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Sec 1.4 - 1
Chapter 1
Review of the Real Number System
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Sec 1.4 - 2
1.4
Properties of Real Numbers
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Sec 1.4 - 3
1.4 Properties of Real Numbers
Objectives
1.
2.
3.
4.
Use the distributive property.
Use the inverse properties.
Use the identity properties.
Use the commutative and associative
properties.
5. Use the multiplication property of 0.
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Sec 1.4 - 4
1.4 Properties of Real Numbers
Using the Distributive Property
The idea of the distributive property can be illustrated
using rectangles.
3(2 + 5) = 3 • 2 + 3 • 5
2
5
3
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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Sec 1.4 - 5
1.4 Properties of Real Numbers
Use the Distributive Property
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 = a(b + c)
ba + ca = (b + c)a
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Sec 1.4 - 6
1.4 Properties of Real Numbers
Use the Distributive Property
The distributive property allows us to rewrite a
product as a sum:
–4(8 + (–3)) = –4(8) + (–4) (–3)
or a sum as a product.
–6(3) + –6(11) = –6(3 + 11)
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Sec 1.4 - 7
1.4 Properties of Real Numbers
Use the Distributive Property
Product
Sum
–6(x + 9) = –6x + (–6)(9)
= –6x + (–54)
= –6x – 54
4(a + b + c) = 4a + 4b + 4c
7(3x – 2y + 13) = 7(3x + (–2y) + 13)
= 21x + (–14y) + 91
= 21x –14y + 91
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Sec 1.4 - 8
1.4 Properties of Real Numbers
Use the Distributive Property
Sum
8c – 12c = (8c + (–12c))
Product
The distributive property
can also be used for
subtraction:
= (8 + (–12))c
a(b – c) = ab – ac
= –4c
6w –2w + 5w = 6w + (–2)w + 5w
= (6 + (–2) + 5)w
= 9w
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Sec 1.4 - 9
1.4 Properties of Real Numbers
Use the Distributive Property
The distributive property may be used to perform
calculations mentally.
Calculate
29 • 92 + 29 • 8.
29 • 92 + 29 • 8 = 29(92 + 8)
= 29(100)
Combining the 92
and 8 makes the
problem much easier!
= 2900
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Sec 1.4 - 10
1.4 Properties of Real Numbers
Using the Inverse Properties
Inverse Properties
For any real number a,
a   a   0 and  a  a  0
1
1
a   1 and  a  1 (a  0).
a
a
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Sec 1.4 - 11
1.4 Properties of Real Numbers
Using the Inverse Properties
Complete the following statements.
a 
5  _____  0
5
b
19
 _____  0
3
5
  _____  1
11
19
c
d
0  _____  1
Copyright © 2010 Pearson Education, Inc. All rights reserved.
–
3
– 11
5
Zero does not have a
multiplicative
inverse.
Sec 1.4 - 12
1.4 Properties of Real Numbers
Use the Identity Properties
Identity Properties
For any real numbers a,
a+0=0+a=a
a · 1 = 1 · a = a.
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Sec 1.4 - 13
1.4 Properties of Real Numbers
Use the Identity Properties
–(3b + b – 7b) = –1(3 + 1 – 7)b
= ((–1)3 + (–1)1 + (–1)(– 7))b
= (–3 + (–1) + 7)b
= 3b
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Sec 1.4 - 14
1.4 Properties of Real Numbers
Terms and Like Terms
Terms consist of a number or a product of a
number and one or more variables.
2 and 28
Like Terms
227k and 2k
Like Terms
2
y and 4y
2
Like Terms
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.
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Sec 1.4 - 15
1.4 Properties of Real Numbers
Use the Commutative Property
Commutative Properties
For any real numbers a and b,
and
a+b=b+a
ab = ba.
Interchange the order of the two terms or factors.
The commutative properties are used to change
the order of the terms or factors in an expression.
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Sec 1.4 - 16
1.4 Properties of Real Numbers
Use the Associative Properties
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.
The associative properties are used to regroup
(associate) the terms or factors in an expression, where
the order stays the same.
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Sec 1.4 - 17
1.4 Properties of Real Numbers
Use the Commutative and Associative Properties
Simplify.
–5x + 8x + 7 – 9x + 3
Order of Operations
= (–5x + 8x) + 7 – 9x + 3
Distributive Property
= (–5 + 8) x + 7 – 9x + 3
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Sec 1.4 - 18
1.4 Properties of Real Numbers
Use the Commutative and Associative Properties
Continued:
= [3x + (7 – 9x)] + 3
Commutative Property
= [3x + (–9x + 7)] + 3
Associative Property
= [(3x + [–9x]) + 7] + 3
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Sec 1.4 - 19
1.4 Properties of Real Numbers
Use the Commutative and Associative Properties
Continued:
= [(3x + [–9x]) + 7] + 3
Combine like terms
= (–6x + 7) + 3
Associative Property
= –6x + (7 + 3)
Add like terms
= –6x + 10
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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.
Sec 1.4 - 20
1.4 Properties of Real Numbers
Use the Commutative and Associative Properties
Simplify.
4 –1(3g – 7) + 2g(h) (–3) + g
Distributive Property
= 4 –3g + 7 + 2g(h)(–3) + g
Commutative and Associative
Properties; Multiplying
= 4 –3g + 7 + (–6gh) + g
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Sec 1.4 - 21
1.4 Properties of Real Numbers
Use the Commutative and Associative Properties
Continued:
= 4 –3g + 7 + (–6gh) + g
Commutative and
Associative Properties
= 4 + 7 –3g + g + (–6gh)
Adding like terms
=11 –2g – 6gh
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Sec 1.4 - 22
1.4 Properties of Real Numbers
Use the Distributive Property with Caution
Contined — A Second Look:
4 –1(3g – 7) + 2g(h) (–3) + g
Distributive property applies here
since there is subtraction.
= 4 –3g + 7 + 2g(h)(–3) + g
¹
Distributive property does not
apply since there is no addition
or subtraction.
(2g)(h) + (2g)(–3)
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Sec 1.4 - 23
1.4 Properties of Real Numbers
Use the Multiplication Property of 0
Multiplication Property of 0
For any real number a,
a•0=0
and
0 • a = 0.
The product of any real number and 0 is 0.
–4 • 0 = 0
0 • 100 = 0
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0•0=0
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