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1.5 Amount (thousands of dollars) Growth of a $10,000 investment at 6.2% annual rate 120 (1-31) 31 P(1 r)n to find the actual amount of the debt at the time the payments start. $5,500, $5,441.96 Amount of debt (dollars) a) Use the accompanying graph to estimate the amount of a $10,000 investment in corporate bonds after 30 years. $60,000 b) Use the given expression to calculate the value of a $10,000 investment after 30 years of growth at 6.2% compounded annually. $60,776.47 Properties of the Real Numbers 6,000 5,000 4,000 3,000 2,000 1,000 0 100 80 60 0 1 2 3 4 Time after making the loan (years) FIGURE FOR EXERCISE 107 40 20 0 10 20 30 40 FIGURE FOR EXERCISE 105 106. Saving for college. The average cost of a B.A. in 2005 will be $252,000 at an Ivy League school (Fortune Investors Guide). The principal that must be invested at interest rate r compounded annually to have A dollars n years in the future is given by the algebraic expression A . (1 r)n What investment in 1987 would amount to $252,000 in 2005 at 7% compounded annually? $74,557.71 107. Student loan. A college student borrowed $4,000 at 8% compounded annually in her freshman year and did not have to make payments until 4 years later. Use the accompanying graph to estimate the amount that she owes at the time the payments start. Use the expression 1.5 In this section ● Commutative Properties ● Associative Properties ● Distributive Property ● Identity Properties ● Inverse Properties ● Multiplication Property of Zero 108. High cost of nursing care. The average cost for a oneyear stay in a nursing home in 1990 was $29,930 (Fortune Investors Guide). In n years from 1990 the average cost will be 29,930(1.05)n dollars. Find the projected cost for a one-year stay in 2005. $62,222 109. Soaring cost of nursing care. Some economists project that the average cost of a one-year stay in a nursing home n years from 1990 will be 29,930(1.08)n dollars. How much more would you pay for a one-year stay in 2005 using this expression rather than the expression in the last exercise? $32,721 GET TING MORE INVOLVED 110. Discussion. Evaluate 5(5(5 3 6) 4) 7 and 3 53 6 52 4 5 7. Explain why these two expressions must have the same value. 111. Cooperative learning. Find some examples of algebraic expressions that are not mentioned in this text and explain to your class what they are used for. PROPERTIES OF THE REAL NUMBERS You know that the price of a hamburger plus the price of a Coke is the same as the price of a Coke plus the price of a hamburger. But, do you know which property of the real numbers is at work in this situation? In arithmetic we may be unaware when to use properties of the real numbers, but in algebra we need a better understanding of those properties. In this section we will study the properties of the basic operations on the set of real numbers. Commutative Properties We get the same result whether we evaluate 3 7 or 7 3. With multiplication, we have 4 5 5 4. These examples illustrate the commutative properties. 32 (1-32) Chapter 1 The Real Numbers Commutative Property of Addition For any real numbers a and b, a b b a. Commutative Property of Multiplication For any real numbers a and b, ab ba. In writing the product of a number and a variable, it is customary to write the number first. We write 3x rather than x3. In writing the product of two variables, it is customary to write them in alphabetical order. We write cd rather than dc. Addition and multiplication are commutative operations, but what about subtraction and division? Because 7 3 4 and 3 7 4, subtraction is not commutative. To see that division is not commutative, consider the amount each person gets when a $1 million lottery prize is divided between two people and when a $2 prize is divided among 1 million people. Associative Properties study tip If you need help, don’t hesitate to get it. Math has a way of building upon the past. What you learn today will be used tomorrow, and what you learn tomorrow will be used the day after. If you don’t straighten out problems immediately, then you can get hopelessly lost. If you are having trouble, see your instructor to find what help is available. Consider the expression 2 3 7. Using the order of operations, we add from left to right to get 12. If we first add 3 and 7 to get 10 and then add 2 and 10, we also get 12. So (2 3) 7 2 (3 7). Now consider the expression 2 3 5. Using the order of operations, we multiply from left to right to get 30. However, we can first multiply 3 and 5 to get 15 and then multiply by 2 to get 30. So (2 3) 5 2 (3 5). These examples illustrate the associative properties. Associative Property of Addition For any real numbers a, b, and c, (a b) c a (b c). Associative Property of Multiplication For any real numbers a, b, and c, (ab)c a(bc). Consider the expression 4 9 8 5 8 6 13. According to the accepted order of operations, we could evaluate this expression by computing from left to right. However, if we use the definition of subtraction, we can rewrite this expression as 4 (9) 8 (5) (8) 6 (13). The commutative and associative properties of addition allow us to add these numbers in any order we choose. A good way to add them is to add the positive numbers, 1.5 Properties of the Real Numbers (1-33) 33 add the negative numbers, and then combine the two totals: 4 8 6 (9) (5) (8) (13) 18 (35) 17 For speed we usually do not rewrite the expression. We just sum the positive numbers and sum the negative numbers, and then combine their totals. E X A M P L E 1 Using commutative and associative properties Evaluate. a) 4 7 10 5 b) 6 5 9 7 2 5 8 Solution a) 4 7 10 5 14 (12) 2 ↑ Sum of the positive numbers ↑ Sum of the negative numbers b) 6 5 9 7 2 5 8 18 (24) 6 ■ Not all operations are associative. Using subtraction, for example, we have (8 4) 1 8 (4 1) because (8 4) 1 3 and 8 (4 1) 5. For division we have (8 4) 2 8 (4 2) because (8 4) 2 1 and 8 (4 2) 4. So subtraction and division are not associative. Distributive Property helpful hint Imagine a parade in which 6 rows of horses are followed by 4 rows of horses with 3 horses in each row. There are 10 rows of 3 horses or 30 horses, or there are 18 horses followed by 12 horses for a total of 30 horses. Using the order of operations, we evaluate the product 3(6 4) first by adding 6 and 4 and then multiplying by 3: 3(6 4) 3 10 30 Note that we also have 3 6 3 4 18 12 30. Therefore 3(6 4) 3 6 3 4. Note that multiplication by 3 from outside the parentheses is distributed over each term inside the parentheses. This example illustrates the distributive property. Distributive Property For any real numbers a, b, and c, a(b c) ab ac. 34 (1-34) Chapter 1 The Real Numbers Because subtraction is defined in terms of addition, multiplication distributes over subtraction as well as over addition. For example, 3(x 2) 3(x (2)) 3x (6) 3x 6. Because multiplication is commutative, we can write the multiplication before or after the parentheses. For example, (y 6)3 3(y 6) 3y 18. The distributive property is used in two ways. If we start with the product 5(x 4) and write 5(x 4) 5x 20, we are writing a product as a sum. We are removing the parentheses. If we start with the difference 6x 18 and write 6x 18 6(x 3), we are using the distributive property to write a difference as a product. E X A M P L E study 2 tip Get to class early so that you are relaxed and ready to go when class starts. Collect your thoughts and get your questions ready. If your instructor arrives early, you might be able to have your questions answered before class. Take responsibility for your education. Many come to learn, but not all learn. Using the distributive property Use the distributive property to rewrite each sum or difference as a product and each product as a sum or difference. a) 9x 9 b) b(2 a) c) 3a ac d) 2(x 3) Solution a) 9x 9 9(x 1) b) b(2 a) 2b ab Note that b 2 2b by the commutative property. c) 3a ac a(3 c) d) 2(x 3) 2x (2)(3) 2x (6) 2x 6 ■ Identity Properties The numbers 0 and 1 have special properties. Addition of 0 to a number does not change the number, and multiplication of a number by 1 does not change the number. For this reason, 0 is called the additive identity and 1 is called the multiplicative identity. Additive Identity Property For any real number a, a 0 0 a a. Multiplicative Identity Property For any real number a, a 1 1 a a. 1.5 Properties of the Real Numbers (1-35) 35 Inverse Properties The ideas of additive inverses and multiplicative inverses were introduced in Section 1.3. Every real number a has a unique additive inverse or opposite, a, such that a (a) 0. Every nonzero real number a also has a unique multiplicative inverse (reciprocal), written 1, such that a1 1. For rational numbers the multiplia a cative inverse is easy to find. For example, the multiplicative inverse of 2 is 5 because 5 2 2 5 10 1. 5 2 10 Additive Inverse Property For any real number a, there is a unique number a such that calculator a (a) a a 0. close-up Multiplicative Inverse Property Most scientific calculators have a key labeled 1x, which gives the reciprocal of the number on the display. Graphing calculators do not have a reciprocal key, but you can find reciprocals as shown here. For any nonzero real number a, there is a unique number 1 such that a 1 1 a a 1. a a Reciprocals are used in problems involving rates. For example, if Brandon washes one car in 1 of an hour, then he is washing cars at the rate of 11 or 3 3 3 cars hour (3 cars per hour). If Gilda washes one car in 1 of an hour, then she is 4 washing at the rate of 11 or 4 cars hour. In general, if one task is completed in 4 x hours, then the rate is 1 tasks hour. If Brandon and Gilda maintain the same rates x when working together, then their rate together is the sum of their individual rates, or 7 cars hour. E X A M P L E 3 Work rates An old computer system can process one water bill in 0.002 hour. A new computer system can process one water bill in 0.00125 hour. If the old system is used simultaneously with the new one, then at what rate will the processing of the water bills be accomplished? Solution Since the old system does one bill in 0.002 hour, its rate is 1 bills per hour. Since 0.002 the new system does one bill in 0.00125 hour, its rate is 1 bills per hour. Their 0.00125 rate when working together is the sum of their individual rates: 1 1 1300 0.002 0.00125 They are working together at the rate of 1300 bills per hour. ■ 36 (1-36) Chapter 1 The Real Numbers Multiplication Property of Zero Zero has a property that no other number has. Multiplication involving zero always results in zero. It is the multiplication property of zero that prevents 0 from having a reciprocal. Multiplication Property of Zero For any real number a, 0 a a 0 0. E X A M P L E 4 Recognizing properties Identify the property that is illustrated in each case. 1 a) 5 9 9 5 b) 3 1 3 c) 1 865 865 d) 3 (5 a) (3 5) a e) 4x 6x (4 6)x f) 7 (x 3) 7 (3 x) g) 4567 0 0 h) 239 0 239 i) 8 8 0 j) 4(x 5) 4x 20 Solution a) Commutative property of multiplication c) Multiplicative identity property e) Distributive property g) Multiplication property of zero i) Additive inverse property WARM-UPS b) Multiplicative inverse property d) f) h) j) Associative property of addition Commutative property of addition Additive identity property ■ Distributive property True or false? Explain your answer. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Addition is a commutative operation. True 8 (4 2) (8 4) 2 False 10 2 2 10 False 5 3 3 5 False 10 (7 3) (10 7) 3 False 4(6 2) (4 6) (4 2) False The multiplicative inverse of 0.02 is 50. True Division is not an associative operation. True 3 2x 5x for any value of x. False A machine that washes one car in 0.04 hour is washing at the rate of 25 cars per hour. True 1.5 1. 5 Properties of the Real Numbers (1-37) 37 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What are the commutative properties? The commutative property of addition says that a b b a and the commutative property of multiplication says that a b b a. 2. What are the associative properties? The associative property of addition says that (a b) c a (b c). 3. What is the difference between the commutative property of addition and the associative property of addition? The commutative property of addition says that you get the same result when you add two numbers in either order. The associative property of addition deals with which two numbers are added first when adding three numbers. 4. What is the distributive property? The distributive property says that a(b c) ab ac. 5. Why is 0 called the additive identity? Zero is the additive identity because adding zero to a number does not change the number. 6. Why is 1 called the multiplicative identity? One is the multiplicative identity because multiplying a number by 1 does not change the number. Evaluate. See Example 1. 7. 9 4 6 10 1 8. 3 4 12 9 2 9. 6 10 5 8 7 14 10. 5 11 6 9 12 2 1 11. 4 11 6 8 13 20 24 12. 8 12 9 15 6 22 3 33 13. 3.2 1.4 2.8 4.5 1.6 1.7 14. 4.4 5.1 3.6 2.3 8.1 8.7 15. 3.27 11.41 5.7 12.36 5 19.8 16. 4.89 2.1 7.58 9.06 5.34 4.03 Use the distributive property to rewrite each sum or difference as a product and each product as a sum or difference. See Example 2. 17. 4(x 6) 4x 24 18. 5(a 1) 5a 5 19. 2m 10 2(m 5) 20. 3y 9 3(y 3) 21. a(3 t) 3a at 22. b(y w) by bw 23. 2(w 5) 2w 10 24. 4(m 7) 4m 28 25. 2(3 y) 6 2y 26. 5(4 p) 20 5p 27. 5x 5 5(x 1) 28. 3y 3 3(y 1) 29. 1(2x y) 2x y 30. 1(4y w) 4y w 31. 3(2w 3y) 6w 9y 32. 4(x 6) 4x 24 33. 3y 15 3(y 5) 34. 5x 10 5(x 2) 35. 3a 9 3(a 3) 36. 7b 49 7(b 7) 1 37. (4x 8) 2x 4 2 1 39. (2x 4) x 2 2 1 38. (3x 6) x 2 3 1 40. (9x 3) 3x 1 3 Find the multiplicative inverse (reciprocal) of each number. 1 1 41. 2 42. 3 43. 1 1 2 3 1 1 46. 8 44. 1 1 45. 6 6 8 4 10 47. 0.25 4 48. 0.75 49. 0.7 3 7 5 10 5 50. 0.9 51. 1.8 52. 2.6 13 9 9 Use a calculator to evaluate each expression. Round answers to four decimal places. 1 1 53. 0.6200 2.3 5.4 1 4.3 55. 0.7326 1 1 5.6 7.2 57. 58. 59. 60. 1 1 54. 0.1433 13.5 4.6 1 1 4.5 5.6 56. 0.0639 1 1 3.2 2.7 Solve each problem. See Example 3. Fastest airliner. The world’s fastest airliner, the Concorde, travels one mile in 0.0006897 hour and carries 128 passengers (The Doubleday Almanac). Find its rate in miles per hour. 1,450 mph Fastest jet plane. The U.S. Lockheed SR-71 is the world’s fastest jet plane. The SR-71 can travel one mile in 0.000456 hour (The Doubleday Almanac). Find its rate in miles per hour. 2,193 mph Who’s got the button. A small clothing factory has three workers who attach buttons. Rita, Mary, and Sam can attach a single button in 0.01 hour, 0.02 hour, and 0.015 hour, respectively. At what hourly rate are they attaching buttons when working simultaneously? 217 buttons per hour Modern art. Emilio can paint the exterior of a certain house in 36.5 hours. Alex can paint the same house in 30 hours. If they work together without interfering with each other, then at what hourly rate will the house be painted? 0.06 house per hour Name the property that is illustrated in each case. See Example 4. 61. 3 x x 3 Commutative property of addition 62. x 5 5x Commutative property of multiplication 63. 5(x 7) 5x 35 Distributive property 64. a(3b) (a 3)b Associative property of multiplication 38 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. (1-38) Chapter 1 The Real Numbers 3(xy) (3x)y Associative property of multiplication 3(x 1) 3x 3 Distributive property 4(0.25) 1 Multiplicative inverse property 0.3 9 9 0.3 Commutative property of addition y3x xy3 Commutative property of multiplication 0 52 0 Multiplication property of zero 1 x x Multiplicative identity property (0.1)(10) 1 Multiplicative inverse property 2x 3x (2 3)x Distributive property 8 0 8 Additive identity property 7 (7) 0 Additive inverse property 1 y y Multiplicative identity property (36 79)0 0 Multiplication property of zero 5x 5 5(x 1) Distributive property xy x x(y 1) Distributive property ab 3ac a(b 3c) Distributive property Complete each statement using the property named. 81. 5 w _____, commutative property of addition w5 82. 2x 2 _____, distributive property 2(x 1) 83. 5(xy) ____, associative property of multiplication (5x)y 1 84. x _____, commutative property of addition 2 1 x 2 1.6 In this section 1 1 1 85. x _____, distributive property (x 1) 2 2 2 86. 3(x 7) _____, distributive property 3x 21 87. 6x 9 _____, distributive property 3(2x 3) 88. (x 7) 3 _____, associative property of addition x (7 3) 89. 8(0.125) _____, multiplicative inverse property 1 90. 1(a 3) _____, distributive property a 3 91. 0 5(_____), multiplication property of zero 0 92. 8 (_____) 8, multiplicative identity property 1 93. 0.25 (_____) 1, multiplicative inverse property 4 94. 45(1) _____, multiplicative identity property 45 GET TING MORE INVOLVED 95. Discussion. Does the order in which your groceries are placed on the checkout counter make any difference in your total bill? Which properties are at work here? 96. Discussion. Suppose that you just bought 10 grocery items and paid a total bill that included 6% sales tax. Would there be any difference in your total bill if you purchased the items one at a time? Which property is at work here? USING THE PROPERTIES The properties of the real numbers can be helpful when we are doing computations. In this section we will see how the properties can be applied in arithmetic and algebra. ● Using the Properties in Computation ● Like Terms ● Combining Like Terms Consider the product of 36 and 200. Using the associative property of multiplication, we can write ● Products and Quotients (36)(200) (36)(2 100) (36 2)(100). ● Removing Parentheses E X A M P L E Using the Properties in Computation To find this product mentally, first multiply 36 by 2 to get 72, then multiply 72 by 100 to get 7200. 1 Using properties in computation Evaluate each expression mentally by using an appropriate property. 1 c) 7 45 3 45 a) 536 25 75 b) 5 426 5