 65 34 160 True or False the statement is true or false Explain your reasoning In Exercises 95 and 96 decide whether 64 66 225 52 67 2 68 4 69 3 70 71 4 7 72
49. es 53 56 evaluate the expression 53 120 52 4 In Exercises 71 74 rewrite the expression by using the Distributive Property 54 45 45 3 55 8 3 6 2 7 4 71 u 3v 56 24 10 6 1 3 72 5 2x 4y 73 a 8 3a A gt Evaluate expressions using a calculator and order of 74 x 3x 4y operations In Exercises 57 and 58 evaluate the expression using a calculator Round your answer to two decimal places 57 7 408 27 39 5 0 3 58 59 4 6 5 8 13 4 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 54 Chapter 1 Fundamentals of Algebra 1 4 Algebraic Expressions 1 gt Identify the terms and coefficients of algebraic expressions In Exercises 75 78 identify the terms and coefficients of the algebraic expression 17 75 4y y D x 1 76 2xy2 tees Tis 52 122 1 x 78 EE ae a 2 Simplify algebraic expressions by combining like terms and removing symbols of grouping In Exercises 79 88 simplify the expression 79 6x 3x 80 10y 7y 81 3u 2v 7v 3u 82 9m 4n m 3n 83 5 x 4 10 84 15 7 z 2 85 3x y 2x 86 30x 10x 80 87 3 b 5 b a 88 28 6 t 5t 2 Evaluate algebraic expressions by substituting values for the variables In Exercises 89
50. ession 0 15717 1 19t 11 0 for each integer value of from t 9 tor 15 Year 1999 2000 2001 2002 2003 2004 2005 Revenue 13 0 14 8 16 9 19 3 22 1 25 1 28 5 wv CHECKPOINT Now try Exercises 101 and 102 Technology Tip Most graphing calculators can be used to evaluate an algebraic expression for several values of x and display the results in a table For instance to evaluate 2x 3x 2 when x is 0 1 2 3 4 5 and 6 you can use the following steps 1 Enter the expression into the graphing calculator 2 Set the minimum value of the table to 0 3 Set the table step or table increment to 1 4 Display the table The results are shown below E AFT aA Po jeh E fek Pat id a Consult the user s guide for your graphing calculator for specific instructions Then complete a table for the expression 4x2 5x 8 when x is 0 1 2 3 4 5 and 6 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Explain how you can use the Distributive Property to simplify the expression 5x 3x Go to pages 50 51 to record your assignments Section 1 4 Algebraic Expressions 37 Concept Check Explain the difference between terms and factors in 3 Explain how to combine like terms in an algebraic an algebraic expression expression Give an example 4 E
51. f a food product that is 65 water The amount of wage tax due for a taxable income of I dollars that is taxed at the rate of 1 25 The amount of sales tax on a purchase valued at L dollars if the tax rate is 5 5 The sale price of a coat that has a list price of L dollars if it is a 20 off sale The total bill for a meal that cost C dollars if you plan on leaving a 15 tip The total hourly wage for an employee when the base pay is 8 25 per hour plus 60 cents for each of q units produced per hour The total hourly wage for an employee when the base pay is 11 65 per hour plus 80 cents for each of q units produced per hour The sum of a number n and five times the number The sum of three consecutive integers the first of which is n The sum of three consecutive odd integers the first of which is 2n 1 The sum of three consecutive even integers the first of which is 2n The product of two consecutive even integers divided by 4 The absolute value of the difference of two consecu tive integers divided by 2 Solving gt E A Geometry In Exercises 65 68 write an expression for the area of the figure Simplify the expression 65 66 lt _ _ w 3x 3 gt lt s gt rar D 2 sh 6 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 5 C
52. g the Properties of Real Numbers In the solution of the equation b 2 6 identify the property of real numbers that justifies each step Solution b 2 6 Original equation Solution Step Property b 2 2 6 2 Addition Property of Equality b 2 2 6 2 Associative Property of Addition b 0 4 Additive Inverse Property b 4 Additive Identity Property V CHECKPOINT Now try Exercise 63 J MPLE 6 Applying the Properties of Real Numbers In the solution of the equation 3x 15 identify the property of real numbers that justifies each step Solution 3x 15 Original equation Solution Step Property 1 1 ae 3 3x 3 15 Multiplication Property of Equality 1 N KEE 3 3 x 5 Associative Property of Multiplication 1 x 5 Multiplicative Inverse Property x Multiplicative Identity Property V CHECKPOINT Now try Exercise 65 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Are subtraction and division commutative If not show a counterexample 28 Chapter 1 Fundamentals of Algebra Concept Check 1 Does every real number have an additive inverse 3 Explain 2 Does every real number have a multiplicative 4 inverse Explain Go to pages 50 51 to record your assignments Developing In Exercises 1 20 identify the property of real numbers illustrated by the statement See Example 1 1 18 18 0 25 0 5 3
53. gebraic expression The terms of an algebraic expression are those parts that are separated by addition For example the algebraic expression x 3x 6 has three terms x 3x and 6 Note that 3x is a term rather than 3x because x 3x 6 27 3x 6 Think of subtraction as a form of addition The terms x and 3x are the variable terms of the expression and 6 is the constant term The numerical factor of a term is called the coefficient For instance the coefficient of the variable term 3x is 3 and the coefficient of the variable term x is 1 Example 1 identifies the terms and coefficients of three different algebraic expressions AMPLE 1 Identifying Terms and Coefficients Algebraic Expression Terms Coefficients 1 1 1 BPO dee I SS Oe a 5x 3 x 3 3 b 4y 6x 9 4y 6x 9 4 6 9 2 2 e 5x4 y 5x4 y 2 5 1 V CHECKPOINT Now try Exercise 1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 4 Algebraic Expressions 33 2 Simplify algebraic expressions by combining like terms and removing symbols of grouping Simplifying Algebraic Expressions In an algebraic expression two terms are said to be like terms if they are both constant terms or if they have the same variable factor For example 2x y xy and 5 x2y are like terms because
54. gure 1 10 Figure 1 11 4 units 14 units f 5 x p x t t t o gt t e gt 4 3 2 I 0 al 0 1 2 Figure 1 12 Figure 1 13 V CHECKPOINT Now try Exercise 29 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 4 Determine the absolute value of a real number Study Tip Because opposite numbers lie the same distance from 0 on the real number line they have the same absolute value So 5 5 and 5 5 Section 1 1 The Real Number System 7 Absolute Value Two real numbers are called opposites of each other if they lie the same distance from but on opposite sides of 0 on the real number line For instance 2 is the opposite of 2 see Figure 1 14 Opposite numbers gemm ce 4 3 2 1 0 1 2 3 4 Figure 1 14 The opposite of a negative number is called a double negative see Figure 1 15 Opposite numbers 3 3 S t _ gt 4 3 2 1 0 1 2 3 4 Figure 1 15 Opposite numbers are also referred to as additive inverses because their sum is zero For instance 3 3 0 In general you have the following The distance between a real number a and 0 the origin is called the absolute value of a Absolute value is denoted by double vertical bars For example 5 distance between 5 and 0 5 and 8 distance between 8 and 0 8 Be sure y
55. hapter 1 2 Use the real number line to order real numbers Figure 1 4 Fundamentals of Algebra The Real Number Line The picture that represents the real numbers is called the real number line It consists of a horizontal line with a point the origin labeled 0 Numbers to the left of zero are negative and numbers to the right of zero are positive as shown in Figure 1 2 Origin gt i J T T 3 2 1 0 1 2 3 4 Negative numbers Positive numbers Figure 1 2 The Real Number Line Zero is neither positive nor negative So to describe a real number that might be positive or zero you can use the term nonnegative real number Each point on the real number line corresponds to exactly one real number and each real number corresponds to exactly one point on the real number line as shown in Figure 1 3 When you draw the point on the real number line that corresponds to a real number you are plotting the real number 5 2 1 t e t He t m 2 1 0 1 2 2 1 0 1 2 Each point on the real number line Each real number corresponds to a corresponds to a real number point on the real number line Figure 1 3 AMPLE 2 Plotting Points on the Real Number Line Plot the real numbers on the real number line 5 9 a b 2 3 G d 0 3 3 4 Solution All four points are shown in Figure 1 4 a The point representing the real number 3 1 666 lies between 2 and 1 but closer to 2
56. he even integers between 5 5 and 2 5 2 Use the real number line to order real numbers In Exercises 5 and 6 plot the real numbers on the real number line 5 a 4 3 er 6 a 9 br 2 d 2 4 d 5 25 In Exercises 7 10 place the correct inequality symbol lt or gt between the numbers 7 5 3 8 2 B 8 9 5 B 10 84 3 2 2 Use the real number line to find the distance between two real numbers In Exercises 11 14 find the distance between the pair of real numbers 11 11 and 3 12 4 and 13 13 13 5 and 6 2 14 8 4 and 0 3 4 Determine the absolute value of a real number In Exercises 15 18 evaluate the expression 15 5 16 6 17 7 2 18 3 6 1 2 Operations with Real Numbers 1 gt Add subtract multiply and divide real numbers In Exercises 19 40 evaluate the expression If it is not possible state the reason Write all fractions in simplest form 19 15 4 20 12 3 21 340 115 5 22 154 86 240 23 63 5 21 7 24 14 35 10 3 25 4 4 26 B 27 2 1 28 3 5 29 84 6 30 28 54 31 7 4 32 9 5 33 120 5 7 34 16 15 4 35 5 7 36 5 37 38 3 5 a 40 3 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Review Exercise
57. he set also contains the numbers 5 6 7 and so on Positive integers can be used to describe many quantities in everyday life For instance you might be taking four classes this term or you might be paying 240 dollars a month for rent But even in everyday life positive integers cannot describe some concepts accurately For instance you could have a zero balance in your checking account To describe such a quantity you need to expand the set of positive integers to include zero forming the set of whole numbers To describe a quantity such as 5 you need to expand the set of whole numbers to include negative integers This expanded set is called the set of integers The set of integers is also a subset of the set of real numbers Whole numbers SSS SSS 3 2 1 0 15 2 3 2 amp of J The set of integers ae Negative integers Positive integers Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Technology Tip You can use a calculator to round decimals For instance to round 0 2846 to three decimal places on a scientific calculator enter 2846 On a graphing calculator enter round 2846 3 Consult the user s manual for your graphing calculator for specific keystrokes or instructions Then use your calculator to round 0 38174 to four decimal places Real numbers
58. hed order of operations 5 O04 3 amp 2 0 ENTER 5 Graphing it will display 18 CHECKPOINT Now try Exercise 125 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 2 Operations with Real Numbers 19 Concept Check 1 Is the reciprocal of every nonzero integer an integer 3 Explain how to subtract one real number from another 2 Can the sum of two real numbers be less than either 4 If a gt 0 state the values of n such that number If so give an example a a Go to pages 50 51 to record your assignments Developing Skills the expression See 40 15 15 15 15 In Exercises 1 38 evaluate Examples 1 7 4 4 44 i44 H 1 13 32 2 16 84 42 3 54545 3 8 12 4 5 9 43 4 3 4 4 5 6 4 3 7 6 5 1 0 9 44 5 5 3 Z 7 13 6 8 12 10 9 12 6 38 5 10 10 4 43 5 In Exercises 45 62 find the product See Examples 8 u 8 12 12 3 17 and 9 13 21 5 6 3 14 13 2 9 6 amp 45 5 6 46 3 9 15 4 11 9 16 17 6 24 47 8 6 48 4 7 17 5 3 2 2 6 9 18 46 08 35 1 16 25 49 2 4 5 50 3 7 10 19 15 6 31 18 ae hes pe GAN 20 6 26 17 10 53 8 5 54 7 3 21 2 2 ry po 55
59. ies 27 nickels 17 dimes 15 quarters 98 111 pennies 22 nickels 2 dimes 42 quarters A Geometry In Exercises 99 and 100 write and simplify an expression for the area of the figure Then evaluate the expression for the given value of the variable 99 b 15 100 h 12 4 h b 3 j 4 29410 Using a Model n Exercises 101 and 102 use the following model which approximates the annual sales in millions of dollars of sports equipment in the United States from 2001 to 2006 see figure where t represents the year with t 1 corresponding to 2001 Source National Sporting Goods Association Sales 607 6t 20 737 1 lt t lt 6 25 000 E 23 000 a 21 000 Annual sales in millions of dollars 19 000 17 000 1 2 3 4 5 6 Year 1 2001 A 101 Graphically approximate the sales of sports equip ment in 2005 Then use the model to confirm your estimate algebraically CA 102 Use the model and a calculator to complete the table showing the sales from 2001 to 2006 Year 2001 2002 2003 Sales Year 2004 2005 2006 Sales Using a Model n Exercises 103 and 104 use the following model which approximates the total yearly disbursements in billions of dollars of Federal Family Education Loans FFEL in the United States from 1999 to 2005 see figure where t represents the year with t 9 corresponding to 1999 Source U S Department of Education
60. ifference of a number and 5 divided by 4 0 187 103 2 5 P 51 Chapter Test page 55 1 4 8 11 12 15 17 18 a lt b gt 2113 3 20 1 1 3 5 150 6 60 7 6 4 9 7 10 15 a Associative Property of Multiplication b Multiplicative Inverse Property 12x 6 13 2x7 5x 1 14 x 26 a 16 11f 7 Evaluating an expression is solving the expression when values are provided for its variables a 23 b 7 6 inches 19 640 cubic feet 20 5n 8 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part
61. imes its height Find the number of cubic feet in 5 cords of wood 20 Translate the phrase into an algebraic expression 0 61 The product of a number n and 5 decreased by 8 21 Write an algebraic expression for the sum of two consecutive even integers the first of which is 2n 7 l 22 Write expressions for the perimeter and area of the rectangle shown at the Figure for 22 left Simplify the expressions and evaluate them when 45 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Improving Your Memory Have you ever driven on a highway for ten minutes when all of a sudden you kind of woke up and won dered where the last ten miles had gone The car was on autopilot The same thing happens to many college students as they sit through back to back classes The longer students sit through classes on autopilot the more likely they will crash when it comes to studying outside of class on their own While on autopilot you do not process and retain new information effectively Your memory can be improved by learning how to focus during class and while studying on your own VP Academic Hm baby Mobi expert in developmental education Smart Study Strategy rate Cost Marku 8 J ofo z E g While gallin o Z395 4 e Amoun First rate A e Fi e eee classes Per nour work va comp ajo 56 Keep Your Mind
62. irrational For instance J2 1 4142135 and m 3 1415926 are irrational The decimal representation of an irrational number neither terminates nor repeats When you perform calculations using decimal representations of nonterminating nonrepeating decimals you usually use a decimal approximation that has been rounded to a certain number of decimal places The rounding rule used in this text is to round up if the succeeding digit is 5 or more or to round down if the succeeding digit is 4 or less For example to one decimal place 7 35 would round up to 7 4 Similarly to two decimal places 2 364 would round down to 2 36 Rounded to four decimal places the decimal approximations of the rational number Z and the irrational number 7 are 2 37 0 6667 and m 3 1416 The symbol means is approximately equal to Figure 1 1 shows several commonly used subsets of real numbers and their relationships to each other Classifying Real Numbers Which of the numbers in the set 9 3 0 2 V2 T 5 are a natural numbers b integers c rational numbers and d irrational numbers Solution a Natural numbers 5 b Integers 7 1 0 5 c Rational numbers 7 i 4 0 3 5 d Irrational numbers 3 2 a V CHECKPOINT Now try Exercise 1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 4 C
63. l to know how to represent certain types of integers algebraically For instance consider the set 2 4 6 8 of even integers Because every even integer has 2 as a factor 2 2 1 4 2 2 6 2 3 8 2 4 it follows that any integer n multiplied by 2 is sure to be the even number 2n Moreover if 2n is even then 2n 1 and 2n 1 are sure to be odd integers Two integers are called consecutive integers if they differ by 1 For any integer n its next two larger consecutive integers are n 1 and n 1 1 or n 2 So you can denote three consecutive integers by n n 1 andn 2 These results are summarized below Labels for Integers Let n represent an integer Then even integers odd integers and consecutive integers can be represented as follows 1 2n denotes an even integer for n 1 2 3 2a anden denote odad integers Oma PRET 3 nn 1 n 2 denotes a set of consecutive integers Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 46 Chapter 1 Fundamentals of Algebra MPLE 10 Constructing a Mathematical Model Write an expression for the following phrase The sum of two consecutive integers the first of which is n Solution The first integer is n The next consecutive integer is n 1 So the sum of two consecutive integers is n n 1 2n 1 V CHECKPOINT Now try
64. less than the product of 5 and a number b The sum of 3 and a number all divided by 4 c Twice the sum of 3 times a number and 4 d Four divided by a number reduced by 2 wv CHECKPOINT Nov try Exercise 25 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 5 Constructing Algebraic Expressions 43 2 gt Construct algebraic expressions with cay Constructing Mathematical Models idden products Study Tip Most real life problems do not contain verbal expressions that clearly identify the arithmetic operations involved in the problem You need to rely on past experience and the physical nature of the problem in order to identify the operations hidden in the problem statement Watch for hidden products in Examples 5 and 6 Translating a verbal phrase into a mathematical model is critical in problem solving The next four examples will demonstrate three steps for creating a mathematical model 1 Construct a verbal model that represents the problem situation 2 Assign labels to all quantities in the verbal model 3 Construct a mathematical model algebraic expression AMPLE 5 Constructing a Mathematical Model A cash register contains x quarters Write an algebraic expression for this amount of money in dollars Solution Verbal Value Number Model of coin of coins Labels Value of coin 0 25
65. lif 44 implify V CHECKPOINT Now try Exercise 71 2 Write repeated multiplication in Positive Integer Exponents exponential form and evaluate exponential expressions Repeated multiplication can be written in what is called exponential form Repeated Multiplication Exponential Form TELETA 7 m Technology Discovery 4 factors of 7 When a negative number Is raised N i J B to a power the use of parentheses 4 4 4 4 is very important To discover why use a calculator to evaluate 4 and 4 Write a statement 3 3 factors of z explaining the results Then use a calculator to evaluate 4 and 45 If necessary write a new statement explaining your discoveries Exponential Notation Let n be a positive integer and let a be a real number Then the product of n factors of a is given by ai Oogeg 8 ee n factors In the exponential form a a is the base and n is the exponent Writing the exponential form a is called raising a to the nth power When a number say 5 is raised to the first power you would usually write 5 rather than 5 Raising a number to the second power is called squaring the number Raising a number to the third power is called cubing the number Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Use order of operations to evaluate expressions Stud
66. lify the expression for a the perimeter and b the area of the rectangle TI x 6 78 5x 2x 1 Explaining Concepts 79 What is the additive inverse of a real number Give an example of the Additive Inverse Property 80 What is the multiplicative inverse of a real number Give an example of the Multiplicative Inverse Property 81 amp In your own words give a verbal description of the Commutative Property of Addition 82 amp Explain how the Addition Property of Equality can be used to allow you to subtract the same number from each side of an equation 83 You define a new mathematical operation using the symbol This operation is defined as a b 2 a b Give examples to show that this operation is neither commutative nor associative 84 You define a new mathematical operation using the symbol This operation is defined as atb a b 1 Give examples to show that this operation is neither commutative nor associative Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Mid Chapter Quiz 31 Figure for 21 Take this quiz as you would take a quiz in class After you are done check your work against the answers in the back of the book In Exercises 1 and 2 plot the two real numbers on the real number line and place the correct inequality symbol lt or gt between the two num
67. merators over the common denominator 1 3 Properties of Real Numbers Assignment 7 To multiply fractions Multiply the numerators and the denominators 8 To raise a to the nth power n is an integer GHA a a a a n factors Use order of operations to evaluate expressions 1 First do operations within symbols of grouping 2 Then evaluate powers 3 Then do multiplications and divisions from left to right 4 Finally do additions and subtractions from left to right Due date Use properties of real numbers Let a b and c represent real numbers variables or algebraic expressions Commutative Properties at b bt a ab ba Associative Properties a b c a b c ab c a bc 1 4 Algebraic Expressions Assignment LI Identify the terms and coefficients of algebraic expressions The terms of an algebraic expression are those parts that are separated by addition The coefficient of a term is its numerical factor 1 5 Constructing Algebraic Expressions Distributive Properties alb c ab ac a b c ac be alb c ab ac a b c ac be Identity Properties a 0 0 a a a l l a a Inverse Properties a a 0 a 1 a 0 See page 25 for additional properties of real numbers Due date Simplify algebraic expressions To simplify an algebraic expression remove the symbols of grouping and combine like terms Evalu
68. n a number b and 0 is 6 Explain 9 2 How are the numbers connected by each brace related what you know about the number b A r 3 n i eeo eo _ __ _ 4 3 2 l 0 il 2 3 4 Go to pages 50 51 to record your assignments Developing Skills In Exercises 1 4 which of the real numbers in the In Exercises 15 18 approximate the two numbers and set are a natural numbers b integers c rational order them numbers and d irrational numbers See Example 1 15 a i 1 6 V6 0 3 1 V2 2 m 6 U 4 ee 2 7 73 1 0 2 V3 3 5 101 a te ee he _ 16 a b 3 4 2 V4 0 3 V11 5 5 5 543 i 7 5 4 25 6 0 T 3 0 0 85 3 110 i E LER f 0 1 2 3 4 In Exercises 5 8 use an overbar symbol to rewrite the 17 j b decimal using the smallest number of digits possible U 5 0 2222 6MI 5SS5d D i o OMENA 7 2 121212 8 0 436436436 SE TA EL eTA 18 a b In Exercises 9 12 list all members of the set U 9 The integers between 5 8 and 3 2 i E 60 6l 62 63 64 65 66 Hy The evef intenet Between lt 2 kane 10 3 In Exercises 19 28 place the correct inequality symbol lt or gt between the pair of numbers See Example 3 11 The odd integers between 7 and 10 4 5 12 The prime numbers between 4 and 25 19 5 l 20 2 3 21 5 2 22 9 j 23 5 2 24 8 3 In Exercises 13 and 14 plot the real numb
69. nd each of the two numbers is a factor of the product Multiplication of Two Real Numbers To multiply two real numbers with like signs find the product of their absolute values The product is positive To multiply two real numbers with unlike signs find the product of their absolute values and attach a minus sign The product is negative The product of zero and any other real number is zero Multiplying Integers Unlike signs p 6 9 Study Tip a 54 The product is negative To find the product of two or more Iai numbers first find the product of their absolute values If there is an b 5 7 35 The product is positive even number of negative factors as Like signs in Example 8 c the product is positive If there is an odd number c 5 3 4 7 420 The product is positive of negative factors as in Example ji ji 8 a the product is negative V CHECKPOINT Now try Exercise 45 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Study Tip When operating with fractions you should check to see whether your answers can be simplified by dividing out factors that are common to the numerator and denominator For instance the fraction 2 can be written in simplified form as Note that dividing out a common factor is the division of a number by itself and what remains is a fact
70. nderstand the set of real numbers and the subsets of real numbers 2 gt Use the real number line to order real numbers 3 gt Use the real number line to find the distance between two real numbers 4 gt Determine the absolute value of a real number Sets and Real Numbers This chapter introduces the basic definitions operations and rules that form the fundamental concepts of algebra Section 1 1 begins with real numbers and their representation on the real number line Sections 1 2 and 1 3 discuss operations and properties of real numbers and Sections 1 4 and 1 5 discuss algebraic expressions The formal term that is used in mathematics to refer to a collection of objects is the word set For instance the set 1 2 3 A set with three members contains the three numbers 1 2 and 3 Note that the members of the set are enclosed in braces Parentheses and brackets are used to represent other ideas The set of numbers that is used in arithmetic is called the set of real numbers The term real distinguishes real numbers from imaginary or complex numbers a type of number that you will study later in this text If all members of a set A are also members of a set B then A is a subset of B One of the most commonly used subsets of real numbers is the set of natural numbers or positive integers 1 2 3 4 The set of positive integers Note that the three dots indicate that the pattern continues For instance t
71. ne distance between two real numbers Once you know how to represent real numbers as points on the real number line it is natural to talk about the distance between two real numbers Specifically if a and b are two real numbers such that a lt b then the distance between a and b is defined as b a Distance Between Two Real Numbers If a and b are two real numbers such that a lt b then the distance between a and b is given by Distance between a and b b a Note from this definition that if a b the distance between a and b is zero If a b then the distance between a and b is positive Finding the Distance Between Two Real Numbers Find the distance between each pair of real numbers a 2 and 3 b 0 and 4 c 4and 0 d 1 and Solution a Because 2 lt 3 the distance between 2 and 3 is Study Tip 3 2 3 2 5 See Figure 1 10 Recall that when you subtract a negative number as in Example 4 a you add the opposite of the f See Figure 1 11 second number to the first Because the opposite of 2 is 2 you add 2 b Because 0 lt 4 the distance between 0 and 4 is c Because 4 lt 0 the distance between 4 and 0 is to 3 0 4 04 4 4 See Figure 1 12 d Because 4 lt l leta 5 and b 1 So the distance between 1 and 4 is 1 1 1 1 z 7 I5 See Figure 1 13 5 units 4 units f an c A e a o 2 i 0 1 2 3 0 1 2 3 4 Fi
72. onstructing Algebraic Expressions 49 75 Finding a Pattern Complete the table The third row contains the differences between consecutive entries of the second row Describe the pattern of the A Geometry In Exercises 69 72 write expressions for the perimeter and area of the region Simplify the expressions See Example 11 third row Z 69 n 70 4 if 0 57 n Oat ae E eS y Sn 3 2w l Differences 71 3 72 1 1 p x42 gt ie bo 76 Finding a Pattern Complete the table The third 3 row contains the differences between consecutive 2x 5 entries of the second row Describe the pattern of the x third row n ae esa eales 3n 1 73 AX Geometry Write an expression for the area of the soccer field shown in the figure What is the unit Differences of measure for the area A 77 Finding a Pattern Using the results of Exercises 75 and 76 guess the third row difference that would result in a similar table if the algebraic expression b 50 m were an b 78 Think About It Find a and b such that the expres sion an b would yield the following table bii n lil al alkal s anea 3 7 11 15 19 23 74 AX Geometry Write an expression for the area of the advertising banner shown in the figure What is the unit of measure for the area a lt
73. or of 1 Study Tip Division by 0 is not defined because 0 has no reciprocal If 0 had a reciprocal value b then you would obtain the false result b The reciprocal of zero is b lZ 0 1 b 0 Multiply each side by 0 1 0 False result 1 0 Section 1 2 Operations with Real Numbers 15 Multiplying Fractions Find the product Cako Solution 16 40 3 11 8 2 3 Multiply numerators and denominators Factor and divide out common factor Simplify CHECKPOINT Now try Exercise 57 The reciprocal of a nonzero real number a is defined as the number by which a must be multiplied to obtain 1 For instance the reciprocal of 3 is because 1 sJ 1 3 Similarly the reciprocal of 3 is 3 because 4 5 _ 5 4 In general the reciprocal of a b is b a Note that the reciprocal of a positive number is positive and the reciprocal of a negative number is negative Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 16 Chapter 1 Fundamentals of Algebra AMPLE 10 Division of Real Numbers 1 a 30 5 30 5 Invert divisor and multiply Multipl 5 ultiply 6 38 a z Factor and divide out common factor 6 Simplify gt nt E iN b 16 gt 24 16 4 Write 27 as 7 aapa Invert divi d multipl 16 11 nvert divisor and multiply AA Multipl 16 11 AA 2 Simp
74. ou see from the following definition that the absolute value of a real number is never negative For instance if a 3 then 3 3 3 Moreover the only real number whose absolute value is zero is 0 That is Jo 0 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 8 Chapter 1 Fundamentals of Algebra MPLE 5 Finding Absolute Values a 10 10 The absolute value of 10 is 10 3 3 b 3 Fi The absolute value of 3 is 3 C 3 2 3 2 The absolute value of 3 2 is 3 2 d 6 6 6 The opposite of 6 is 6 Note that part d does not contradict the fact that the absolute value of a number cannot be negative The expression 6 calls for the opposite of an absolute value and so it must be negative V CHECKPOINT Now try Exercise 41 For any two real numbers a and b exactly one of the following orders must be true a lt b a b or a gt b This property of real numbers is called the Law of Trichotomy In words this property tells you that if a and b are any two real numbers then a is less than b a is equal to b or a is greater than b MPLE 6 Comparing Real Numbers Place the correct symbol lt gt or between each pair of real numbers a 2 1 b 4 4 c 12 15 d 3 3 e 2 2 f 3 3 Solution 2 gt 1 be
75. p can also be described by saying that b is greater than a and writing b gt a The expression a lt b means that a is less than or equal to b and the expression b a means that b is greater than or equal to a The symbols lt gt lt and 2 are called inequality symbols When asked to order two numbers you are simply being asked to say which of the two numbers is greater Solution a Because 4 lies to the left of 0 on the real number line as shown in Figure 1 6 you can say that 4 is less than 0 and write 4 lt 0 b Because 3 lies to the right of 5 on the real number line as shown in Figure 1 7 you can say that 3 is greater than 5 and write 3 gt 5 c Because lies to the left of on the real number line as shown in Figure 1 8 you can say that is less than re and write lt ie d Because lies to the right of 5 on the real number line as shown in Figure 1 9 you can say that 4 is greater than 5 and write j gt 4 V CHECKPOINT Now try Exercise 19 One effective way to order two fractions such as D and 5 is to compare their decimal equivalents Because a 0 416 and A 0 391 you can write 5 9 pee Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 6 Chapter 1 Fundamentals of Algebra 2 Use the real number line to find the Distance on the Real Number Li
76. pression watch for key Translating verbal phrases into words and phrases that indicate the four different operations of arithmetic algebraic expressions enables you to model real life problems For instance in Exercise 73 on page 49 you will Translating Key Words and Phrases write an algebraic expression that Key Words Verbal Algebraic models the area of a soccer field and Phrases Description Expression Addition 1 gt Translate verbal phrases into Sum plus greater than The sum of 5 and x 5 x algebraic expressions and vice versa increased by more than exceeds total of Seven more than y Subtraction Difference minus b is subtracted from 4 less than decreased by subtracted from Three less than z reduced by the remainder Multiplication Product multiplied by Two times x twice times percent of Division Quotient divided by The ratio of x and 8 ratio per Verbal Description Algebraic Expression a Seven more than three times x 3x 7 b Four less than the product of 6 and n 6n 4 c The quotient of x and 3 decreased by 6 V CHECKPOINT Now try Exercise 1 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 42 Chapter 1 Study Tip When you write a verbal model or construct an algebraic expression watch out for statements that may be interpreted in more than one way For instance the statemen
77. r of 867 132 87 Find the profit or loss of the company for the second 6 months of the year Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 135 Stock Values On Monday you purchased 500 worth of stock The values of the stock during the remainder of the week are shown in the bar graph a b ma 136 Net Profit A 525 7 _ 500 il l y Fa gt s os 500 475 Ny Stock value in dollars Use the graph to complete the table Day Daily gain or loss Tuesday Wednesday Thursday Friday Find the sum of the daily gains and losses Interpret the result in the context of the problem How could you determine this sum from the graph The net profits for Columbia Sportswear in millions of dollars for the years 2001 to 2006 are shown in the bar graph Use the graph to create a table that shows the yearly gains or losses Source Columbia Sportswear Company A 160 5 140 peresse S 120 100 S o g 801m Do Z op al a 40 204 2001 2002 2003 2004 2005 2006 Year Section 1 2 Operations with Real Numbers 21 137 Savings Plan a b wm c You save 50 per month for 18 years How much money has been set aside during the 18 years If the money in part a is deposited in a savings account earning
78. rs Once you have determined the basic properties or axioms of a mathematical system you can go on to develop other properties These additional properties are theorems and the formal arguments that justify the theorems are proofs Additional Properties of Real Numbers Let a b and c be real numbers variables or algebraic expressions Properties of Equality Addition Property of Equality Ifa b thena c b c Multiplication Property of Equality If a b then ac be Cancellation Property of Addition Ifa c b c thena b Cancellation Property of Multiplication If ac bc and c 0 then a b Properties of Zero Multiplication Property of Zero 0 a 0 Division Property of Zero 0 a 0 Division by Zero Is Undefined 5 is undefined Properties of Negation Multiplication by 1 1 a 1 a a Placement of Negative Signs a b ab a b Product of Two Opposites a b ab Study Tip In Section 2 1 you will see that the properties of equality are useful for solving equations as shown below Note that the Addition and Multiplication When the properties of real numbers Properties of Equality can be used to subtract the same nonzero quantity from are used in practice the process is each side of an equation or to divide each side of an equation by the same nonzero usually less formal than it would quantity appear from the list of properties on this page For instance the steps
79. s 53 In Exercises 41 and 42 write the expression as a 59 Total Charge You purchased an entertainment repeated addition problem system and made a down payment of 395 plus nine monthly payments of 45 each What is the total 41 7 3 i amount you paid for the system 2 60 Savings Plan You deposit 80 per month in a 42 565 savings account for 10 years The account earns 2 interest compounded monthly The total amount in In Exercises 43 and 44 write the expression as a the account after 10 years will be multiplication problem alea 0 02120 i 12 43 8 8 8 8 8 8 4 8 8 12 0 0277 44 5 5 5 5 5 Use a calculator to determine this amount 2 gt Write repeated multiplication in exponential form and evaluate exponential expressions 1 3 Properties of Real Numbers In Exercises 45 and 46 write the expression using 1 gt Identify and use the properties of real numbers exponential notation In Exercises 61 70 identify the property of real 45 6 6 6 6 6 6 6 numbers illustrated by the statement 1 _1 _1 a6 DEDY lg T In Exercises 47 52 evaluate the exponential 62 16 a expression 63 719 3 7 9 7 3 47 6 4 64 15 4 4 15 48 3 4 65 5 4 y 5 4 y 49 4 66 6 4z 6 4 z 50 25 51 4 67 u v 2 2 u v 2 4 52 3 68 xy 1 xy 2 Use order of operations to evaluate expressions 69 8S y 8 x 8 y 70 xz yz x y z In Exercis
80. t The sum of x and 1 divided by 5 is ambiguous because it could mean 1 or xX 5 5 Notice in Example 4 b that the verbal description for SEX 4 contains the phrase all divided by 4 Fundamentals of Algebra i MPLE 2 Translating Verbal Phrases Verbal Description Algebraic Expression a Eight added to the product of 2 and n 2n 8 b Four times the sum of y and 9 4 y 9 c The difference of a and 7 all divided by 9 at V CHECKPOINT Now try Exercise 7 In Examples 1 and 2 the verbal description specified the name of the variable In most real life situations however the variables are not specified and it is your task to assign variables to the appropriate quantities Translating Verbal Phrases Verbal Description Label Algebraic Expression a The sum of 7 and a number The number x TEY b Four decreased by the The number n 4 2n product of 2 and a number c Seven less than twice the The number y 2 y 5 7 sum of a number and 5 V CHECKPOINT Now try Exercise 19 A good way to learn algebra is to do it forward and backward For instance the next example translates algebraic expressions into verbal form Keep in mind that other key words could be used to describe the operations in each expression i MPLE 4 Translating Expressions into Verbal Phrases Without using a variable write a verbal description for each expression a 5x 10 d 3 Solution a 10
81. t form Commutative Property Associative Property Distributive Property Simplest form Group like terms Combine like terms Group like terms Combine like terms Group like terms Combine like terms Licensed to iChapters User 34 Chapter 1 Fundamentals of Algebra Study Tip The exponential notation described in Section 1 2 can also be used when the base is a variable or an algebraic expression For instance in Example 4 b x2 x can be written as X X X X X X 3 factors of x Another way to simplify an algebraic expression is to remove symbols of grouping Remove the innermost symbols first and combine like terms Repeat this process as needed to remove all the symbols of grouping A set of parentheses preceded by a minus sign can be removed by changing the sign of each term inside the parentheses For instance 3x 2x 7 3x 2x 7 This is equivalent to using the Distributive Property with a multiplier of 1 That is 3x 2x 7 3x 1 2x 7 3x 2x 7 A set of parentheses preceded by a plus sign can be removed without changing the signs of the terms inside the parentheses For instance 3x 2x 7 3x 2x 7 APLE 4 Removing Symbols of Grouping Simplify each expression a 3 x 5 2x 7 b 4 x2 4 x2 x 4 Solution a 3 x 5 2x 7 3x 15 2x 7 Distributive Property 3x 2x 15 7 Gro
82. they have the same variable factor xy Note that 4x y and xy are not like terms because their variable factors x y and xy are Study Tip As you gain experience with the rules of algebra you may want to combine some of the steps in your work For instance you might feel comfortable listing only the following steps to solve Example 2 0 5x 3y 4x 3y 5x 4x 3y x different One way to simplify an algebraic expression is to combine like terms MPLE 2 Combining Like Terms Simplify each expression by combining like terms a 2x 3x 4 be 3 5 2y Ty Solution a 2x 3x 4 2 3 x 4 3x 4 b 3 5 2y 7y 3 5 2 7 y 2 5y c 5x 3y 4x 3y 5x 4x 3y 5x 4x 3y 5 4 x 3y x V CHECKPOINT Now try Exercise 25 MPLE 3 Combining Like Terms Simplify each expression by combining like terms a 7x Ty 4x y b 2x 3x 5x7 x c 3xy 4x y 2xy x y Solution a 7x Ty 4x y 7x 4x Ty y 3x 6y b 2x 3x 5x x 2x 5x 3x x 3x 2x c 3xy 4x y 2xy xy 3xy 2xy 4x32 xy 5xy 3x V CHECKPOINT Now try Exercise 33 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part ce Sx 3y 4x Distributive Property Simplest form Distributive Property Simples
83. to Right Rule Order of Operations To evaluate an expression involving more than one operation use the following order 1 First do operations that occur within symbols of grouping Then evaluate powers 2 3 Then do multiplications and divisions from left to right 4 Finally do additions and subtractions from left to right Order of Operations Without Symbols of Grouping a 20 2 37 20 2 9 Evaluate power 20 18 2 Multiply then subtract b 5 6 2 5 6 2 Left to Right Rule 1 2 3 Subtract c 8 2 2 8 2 2 Left to Right Rule 4 2 8 Divide then multiply 4 CHECKPOINT Now try Exercise 105 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User 18 Chapter 1 Fundamentals of Algebra When you want to change the established order of operations you must use parentheses or other symbols of grouping Part d in the next example shows that a fraction bar acts as a symbol of grouping MPLE 13 Order of Operations with Symbols of Grouping Subtract within symbols of grouping a 7 3 4 2 7 3 2 7 6 1 Multiply then subtract b 4 3 2 4 3 8 Evaluate power 4 24 20 Multiply then subtract amp I 4 E 5 3 1 4 2 Subtract within symbols of grouping 2 1 Subtract within symbols of grouping d H 2 5 10 gt 32 4 esac parentheses 50
84. up like terms x 8 Combine like terms b 4 x2 4 x x 4 4x 16 x x 4 Distributive Property 4 16 23 4 Exponential form 4x2 4x 16 Group like terms x 0 16 Combine like terms x 16 Additive Identity Property iv CHECKPOINT Now try Exercise 53 Removing Symbols of Grouping a 5x 2x 3 2 x 7 5x 2x 3 2x 14 Distributive Property 5x 2x 2x 11 Combine like terms 5x 4x 22x Distributive Property 4x 27x Combine like terms b 3x 5x4 2x5 15x x4 2x5 Multiply 15x 2x5 Exponential form 13x5 Combine like terms V CHECKPOINT Now try Exercise 65 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Evaluate algebraic expressions by substituting values for the variables Section 1 4 Algebraic Expressions 35 Evaluating Algebraic Expressions To evaluate an algebraic expression substitute numerical values for each of the variables in the expression Note that you must substitute the value for each occurrence of the variable Evaluate each algebraic expression when x 2 a 5 x b 5 x Evaluating Algebraic Expressions Solution a When x 2 the expression 5 x has a value of 5 2 5 4 9 b When x 2 the expression 5 x has a value of
85. ur local office at www cengage com global Cengage Learning products are represented in Canada by Nelson Education Ltd For your course and learning solutions visit www cengage com Purchase any of our products at your local college store or at our preferred online store www ichapters com Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Chapter I Fundamentals of Algebra 1 1 The Real Number System 1 2 Operations with Real Numbers 1 3 Properties of Real Numbers 1 4 Algebraic Expressions 1 5 Constructing Algebraic Expressions eae p A Copyright 2010 Cengage Learning All Rights Reserved May not be cop ied scanned or duplicated in whole or in part Licensed 2 Duomo CORBIS 1 an to iChapters User Chapter 1 Why You Should Learn It Inequality symbols can be used to represent many real life situations such as bicycling speeds see Exercise 87 on page 10 gt Understand the set of real numbers d the subsets of real numbers Study Tip In this text whenever a mathematical term is formally introduced the word will appear in boldface type Be sure you understand the meaning of each new word it is important that each word become part of your mathematical vocabulary It may be helpful to keep a vocabulary journal Fundamentals of Algebra What You Should Learn 1 gt U
86. ution At first glance it is a little difficult to see what you are being asked to prove However a good way to start is to consider carefully the definitions of the three numbers in the equation a given real number ol the additive inverse of 1 a the additive inverse of a Now by showing that 1 a has the same properties as the additive inverse of a you will be showing that 1 a must be the additive inverse of a 1 a a 1 a 1 a Multiplicative Identity Property 1 1 a Distributive Property O a Additive Inverse Property 0 Multiplication Property of Zero Because 1 a a 0 you can use the fact that a a 0 to conclude that 1 a a a a From this you can complete the proof as follows lhata ata Shown in first part of proof l a a Cancellation Property of Addition V CHECKPOINT Now try Exercise 61 Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Section 1 3 Properties of Real Numbers 27 The list of additional properties of real numbers on page 25 forms a very important part of algebra Knowing the names of the properties is useful but knowing how to use each property is extremely important The next two examples show how several of the properties can be used to solve equations You will study these techniques in detail in Section 2 1 MPLE 5 Applyin
87. wo consecutive integers the first of which is n Copyright 2010 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Licensed to iChapters User Chapter Test 55 Take this test as you would take a test in class After you are done check your work against the answers in the back of the book 1 Place the correct inequality symbol lt or gt between each pair of numbers 5 2 3 on 3 ma 6 9 2 Find the distance between 4 4 and 6 9 In Exercises 3 10 evaluate the expression 3 14 9 15 4 3 2 5 2 225 150 6 3 4 5 1 A A TE 3 y tans 9 2 10 4 13 11 Identify the property of real numbers illustrated by each statement a 3 5 6 3 5 6 1 b b 3y 3y 12 Rewrite the expression 6 2x 1 using the Distributive Property In Exercises 13 16 simplify the expression 13 3x 2x 5x7 7x 1 14 x x 2 2 x x 13 15 a 5a 4 2 2a 2a 16 4t 3t 10t 7 17 Explain the meaning of evaluating an algebraic expression Evaluate the expression 7 x 3 for each value of x a x 1 b x 3 18 An electrician wants to divide 102 inches of wire into 17 pieces with equal lengths How long should each piece be 19 A cord of wood is a pile 4 feet high 4 feet wide and 8 feet long The volume of a rectangular solid is its length times its width t
88. xplain the difference between simplifying an algebraic expression and evaluating an algebraic expression Developing Skills In Exercises 1 14 identify the terms and coefficients of the algebraic expression See Example 1 1 10x 5 2 4 17y 3 12 6x 4 167 48 5 3y 2y 8 6 912 2t 10 7 1 2a 4a3 8 2523 4 827 9 4x 3y 5x 21 10 11 12 13 Ta 4a b 19 xy 5x y 2y 14u 25uv 3v 5x2 3y 5 14 Bz 2 In Exercises 15 20 identify the property of algebra illustrated by the statement 15 16 17 18 19 20 4 3x 3x 4 0 x y 10 x y 5 2x 5 2 x x 2 3 3 2 5 2 x 5x 2x Ty 2y 7 2 y In 21 22 23 24 Exercises 21 24 use the indicated property to rewrite the expression Distributive Property 5 x 6 Distributive Property 6x 6 Commutative Property of Multiplication 5 x 6 Commutative Property of Addition 6x 6 In Exercises 25 40 simplify the expression by combining like terms See Examples 2 and 3 o 25 a7 29 31 amp 33 34 35 36 37 38 39 40 3x 4x 26 18z 14z 2x 4x 28 20a 5a 7x 11x 30 23t 11t 9y 5y 4y 32 8y Ty y 3x 2y 5x 20y 2a 4b Ta b Tage E A 9y y 6y 3z4 6z z 8 z 42 5y 3y 6y By y 4
89. y Tip The order of operations for multiplication applies when multiplication is written with the symbol x or When multiplication is implied by parentheses it has a higher priority than the Left to Right Rule For instance 8 4 2 8 8 1 but 8 4 2 2 2 4 Section 1 2 Operations with Real Numbers 17 MPLE 11 Evaluating Exponential Expressions a 3 4 3 3 3 3 81 Negative sign is part of the base b 34 3 3 3 3 l Negative sign is not part of the base 3 G 5 d 5 5 5 hes 125 Negative raised to odd power e 5 5 5 5 V CHECKPOINT Now try Exercise 91 5 625 Negative raised to even power In parts d and e of Example 11 note that when a negative number is raised to an odd power the result is negative and when a negative number is raised to an even power the result is positive Order of Operations One of your goals in studying this book is to learn to communicate about algebra by reading and writing information about numbers One way to help avoid confusion when communicating algebraic ideas is to establish an order of operations This is done by giving priorities to different operations First priority is given to exponents second priority is given to multiplication and division and third priority is given to addition and subtraction To distinguish between operations with the same priority use the Left

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