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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
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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.
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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.
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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.
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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.
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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.
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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
#