Properties of Signed Numbers

Transcription

1 .2 Properties of Signed Numbers.2 OBJECTIVES. Recognize applications of the commutative property 2. Recognize applications of the associative property. Recognize applications of the distributive property All that we do in algebra is based on the rules for the operations introduced in Section.. We call these rules properties of the real numbers. In this section we consider those properties that we will use in the remainder of this chapter. The commutative properties tell us that we can add or multiply in any order. Rules and Properties: The Commutative Properties If a and b are any numbers,. a b b a Commutative property of addition 2. a b b a Commutative property of multiplication NOTE All integers, decimals, and fractions that we see in this course are real numbers. Example Identifying the Commutative Properties (a) and x 7 7 x These are applications of the commutative property of addition. (b) This is an application of the commutative property of multiplication. CHECK YOURSELF Identify the property being applied. (a) 7 7 (b) 7 7 (c) a 4 4 a (d) x 2 2 x 200 McGraw-Hill Companies We also want to be able to change the grouping in simplifying expressions. This is possible because of the associative properties. Numbers or variables can be grouped in any manner to find a sum or a product. Rules and Properties: If a, b, and c are any numbers, The Associative Properties. a (b c) (a b) c Associative property of addition 2. a (b c) (a b) c Associative property of multiplication 6

2 64 CHAPTER THE LANGUAGE OF ALGEBRA Example 2 Demonstrating the Associative Properties NOTE Remember, as we saw in Section 0., we always do the operation in the parentheses first. (a) Show that 2 ( 8) (2 ) 8. 2 ( 8) (2 ) 8 Add first. Add first So 2 ( 8) (2 ) 8 (b) Show that (6 5) 6 5. (6 5) Multiply first. 6 5 Multiply first. (0) (2) So (6 5) 6 5 CHECK YOURSELF 2 Show that the following statements are true. (a) (4 7) ( 4) 7 (b) (4 7) ( 4) 7 (c) (0 4) 5 The distributive property involves addition and multiplication together. We can illustrate this property with an application. 200 McGraw-Hill Companies

3 PROPERTIES OF SIGNED NUMBERS SECTION.2 65 REMEMBER: The area of a rectangle is the product of its length and width: A L W Suppose that we want to find the total of the two areas shown in the following figure. 0 0 Area 5 Area 2 We can find the total area by multiplying We can find the total area as a sum the length by the overall width, which is [or] of the two areas. found by adding the two widths. (Area ) (Area 2) Length Overall Width Length Width Length Width 0 (0 5) So 0 (0 5) This leads us to the following property. Rules and Properties: The Distributive Property NOTE Notice the pattern. a(b c) a b a c We distributed the multiplication over the addition. NOTE 5( 4) or If a, b, and c are any numbers, a(b c) a b a c and (b c)a b a c a Example Using the Distributive Property Use the distributive property to simplify (remove the parentheses in) the following. (a) 5( 4) 5( 4) McGraw-Hill Companies NOTE Because the variables are different, 8x 8y cannot be simplified further. (b) 8(x y) 8(x y) 8x 8y (c) 2(x 5) 2(x 5) 2 x 2 5 6x 0

4 66 CHAPTER THE LANGUAGE OF ALGEBRA NOTE It is also true that (9 2) (2) 7 (d) (9 2) CHECK YOURSELF Use the distributive property to simplify (remove the parentheses). (a) 4(6 7) (b) 9(m n) (c) (5a 7) (d) (0 5) 5 Example 4 requires that you identify which property is being demonstrated. Look for patterns that will help you remember each of the properties. Example 4 Identifying Properties Name the property demonstrated. (a) (x 2) x 2 demonstrates the distributive property. (b) 2 ( 5) (2 ) 5 demonstrates the associative property of addition. (c) 5 5 demonstrates the commutative property of multiplication. CHECK YOURSELF 4 Name the property demonstrated. (a) 2 ( 5) (2 ) 5 (b) 4(a b) 4a 4b (c) x 8 8 x CHECK YOURSELF ANSWERS. (a) Commutative property of addition; (b) commutative property of multiplication; (c) commutative property of addition; (d) commutative property of multiplication 2. (a) (4 7) 4 (b) (4 7) ( 4) ( 4) (c) (0 4) (a) ; (b) 9m 9n; (c) 5a 2; (d) (a) Associative property of multiplication; (b) distributive property; (c) commutative property of addition 200 McGraw-Hill Companies

5 Name.2 Exercises Section Date Identify the property that is illustrated by each of the following statements ( 5) (2 ) 5 4. (5 6) ( 5) (5 7) 2 5 (7 2) 0. (8 9) 2 8 (9 2) ( 5) (4 6) (7 8) (5 7) (2 9) (8 2) 9 7. (0 5) 9 0 (5 9) 8. (5 5) 5 (5 ) 9. 7 ( 8) (6 8) ANSWERS McGraw-Hill Companies Verify that each of the following statements is true by evaluating each side of the equation separately and comparing the results ( 4) (5 ) (9 8) (2 9) (5 ) (6 5) (6 ) (5 6) (9 0) (2 9)

6 ANSWERS (2 8) (0 + 2) = ( 2) 8 (2 8) 0. (8 2) 7 8 (2 7) 0... (4 7) 2 4 (7 2) 2. (6 5) 6 (5 ) (2 6) (6 9) (2..9) (.9 4.) 8. (.7 4.) (4. 7.6) (2 8) (5 ) (4 5) (2.5 4) (5 2) (4.2 5) 2 Use the distributive property to remove the parentheses in each of the following expressions. Then simplify your result where possible ( 5) 46. 5(4 6) 47. (x 5) 48. 5( y 8) (w v) 50. 7(c d) 5. 2(x 5) 52. (7a 4) 5. (5 9) 54. (6 24) McGraw-Hill Companies 68

7 ANSWERS Use the properties of addition and multiplication to complete each of the following statements (5 ) 4 5 ( 4) 57. (8)() () ( ) 58. 8( 4) (2 5) (2 4) ( 2) 4 Use the indicated property to write an expression that is equivalent to each of the following expressions (commutative property of addition) 62. 2( 4) (distributive property) 6. 5 ( 2) (associative property of multiplication) 64. ( 5) 2 (associative property of addition) (distributive property) (commutative property of multiplication) Evaluate each of the following pairs of expressions. Then answer the given question and 5 8 Do you think subtraction is commutative? and 2 Do you think division is commutative? (2 8) 4 and 2 (8 4) Do you think subtraction is associative? 70. (48 6) 4 and 48 (6 4) Do you think division is associative? 7. (6 2) and 6 2 Do you think multiplication is distributive over subtraction? 200 McGraw-Hill Companies (6 0) and Do you think multiplication is distributive over subtraction? In Exercises 7 and 74, complete the statement using (a) the distributive property, (b) the commutative property of addition, (c) the commutative property of multiplication. 69

8 ANSWERS ( 4) (5 4) In Exercises 75 to 78, identify the property that is used (6 7) (5 6) (6 7) 5 (7 6) ( 2) 4 (2 ) ( 2) ( 2) a. b. c. d. e. f. Find each sum. (a) (8 9) (b) 6 (2 ) (c) ( 8) (9 4) (d) 5 (2 ) (e) Answers Getting Ready for Section. [Section.2] Commutative property of addition. Associative property of multiplication 5. Commutative property of multiplication 7. Commutative property of addition 9. Associative property of multiplication. Commutative property of multiplication. Distributive property 5. Associative property of addition 7. Associative property of addition 9. Distributive property x w 4v 5. 6x (5 ) (4 5) 67. No 69. No 7. Yes 7. (a) (b) 5 (4 ) (c) ( 4) Associative property of addition 77. Commutative property of addition a. 20 b. 2 c. 24 d. e. 2 f. 5 9 (f) McGraw-Hill Companies 70