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Jan 03, 2016
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Review of the Real Number System
Chapter 1
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1.4
Properties of Real Numbers
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1.4 Properties of Real Numbers
Objectives
1. Use the distributive property.
2. Use the inverse properties.
3. Use the identity properties.
4. Use the commutative and associative
properties.
5. Use the multiplication property of 0.
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Using the Distributive Property
The idea of the distributive property can be illustrated
using rectangles.
1.4 Properties of Real Numbers
3(2 + 5) = 3 • 2 + 3 • 52
3
5
3
Area of left part is 3 • 2 = 6
Area of right part is 3 • 5 = 15
Area of total rectangle is 3(2 + 5) = 21
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Use the Distributive Property
1.4 Properties of Real Numbers
Distributive Property
For any real numbers a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + ca.
The distributive property can also be written as:
ab + ac
ba + ca
= a(b + c)
= (b + c)a
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Use the Distributive Property
1.4 Properties of Real Numbers
The distributive property allows us to rewrite a product as a sum:
or a sum as a product.
–4(8 + (–3)) =
–4(8) + (–4) (–3)
–6(3) + –6(11) =
–6(3 + 11)
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Use the Distributive Property
1.4 Properties of Real Numbers
–6(x + 9) =
4(a + b + c) =
7(3x – 2y + 13) =
–6x + (–6)(9)
4a + 4b + 4c
7(3x + (–2y) + 13)
= 21x + (–14y) + 91
= 21x –14y + 91
= –6x + (–54)
= –6x – 54
Product Sum
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Use the Distributive Property
1.4 Properties of Real Numbers
Sum Product
6w –2w + 5w = 6w + (–2)w + 5w
= (6 + (–2) + 5)w
= 9w
8c – 12c = (8c + (–12c))
= (8 + (–12))c
= –4c
The distributive property can also be used for subtraction:
a(b – c) = ab – ac
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Use the Distributive Property
1.4 Properties of Real Numbers
The distributive property may be used to perform calculations mentally.
Calculate 29 • 92 + 29 • 8.29 • 92 + 29 • 8 =
29(92 + 8)
= 29(100)= 2900
Combining the 92 and 8 makes the problem much easier!
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Using the Inverse Properties
1.4 Properties of Real Numbers
Inverse Properties
For ,
and
and
0 0
1 11 1 ( 0
any real nu
. )
mber
a
a a a a
a a aa a
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Using the Inverse Properties
1.4 Properties of Real Numbers
Complete the following statements.
5 _____ 0
19_____ 0
35
_____ 111
a
b
c
d 0 _____ 1
5
19
3–
– 115
Zero does not have a multiplicative inverse.
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Use the Identity Properties
1.4 Properties of Real Numbers
Identity Properties
For any real numbers a,
a + 0 = 0 + a = a
a · 1 = 1 · a = a.
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Use the Identity Properties
1.4 Properties of Real Numbers
–(3b + b – 7b) =
–1(3 + 1 – 7)b= ((–1)3 + (–1)1 + (–1)(– 7))b= (–3 + (–1) + 7)b
= 3b
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Terms and Like Terms
1.4 Properties of Real Numbers
Terms consist of a number or a product of a number and one or more variables.
2 and 28 227k and 2k y2 and 4y2
Like terms are numbers or numbers times variables raised to exactly the same power. Simplifying expressions is called combining like terms. Only like terms can be combined.
Like Terms
Like Terms
Like Terms
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Use the Commutative Property
1.4 Properties of Real Numbers
The commutative properties are used to change the order of the terms or factors in an expression.
Commutative Properties
For any real numbers a and b,
a + b = b + aand ab = ba.
Interchange the order of the two terms or factors.
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Use the Associative Properties
1.4 Properties of Real Numbers
The associative properties are used to regroup (associate) the terms or factors in an expression, where the order stays the same.
Associative Properties
For any real numbers a, b and c,
a + (b + c) = (a + b) + c
and a(bc) = (ab)c.
Shift parentheses among three terms or factors; order stays the same.
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Use the Commutative and Associative Properties
1.4 Properties of Real Numbers
Simplify.
–5x + 8x + 7 – 9x + 3
= (–5x + 8x) + 7 – 9x + 3
Order of Operations
= (–5 + 8) x + 7 – 9x + 3
Distributive Property
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Use the Commutative and Associative Properties
1.4 Properties of Real Numbers
Continued:
Commutative Property
Associative Property
= [3x + (7 – 9x)] + 3
= [3x + (–9x + 7)] + 3
= [(3x + [–9x]) + 7] + 3
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Use the Commutative and Associative Properties
1.4 Properties of Real Numbers
Continued:
Combine like terms
Associative Property
= (–6x + 7) + 3
= [(3x + [–9x]) + 7] + 3
= –6x + (7 + 3)
= –6x + 10
Add like terms
In actual practice many of these steps are not actually written down, but you should mentally justify each step whether it is written down or not.
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Use the Commutative and Associative Properties
1.4 Properties of Real Numbers
Simplify.
4 –1(3g – 7) + 2g(h) (–3) + g
= 4 –3g + 7 + 2g(h)(–3) + g
= 4 –3g + 7 + (–6gh) + g
Distributive Property
Commutative and Associative Properties; Multiplying
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Use the Commutative and Associative Properties
1.4 Properties of Real Numbers
Continued:
= 4 + 7 –3g + g + (–6gh)
= 4 –3g + 7 + (–6gh) + g Commutative and
Associative Properties
Adding like terms
=11 –2g – 6gh
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Use the Distributive Property with Caution
1.4 Properties of Real Numbers
Contined — A Second Look:
4 –1(3g – 7) + 2g(h) (–3) + g
= 4 –3g + 7 + 2g(h)(–3) + g Distributive property does not
apply since there is no addition or subtraction.
(2g)(h) + (2g)(–3)¹
Distributive property applies here since there is subtraction.