Download Properties of Signed Numbers

1.2
Properties of Signed Numbers
1.2
OBJECTIVES
1. Recognize applications of the commutative
property
2. Recognize applications of the associative property
3. Recognize applications of the distributive property
All that we do in algebra is based on the rules for the operations introduced in Section 1.1.
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,
NOTE All integers, decimals,
and fractions that we see in this
course are real numbers.
1. a b b a
Commutative property of addition
2. a b b a
Commutative property of multiplication
Example 1
Identifying the Commutative Properties
(a) 5 9 9 5
and
x77x
These are applications of the commutative property of addition.
(b) 5 9 9 5
This is an application of the commutative property of multiplication.
CHECK YOURSELF 1
Identify the property being applied.
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(a) 7 3 3 7
(c) a 4 4 a
(b) 7 3 3 7
(d) x 2 2 x
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: The Associative Properties
If a, b, and c are any numbers,
1. a (b c) (a b) c
Associative property of addition
2. a (b c) (a b) c
Associative property of multiplication
63
64
CHAPTER 1
THE LANGUAGE OF ALGEBRA
Example 2
Demonstrating the Associative Properties
(a) Show that 2 (3 8) (2 3) 8.
Add first.
Add first.
2 11
58
13
13
So
2 (3 8) (2 3) 8
(b) Show that
1
1
(6 5) 6 5.
3
3
3 6 5
1
1
(6 5)
3
Multiply first.
Multiply first.
1
(30)
3
(2) 5
10
10
So
1
1
(6 5) 6 5
3
3
CHECK YOURSELF 2
Show that the following statements are true.
(a) 3 (4 7) (3 4) 7
(b) 3 (4 7) (3 4) 7
(c)
5 10 4 5 (10 4)
1
1
The distributive property involves addition and multiplication together. We can
illustrate this property with an application.
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Section 0.3, we always do the
operation in the parentheses
first.
(2 3) 8
2 (3 8)
NOTE Remember, as we saw in
PROPERTIES OF SIGNED NUMBERS
SECTION 1.2
Suppose that we want to find the total of the two areas shown in the following figure.
REMEMBER: The area of a
rectangle is the product of its
length and width:
30
ALW
Area 1
10
Area 2
We can find the total area by multiplying
the length by the overall width, which is
found by adding the two widths.
30
We can find the total area as a sum
of the two areas.
(Area 1)
Length Width
Length Overall Width
[or]
(10 15)
30 10
30 25
300 450
750
750
(Area 2)
Length Width
15
30 15
So
30 (10 15) 30 10 30 15
This leads us to the following property.
Rules and Properties: The Distributive Property
NOTE Notice the pattern.
If a, b, and c are any numbers,
a(b c) a b a c
a(b c) a b a c
We “distributed” the
multiplication “over” the
addition.
and
(b c)a b a c a
Example 3
Using the Distributive Property
Use the distributive property to simplify (remove the parentheses in) the following.
NOTE 5(3 4) 5 7 35
(a) 5(3 4)
or
5 3 5 4 15 20 35
5(3 4) 5 3 5 4
15 20 35
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NOTE Because the variables
are different, 8x 8y cannot be
simplified further.
65
(b) 8(x y)
8(x y) 8x 8y
(c) 2(3x 5)
2(3x 5) 2 3x 2 5
6x 10
66
CHAPTER 1
THE LANGUAGE OF ALGEBRA
NOTE It is also true that
(d)
1
1
1
(9 12) 9 12
3
3
3
347
CHECK YOURSELF 3
Use the distributive property to simplify (remove the parentheses).
(a) 4(6 7)
(c) 3(5a 7)
(b) 9(m n)
1
(d) (10 15)
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) 3(x 2) 3x 3 2 demonstrates the distributive property.
(b) 2 (3 5) (2 3) 5 demonstrates the associative property of addition.
(c) 3 5 5 3 demonstrates the commutative property of multiplication.
CHECK YOURSELF 4
Name the property demonstrated.
(a) 2 (3 5) (2 3) 5
(b) 4(a b) 4a 4b
(c) x 8 8 x
CHECK YOURSELF ANSWERS
1. (a) Commutative property of addition; (b) commutative property of multiplication;
(c) commutative property of addition; (d) commutative property of multiplication
2. (a) 3 (4 7) 3 11 14
(b) 3 (4 7) 3 28 84
(3 4) 7 7 7 14
(3 4) 7 12 7 84
(c)
5 10 4 2 4 8
1
1
1
(10 4) 40 8
5
5
3. (a) 4 6 4 7 24 28 52; (b) 9m 9n; (c) 15a 21;
1
1
(d) 10 15 2 3 5
5
5
4. (a) Associative property of multiplication; (b) distributive property;
(c) commutative property of addition
© 2001 McGraw-Hill Companies
1
1
(9 12) (21) 7
3
3
Name
1.2
Exercises
Section
Date
Identify the property that is illustrated by each of the following statements.
ANSWERS
1. 5 9 9 5
2. 6 3 3 6
1.
3. 2 (3 5) (2 3) 5
4. 3 (5 6) (3 5) 6
2.
3.
5. 10 5 5 10
6. 8 4 4 8
4.
5.
7. 8 12 12 8
8. 6 2 2 6
6.
7.
9. (5 7) 2 5 (7 2)
10. (8 9) 2 8 (9 2)
8.
9.
11. 9 8 8 9
12. 6 4 4 6
10.
11.
13. 2(3 5) 2 3 2 5
14. 5 (4 6) 5 4 5 6
12.
13.
15. 5 (7 8) (5 7) 8
16. 8 (2 9) (8 2) 9
14.
15.
17. (10 5) 9 10 (5 9)
18. (5 5) 3 5 (5 3)
16.
17.
19. 7 (3 8) 7 3 7 8
20. 5 (6 8) 5 6 5 8
18.
19.
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Verify that each of the following statements is true by evaluating each side of the equation
separately and comparing the results.
21. 7 (3 4) 7 3 7 4
22. 4 (5 1) 4 5 4 1
20.
21.
22.
23.
23. 2 (9 8) (2 9) 8
24. 6 (15 3) (6 15) 3
24.
25.
25. 5 (6 3) (5 6) 3
26. 2 (9 10) (2 9) 10
26.
67
ANSWERS
27.
27. 5 (2 8) 5 2 5 8
28. 3 (10 + 2) = 3 10 + 3 2
29. (3 12) 8 3 (12 8)
30. (8 12) 7 8 (12 7)
31. (4 7) 2 4 (7 2)
32. (6 5) 3 6 (5 3)
28.
29.
30.
31.
32.
33.
33.
1
1
1
(2 6) 2 6
2
2
2
35.
3 6 3 3 6 3
34.
35.
2
1
1
2
1
1
34.
1
1
1
(6 9) 6 9
3
3
3
36.
3
5
1
3
5
1
4
8
2
4
8
2
37. (2.3 3.9) 4.1 2.3 (3.9 4.1)
36.
37.
38. (1.7 4.1) 7.6 1.7 (4.1 7.6)
38.
39.
39.
1
1
(2 8) 2 8
2
2
40.
1
1
(5 3) 5 3
5
5
41.
5 6 3 5 6 3
42.
4
21 8
4 21
8
7
16 3
7 16
3
40.
41.
42.
3
5
4
3
5
4
43.
43. 2.5 (4 5) (2.5 4) 5
44.
45.
44. 4.2 (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.
46.
47.
45. 2(3 5)
46. 5(4 6)
47. 3(x 5)
48. 5( y 8)
49. 4(w v)
50. 7(c d)
51. 2(3x 5)
52. 3(7a 4)
49.
50.
51.
52.
53.
54.
53.
68
1
(15 9)
3
54.
1
(36 24)
6
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48.
ANSWERS
Use the properties of addition and multiplication to complete each of the following
statements.
55. 5 7 5
57. (8)(3) (3) (
56. (5 3) 4 5 (
58. 8(3 4) 8 3 )
59. 7(2 5) 7 75
60. 4 (2 4) (
4)
4
2) 4
Use the indicated property to write an expression that is equivalent to each of the
following expressions.
61. 3 7
(commutative property of addition)
62. 2(3 4)
55.
56.
57.
58.
59.
60.
61.
62.
(distributive property)
63.
63. 5 (3 2)
(associative property of multiplication)
64.
64. (3 5) 2
(associative property of addition)
65. 2 4 2 5
(distributive property)
65.
66. 7 9
66.
(commutative property of multiplication)
67.
Evaluate each of the following pairs of expressions. Then answer the given
question.
68.
69.
67. 8 5
and
58
Do you think subtraction is commutative?
70.
71.
68. 12 3
and
3 12
Do you think division is commutative?
72.
69. (12 8) 4
and
12 (8 4)
Do you think subtraction is associative?
70. (48 16) 4
and
48 (16 4)
Do you think division is associative?
and
3632
Do you think multiplication is distributive over subtraction?
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71. 3(6 2)
72.
1
(16 10)
2
and
1
1
16 10
2
2
Do you think multiplication is distributive over subtraction?
In Exercises 73 and 74, complete the statement using
(a) the distributive property,
(b) the commutative property of addition,
(c) the commutative property of multiplication.
69
ANSWERS
73. 5 (3 4) 74. 6 (5 4) 73.
In Exercises 75 to 78, identify the property that is used.
74.
75. 5 (6 7) (5 6) 7
76. 5 (6 7) 5 (7 6)
75.
77. 4 (3 2) 4 (2 3)
78. 4 (3 2) (3 2) 4
76.
Getting Ready for Section 1.3 [Section 1.2]
77.
Find each sum.
78.
(a) 3 (8 9)
(c) (3 8) (9 4)
3
4
(e) 5
15
a.
b.
(b) 6 (12 3)
(d) 15 11 (2 1)
12
2
(f)
27
9
c.
Answers
d.
1. Commutative property of addition
3. Associative property of multiplication
5. Commutative property of multiplication
7. Commutative property of addition
9. Associative property of multiplication
11. Commutative property of multiplication
13. Distributive property
15. Associative property of addition
17. Associative property of addition
19. Distributive property
21. 49 49
23. 19 19
25. 90 90
27. 50 50
29. 23 23
31. 56 56
e.
f.
2
2
3
3
43. 50 50
47. 3x 15
49. 4w 4v
51. 6x 10
53. 8
55. 7
57. 8
59. 2
61. 7 3
63. (5 3) 2
65. 2 (4 5)
67. No
69. No
71. Yes
73. (a) 5 3 5 4
(b) 5 (4 3) (c) (3 4) 5
75. Associative property of addition
77. Commutative property of addition
a. 20
b. 21
c. 24
d. 1
13
2
e.
f.
15
9
7
7
6
6
45. 16
35.
37. 10.3 10.3
39. 8 8
41.
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33. 4 4
70