copyright © 2010 pearson education, inc. all rights reserved sec 1.4 - 1


Review of the Real Number System

Chapter 1


1.4

Properties of Real Numbers

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 4

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.


Using the Distributive Property

The idea of the distributive property can be illustrated

using rectangles.


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


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

ba + ca

= a(b + c)

= (b + c)a




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)




–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




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




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!


Using the Inverse Properties


Inverse Properties

For ,

and

and

0 0

1 11 1 ( 0

any real nu

. )

mber

a

a a a a

a a aa a


Using the Inverse Properties


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.


Use the Identity Properties


Identity Properties

For any real numbers a,

a + 0 = 0 + a = a

a · 1 = 1 · a = a.


Use the Identity Properties


–(3b + b – 7b) =

–1(3 + 1 – 7)b= ((–1)3 + (–1)1 + (–1)(– 7))b= (–3 + (–1) + 7)b

= 3b


Terms and Like Terms


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


Use the Commutative Property


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.


Use the Associative Properties


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.


Use the Commutative and Associative Properties


Simplify.

–5x + 8x + 7 – 9x + 3

= (–5x + 8x) + 7 – 9x + 3

Order of Operations

= (–5 + 8) x + 7 – 9x + 3





Continued:

Commutative Property

Associative Property

= [3x + (7 – 9x)] + 3

= [3x + (–9x + 7)] + 3

= [(3x + [–9x]) + 7] + 3




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.




Simplify.

4 –1(3g – 7) + 2g(h) (–3) + g

= 4 –3g + 7 + 2g(h)(–3) + g

= 4 –3g + 7 + (–6gh) + g


Commutative and Associative Properties; Multiplying




Continued:

= 4 + 7 –3g + g + (–6gh)

= 4 –3g + 7 + (–6gh) + g Commutative and

Associative Properties

Adding like terms

=11 –2g – 6gh


Use the Distributive Property with Caution


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.


Use the Multiplication Property of 0


The product of any real number and 0 is 0.

Multiplication Property of 0

For any real number a,

a • 0 = 0 and 0 • a = 0.

–4 • 0 = 0 0 • 100 = 0 0 • 0 = 0

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