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6749_Lial_CH01_pp001-042.qxd 30 CHAPTER 1 1/12/10 4:47 PM Page 30 Review of the Real Number System 96. An approximation of federal spending on education in billions of dollars from 2001 through 2005 can be obtained using the e xpression y = 9.0499x - 18,071.87, where x represents the year. (Source: U.S. Department of the Treasury.) (a) Use this expression to complete the table. Round answers to the nearest tenth. Year Education Spending (in billions of dollars) 2001 37.0 2002 46.0 2003 2004 2005 (b) How has the amount of federal spending on education changed from 2001 to 2005? 1.4 Properties of Real Numbers The study of an y object is simplif ied when we know the proper ties of the object. F or example, a property of water is that it freezes when cooled to 0°C. Knowing this helps us to predict the behavior of water. The study of numbers is no different. The basic properties of real numbers studied in this section reflect results that occur consistently in work with numbers, so they have been generalized to apply to expressions with variables as well. OBJECTIVES 1 Use the distributive property. 2 Use the inverse properties. 3 Use the identity properties. OBJECTIVE 1 4 Use the commutative and associative properties. This idea is illustrated by the divided rectangle in Figure 17. Similarly, and so 5 2 2 Area of left part is 2 . 3 = 6. Area of right part is 2 . 5 = 10. Area of total rectangle is 2(3 + 5) = 16. FI GU RE 1 7 2(3 + 5) = 2 # 8 = 16 2 # 3 + 2 # 5 = 6 + 10 = 16, 2(3 + 5) = 2 # 3 + 2 # 5. and so 5 Use the multiplication property of 0. 3 Use the distributive property. Notice that - 435 + (- 3)4 = - 4(2) = - 8 - 4(5) + ( - 4)(- 3) = - 20 + 12 = - 8, - 435 + (- 3)4 = - 4(5) + (- 4)(- 3). These examples are generalized to all real numbers as the distributive property of multiplication with respect to addition, or simply the distributive property. Distributive Property For any real numbers a, b, and c, a(b ⴙ c) ⴝ ab ⴙ ac and (b ⴙ c)a ⴝ ba ⴙ ca. The distributive property can also be written ab ⴙ ac ⴝ a(b ⴙ c) and ba ⴙ ca ⴝ (b ⴙ c)a 6749_Lial_CH01_pp001-042.qxd 1/12/10 4:47 PM Page 31 SECTION 1.4 Properties of Real Numbers 31 and can be extended to more than two numbers as well. a(b ⴙ c ⴙ d ) ⴝ ab ⴙ ac ⴙ ad The distrib utive pr operty pr ovides a w ay to r ewrite a pr oduct a(b ⴙ c) as a sum ab ⴙ ac or a sum as a pr oduct. When w e re write a(b + c) as ab + ac, we sometimes refer to the process as “removing” or “clearing” parentheses. EXAMPLE 1 Using the Distributive Property Use the distributive property to rewrite each expression. (a) 3(x + y) = 3x + 3y Use the first form of the property to rewrite the given product as a sum. (b) - 2(5 + k) = - 2(5) + ( - 2)(k) = - 10 - 2k (c) 4x + 8x Use the second form of the property to rewrite the given sum as a product. = (4 + 8)x = 12x (d) 3r - 7r = 3r + ( - 7r) Definition of subtraction = 33 + ( - 7)4r Distributive property = - 4r (e) 5p + 7q Because there is no common number or v ariable here, we cannot use the distributive property to rewrite the expression. (f) 6(x + 2y - 3z) = 6x + 6(2y) + 6(- 3z) = 6x + 12y - 18z NOW TRY Exercises 11, 13, 15, and 19. The distributive property can also be used for subtraction (Example 1(d)), so a(b ⴚ c) ⴝ ab ⴚ ac. OBJECTIVE 2 Use the inverse properties. In Section 1.1, we saw that the additive inverse (or opposite) of a number a is - a and that additive inverses have a sum of 0. 5 and - 5, - 1 2 and 1 , 2 - 34 and 34 Additive inverses (sum of 0) In Section 1.2, we saw that the multiplicative inverse (or reciprocal) of a number a is 1 a (where a Z 0) and that multiplicative inverses have a product of 1. 5 and 1 , 5 - 1 2 and - 2, 3 4 and 4 3 Multiplicative inverses (product of 1) This discussion leads to the inverse properties of addition and multiplication, w hich can be extended to the real numbers of algebra. 6749_Lial_CH01_pp001-042.qxd 32 CHAPTER 1 1/12/10 4:47 PM Page 32 Review of the Real Number System Inverse Properties For any real number a, a ⴙ (ⴚa) ⴝ 0 1 a # ⴝ1 a and and ⴚa ⴙ a ⴝ 0 1 # a ⴝ 1 (a ⴝ 0). a The inverse properties “undo” addition or multiplication. Think of putting on your shoes when you get up in the mor ning and then taking them of f before you go to bed at night. These are inverse operations that undo each other. OBJECTIVE 3 Use the identity properties. The numbers 0 and 1 each have a special property. Zero is the onl y number that can be added to an y number to get that number . Adding 0 to any number leaves the identity of the number unchanged. F or this reason, 0 is called the identity element f or addition, or the additive identity. In a similar w ay, multiplying any number b y 1 lea ves the identity of the number unchanged , so 1 is the identity element for multiplication, or the multiplicative identity. The identity properties summarize this discussion and extend these properties from arithmetic to algebra. Identity Properties For any real number a, aⴙ0ⴝ0ⴙaⴝa a # 1 ⴝ 1 # a ⴝ a. The identity proper ties leave the identity of a real number unchanged. Think of a child wearing a costume on Hallo ween. The child’s appearance is changed , but his or her identity is unchanged. EXAMPLE 2 Using the Identity Property 1 # a ⴝ a Simplify each expression. (a) 12m = = = + m 12m + 1m (12 + 1)m 13m (b) y + = = = y 1y + 1y (1 + 1)y 2y Identity property; m = 1 # m, or 1m Distributive property Add inside parentheses. Identity property Distributive property Add inside parentheses. (c) Multiply each term by ⴚ1. Be careful with signs. - (m - 5n) = - 1(m - 5n) = - 1(m) + ( - 1)(- 5n) = - m + 5n Identity property Distributive property Multiply. NOW TRY Exercises 21 and 23. 6749_Lial_CH01_pp001-042.qxd 1/12/10 4:47 PM Page 33 Properties of Real Numbers SECTION 1.4 - 7y -7 34r 3 34 - 26x5yz4 - k = - 1k - 26 Expressions such as 12m and 5n from Example 2 are examples of terms. A term is a number or the product of a number and one or more v ariables raised to powers. The numerical factor in a ter m is called the numerical coefficient, or just the coefficient. Some examples of terms and their coefficients are shown in the table in the margin. Terms with exactly the same variables raised to exactly the same powers are called like terms. Some examples of like terms are -1 r = 1r 1 3x 3 = x 8 8 3 8 x 1x 1 = = x 3 3 3 1 3 5p and - 21p - 6x 2 and 9x 2. Like terms Some examples of unlike terms are 3m and 16x 7y 3 and - 3y 2. Unlike terms OBJECTIVE 4 Use the commutative and associative properties. Simplifying expressions as in parts (a) and (b) of Example 2 is called combining like terms. Only like terms may be combined. To combine like terms in an expression such as - 2m + 5m + 3 - 6m + 8, we need two more properties. From arithmetic, we know that 3 + 9 = 12 and 9 + 3 = 12 3 9 = 27 and 9 # 3 = 27. # The order of the numbers being added or multiplied does not matter. The same answers result. Also, (5 + 7) + 2 = 12 + 2 = 14 5 + (7 + 2) = 5 + 9 = 14, and (5 # 7) # 2 = 35 # 2 = 70 5(7 # 2) = 5 # 14 = 70. The way in which the numbers being added or multiplied are g rouped does not matter. The same answers result. These arithmetic examples can be extended to algebra. Commutative and Associative Properties For any real numbers a, b, and c, aⴙbⴝbⴙa and ab ⴝ ba. Commutative properties Interchange the order of the two terms or factors. Also, and a ⴙ (b ⴙ c) ⴝ (a ⴙ b) ⴙ c a(bc) ⴝ (ab)c. ⎧ ⎨ ⎩ Numerical Coefficient ⎧ ⎨ ⎩ Term 33 Associative properties Shift parentheses among the three ter ms or factors; the order stays the same. The commutative properties are used to change the order of the terms or factors in an expression. Think of commuting from home to w ork and then from w ork to home. 6749_Lial_CH01_pp001-042.qxd 34 CHAPTER 1 1/12/10 4:47 PM Page 34 Review of the Real Number System The associative proper ties are used to regroup the ter ms or f actors of an e xpression. Remember, to associate is to be part of a group. EXAMPLE 3 Using the Commutative and Associative Properties Simplify - 2m + 5m + 3 - 6m + 8. - 2m + 5m + 3 - 6m + 8 = (- 2m + 5m) + 3 - 6m + 8 = ( - 2 + 5)m + 3 - 6m + 8 = 3m + 3 - 6m + 8 Order of operations Distributive property Add inside parentheses. By the order of operations, the ne xt step w ould be to add 3m and 3, but the y are unlike ter ms. To get 3 m and - 6m together, use the associati ve and commutati ve properties. Be gin b y inser ting parentheses and brack ets according to the order of operations. = 3(3m + 3) - 6m4 + 8 = 33m + (3 - 6m)4 + 8 Associative property = 33m + (- 6m + 3)4 + 8 = 3(3m + 3 - 6m4) + 34 + 8 Commutative property Associative property = ( - 3m + 3) + 8 Combine like terms. = - 3m + (3 + 8) Associative property = - 3m + 11 Add. In practice, many of these steps are not written down, but you should realize that the commutati ve and associati ve proper ties are used w henever the ter ms in an expression are rearranged to combine like terms. NOW TRY EXAMPLE 4 Using the Properties of Real Numbers Simplify each expression. (a) 5y - 8y - 6y + 11y = (5 - 8 - 6 + 11)y Distributive property = 2y Combine like terms. (b) 3x + 4 - 5(x + 1) - 8 Be careful with signs. (c) 8 = = = (3m 8 8 6 - = 3x + 4 - 5x - 5 - 8 Distributive property = 3x - 5x + 4 - 5 - 8 Commutative property = - 2x - 9 Combine like terms. + 2) 1(3m + 2) 3m - 2 3m Identity property Distributive property Combine like terms. Exercise 27. 6749_Lial_CH01_pp001-042.qxd 1/12/10 4:47 PM Page 35 SECTION 1.4 (d) 3x(5)( y) = 33x(5)4 y = 33(x # 5)4 y = 33(5x)4 y = 3(3 # 5)x4 y = (15x)y = 15(xy) = 15xy Properties of Real Numbers 35 Order of operations Associative property Commutative property Associative property Multiply. Associative property As previously mentioned, many of these steps are not usuall y written out. NOW TRY Exercises 29 and 31. CAUTION Be careful. The distributive proper ty does not appl y in Example 4(d), because there is no addition involved. (3x)(5)( y) Z (3x)(5) # (3x)( y) OBJECTIVE 5 Use the multiplication property of 0. The additive identity prop- erty gives a special proper ty of 0, namely, that a + 0 = a for any real number a. The multiplication property of 0 gives a special property of 0 that involves multiplication. The product of any real number and 0 is 0. Multiplication Property of 0 For any real number a, a # 0ⴝ0 0 # a ⴝ 0. and 1.4 Exercises NOW TRY Exercise Multiple Choice Choose the correct response in Exercises 1–4. 2. The identity element for multiplication is 1 A. - a B. 0 C. 1 D. . a 4. The multiplicati ve in verse of a, w here a Z 0, is 1 A. - a B. 0 C. 1 D. . a 1. The identity element for addition is 1 A. - a B. 0 C. 1 D. . a 3. The additive inverse of a is 1 A. - a B. 0 C. 1 D. . a Fill in the Blanks Complete each statement. 5. The multiplication proper ty of 0 sa ys that the is . 6. The commutative property is used to change the 7. The associative property is used to change the 8. Like terms are terms with the 10. The coefficient in the term of two terms or factors. of three terms or factors. variables raised to the 9. When simplifying an expression, only - 8yz 2 of 0 and an y real number is terms can be combined. . powers. 6749_Lial_CH01_pp001-042.qxd 36 CHAPTER 1 1/12/10 4:47 PM Page 36 Review of the Real Number System Simplify each expression. See Examples 1 and 2. 11. 2(m + p) 12. 3(a + b) 13. - 12(x - y) 14. - 10( p - q) 15. 5k + 3k 16. 6a + 5a 17. 7r - 9r 18. 4n - 6n 19. - 8z + 4w 20. - 12k + 3r 21. a + 7a 22. s + 9s 23. - (2d - f ) 24. - (3m - n) Simplify each expression. See Examples 1–4. 25. - 12y + 4y + 3 + 2y 26. - 5r - 9r + 8r - 5 27. - 6p + 5 - 4p + 6 + 11p 28. - 8x - 12 + 3x - 5x + 9 29. 3(k + 2) - 5k + 6 + 3 30. 5(r - 3) + 6r - 2r + 4 31. - 2(m + 1) - (m - 4) 32. 6(a - 5) - (a + 6) 33. 0.25(8 + 4p) - 0.5(6 + 2p) 34. 0.4(10 - 5x) - 0.8(5 + 10x) 35. - (2p + 5) + 3(2p + 4) - 2p 36. - (7m - 12) - 2(4m + 7) - 8m 37. 2 + 3(2z - 5) - 3(4z + 6) - 8 38. - 4 + 4(4k - 3) - 6(2k + 8) + 7 Fill in the Blanks Complete eac h statement so that the indicated pr Simplify each answer if possible. 39. 5x + 8x = 41. 5(9r) = 43. 5x + 9y = 45. 1 # 7 = (distributive property) (associative property) (commutative property) (identity property) 1 1 47. - ty + ty = 4 4 49. 8(- 4 + x) = (inverse property) (distributive property) 51. 0(0.875x + 9y - 88z) = (multiplication property of 0) 40. 9y - 6y = operty is illustr ated. (distributive property) 42. - 4 + (12 + 8) = 44. - 5 # 7 = (commutative property) 46. - 12x + 0 = 9 8 48. - (- ) = 8 9 50. 3(x - y + z) = 52. 0(35t 2 (associative property) (identity property) (inverse property) (distributive proper ty) - 8t + 12) = (multiplication property of 0) 53. Concept Check Give an “everyday” example of a commutative operation and of an operation that is not commutative. 54. Concept Check Give an “everyday” example of inverse operations. The distributive property can be used to mentall y perform calculations. For example, calculate 38 # 17 + 38 # 3 as follows: 38 # 17 + 38 # 3 = 38(17 + 3) = 38(20) = 760. Distributive property Add inside parentheses. Multiply. Use the distributive property to calculate each value mentally. 55. 96 # 19 + 4 # 19 56. 27 # 60 + 27 # 40 58. 8.75(15) - 8.75(5) 59. 4.31(69) + 4.31(31) 3 3 - 8 # 2 2 8 8 60. (17) + (13) 5 5 57. 58 #