Intermediate Algebra (11th edition)

34 CHAPTER 1 Review of the Real Number System

(c)

Identity property

Distributive property

=-m+ 5 n Multiply.

= - 11 m 2 + 1 - 121 - 5 n 2

= - 11 m- 5 n 2

- 1 m- 5 n 2

Multiply eachterm by

Be careful with signs.

• 
  
  
  
  1. NOW TRY
  
  
  
  

OBJECTIVE 3 Use the inverse properties. The additive inverse(or opposite)

of a number ais. Additive inverses have a sum of 0 (the additive identity).

5 and and and 34

The multiplicative inverse (or reciprocal)of a number ais (where ), and mul-

tiplicative inverses have a product of 1 (the multiplicative identity).

5 and and and

Multiplicative inverses

(product of 1)

This discussion leads to the inverse propertiesof addition and multiplication.

4

3

3

4

- -2,

1

2

1

5

,

(^1) a aZ 0

- 34

1

2

- ,

1

2

- 5,

- a

NOW TRY
EXERCISE 2
Simplify each expression.

(a) 7 x+x (b) - 15 p- 3 q 2

NOW TRY ANSWERS

2. (a) 8 x (b)- 5 p+ 3 q

Inverse Properties

For any real number a, the following are true.

and

and

1

a

a #^1 #a 1 1 a 02

a

 1

a  1 a 2  0  aa 0

The inverse properties “undo” addition or multiplication.Think of putting on

your shoes when you get up in the morning and then taking them off before you go to

bed at night. These are inverse operations that undo each other.

Expressions such as 12mand 5nfrom Example 2are examples of terms.A term

is a number or the product of a number and one or more variables raised to powers.

The numerical factor in a term is called the numerical coefficient,or just the

coefficient.Some examples are shown in the table in the margin.

Terms with exactly the same variables raised to exactly the same powers are

called like terms.

Like terms

Unlike terms

OBJECTIVE 4 Use the commutative and associative properties.Simplifying

expressions as in parts (a) and (b) of Example 2is called combining like terms.

Only like terms may be combined.To combine like terms in an expression such as

we need two more properties. From arithmetic, we know that

and

and

The order of the numbers being added or multiplied does not matter. The same

answers result. Also,

5 + 17 + 22 = 5 + 9 =14,

15 + 72 + 2 = 12 + 2 = 14

3 # 9 = 27 9 # 3 =27.

3 + 9 = 12 9 + 3 = 12

- 2 m+ 5 m+ 3 - 6 m+8,

3 m and 16x 7 y^3 and - 3 y^2

5 p and - 21 p - 6 x^2 and 9x^2

Numerical

Term Coefficient

34

1

1

3

x

3

=^1 x

3

=^1

3

x

3

8

3 x

8 =

3

8 x

r= 1 r

• k=- 1 k - 1


• 26 x^5 yz^4 - 26

34 r^3

• 7 y - 7

Additive inverses

(sum of 0)