1.4 Properties of Real Numbers and Algebraic Expressions

30 CHAPTER 1 Real Numbers and Algebraic Expressions

1.4 Properties of Real Numbers and Algebraic Expressions

OBJECTIVES

1 Use Operation and Order

Symbols to Write Mathematical Sentences.

2 Identify Identity Numbers

and Inverses.

3 Identify and Use the Commuta-

tive, Associative, and Distributive Properties.

4 Write Algebraic Expressions.

5 Simplify Algebraic Expressions.

OBJECTIVE

1 Using Symbols to Write Mathematical Sentences

In Section 1.2, we used the symbol = to mean "is equal to." All of the following key words and phrases also imply equality.

Equality

equals gives

is/was yields

represents amounts to

is the same as is equal to

E X A M P L E S Write each sentence as an equation.

1. (T+he+s1u+m)of+x+an+d15* is 20.

T x+5

= 20

2.

Two times (+1)+1*

t(h+e+su+m)of+3 +an+d *y

(a+m1o) un+ts1t*o

4.

T

T

T

2

13 + y2

= 4

3. (T+he+d+if1fe+re1) nce+o+f 8+a1n+d1x* i(s+th+e) sam+e+a*s (th+e+pr1o1+du1c)t o+f +2 a+n1d11x*.

T

T

T

8-x

=

2#x

4. (T+he+q+uo+ti1e)nt+of+z+a+nd1*9 a(+mo1) un+ts 1t*o 9 plus z.

T z,9

T = 9+z

z or

9

= 9+z

PRACTICE

1?4 Write each sentence using mathematical symbols.

1. The product of -4 and x is 20. 2. Three times the difference of z and 3 equals 9. 3. The sum of x and 5 is the same as 3 less than twice x. 4. The sum of y and 2 is 4 more than the quotient of z and 8.

Helpful Hint

Notice that if a 6 b, then b 7 a. For example, since -1 6 7, then 7 7 -1.

If we want to write in symbols that two numbers are not equal, we can use the symbol , which means "is not equal to." For example,

32

Graphing two numbers on a number line gives us a way to compare two numbers. For two real numbers a and b, we say a is less than b if on the number line a lies to the left of b. Also, if b is to the right of a on the number line, then b is greater than a. The symbol 6 means "is less than." Since a is less than b, we write

a

b

b6a The symbol 7 means "is greater than." Since b is greater than a, we write

b7a

Section 1.4 Properties of Real Numbers and Algebraic Expressions 31

E X A M P L E 5 Insert 6 , 7 , or = between each pair of numbers to form a true statement.

a. -1 -2 53

e. 8 8

12

b.

3

4

23 f.

3 4

c. -5 0

d. -3.5 -3.05

Solution

a. -1 7 -2 since -1 lies to the right of -2 on the number line.

3 2 1 0 1

b. 12 = 3. 4

c. -5 6 0 since -5 lies to the left of 0 on the number line.

5 4 3 2 1 0 1

d. -3.5 6 -3.05 since -3.5 lies to the left of -3.05 on the number line.

3.5 3.05

5 4 3 2 1 0

53

53

e. 8 7 8 The denominators are the same, so 8 7 8 since 5 7 3.

23 f. 3 6 4

By dividing, we see that 3 = 0.75 and 2 = 0.666. c

4

3

Thus

2 3

6

3 4

since 0.666c

6

0.75.

PRACTICE

5

a. -6

9 e.

10

Insert 6 , 7 , or = between each pair of numbers to form a true statement.

24

-5

b.

8

3

c. 0 -7

d. -2.76 -2.67

7

27

f.

10

39

Helpful Hint

When inserting the 7 or 6 symbol, think of the symbols as arrowheads that point toward the smaller number when the statement is true.

In addition to 6 and 7, there are the inequality symbols ... and ? . The symbol ... means ;is less than or equal to< and the symbol ? means ;is greater than or equal to<

For example, the following are true statements.

10 ... 10 -8 ... 13 -5 ? -5 -7 ? -9

since since since since

10 = 10 -8 6 13 -5 = -5 -7 7 -9

E X A M P L E 6 Write each sentence using mathematical symbols.

a. The sum of 5 and y is greater than or equal to 7. b. 11 is not equal to z. c. 20 is less than the difference of 5 and twice x.

32 CHAPTER 1 Real Numbers and Algebraic Expressions

Solution

a. 5 + y ? 7

b. 11 z

c. 20 6 5 - 2x

PRACTICE

6 Write each sentence using mathematical symbols.

a. The difference of x and 3 is less than or equal to 5. b. y is not equal to -4. c. Two is less than the sum of 4 and one-half z.

OBJECTIVE

2 Identifying Identities and Inverses

Of all the real numbers, two of them stand out as extraordinary: 0 and 1. Zero is the only number that, when added to any real number, results in the same real number. Zero is thus called the additive identity. Also, one is the only number that, when multiplied by any real number, results in the same real number. One is thus called the multiplicative identity.

Identity Properties

Addition

The additive identity is 0. a+0=0+a=a

Multiplication

The multiplicative identity is 1.

a#1 = 1#a = a

In Section 1.2, we learned that a and - a are opposites.

Another name for opposite is additive inverse. For example, the additive inverse

of 3 is -3. Notice that the sum of a number and its opposite is always 0.

1 In Section 1.3, we learned that, for a nonzero number, b and are reciprocals.

b

Another name for reciprocal is multiplicative inverse. For example, the multiplicative

23

inverse

of

-

3

is

-

. 2

Notice

that

the

product

of

a

number

and

its

reciprocal

is

always

1.

Inverse Properties

Opposite or Additive Inverse

For each number a, there is a unique number -a called the additive inverse or opposite of a such that

a + 1-a2 = 1-a2 + a = 0

Reciprocal or Multiplicative Inverse

For each nonzero a, there is a unique 1

number called the multiplicative a

inverse or reciprocal of a such that

a#1 = 1#a = 1

aa

E X A M P L E 7 Write the additive inverse, or opposite, of each.

3

a. 4

b.

7

Solution

c. -11.2

a. The opposite of 4 is -4.

33

b.

The

opposite

of

7

is

-

. 7

c. The opposite of -11.2 is -1 -11.22 = 11.2.

PRACTICE

7 Write the additive inverse, or opposite, of each.

a. -7

b. 4.7

3 c. - 8

Section 1.4 Properties of Real Numbers and Algebraic Expressions 33

E X A M P L E 8 Write the multiplicative inverse, or reciprocal, of each.

7

a. 11

b. -9

c. 4

Solution

1 a. The reciprocal of 11 is .

11

1

b.

The

reciprocal

of

-9

is

-

. 9

c. The reciprocal of 7 is 4 because 7 # 4 = 1.

4 7

47

PRACTICE

8 Write the multiplicative inverse, or reciprocal, of each.

5 a. - 3

b. 14

c. -2

Helpful Hint

The number 0 has no reciprocal. Why? There is no number that when multiplied by 0 gives a product of 1.

CONCEPT CHECK

Can a number's additive inverse and multiplicative inverse ever be the same? Explain.

Answer to Concept Check: no; answers may vary

OBJECTIVE

3 Using the Commutative, Associative, and Distributive Properties

In addition to these special real numbers, all real numbers have certain properties that allow us to write equivalent expressions--that is, expressions that have the same value. These properties will be especially useful in Chapter 2 when we solve equations.

The commutative properties state that the order in which two real numbers are added or multiplied does not affect their sum or product.

Commutative Properties For real numbers a and b,

Addition: a + b = b + a

Multiplication: a # b = b # a

The associative properties state that regrouping numbers that are added or multiplied does not affect their sum or product.

Associative Properties For real numbers a, b, and c,

Addition: 1a + b2 + c = a + 1b + c2

Multiplication: 1a # b2 # c = a # 1b # c2

34 CHAPTER 1 Real Numbers and Algebraic Expressions

E X A M P L E 9 Use the commutative property of addition to write an expression

equivalent to 7x + 5.

Solution

7x + 5 = 5 + 7x.

PRACTICE

9 Use the commutative property of addition to write an expression equivalent to 8 + 13x.

E X A M P L E 1 0 Use the associative property of multiplication to write an

expression equivalent to 4 # 19y2. Then simplify this equivalent expression.

Solution

4 # 19y2 = 14 # 92y = 36y.

PRACTICE

10 Use the associative property of multiplication to write an expression equiva-

lent to 3 # 111b2. Then simplify the equivalent expression.

The distributive property states that multiplication distributes over addition.

Distributive Property For real numbers a, b, and c,

Also,

a1b + c2 = ab + ac a1b - c2 = ab - ac

E X A M P L E 1 1 Use the distributive property to multiply.

a. 312x + y2

b. -13x - 12

c. 0.7a1b - 22

Solution

# # a. 3(2x+y) = 3 2x + 3 y Apply the distributive property.

= 6x + 3y

Apply the associative property of multiplication.

b. Recall that -13x - 12 means -113x - 12.

?1(3x-1) = -113x2 + 1 -121 -12 = -3x + 1

Answer to Concept Check: no; 612a213b2 = 12a13b2 = 36ab

c. 0.7a(b-2) = 0.7a # b - 0.7a # 2 = 0.7ab - 1.4a

PRACTICE

11 Use the distributive property to multiply.

a. 41x + 5y2

b. -13 - 2z2

c. 0.3x1y - 32

CONCEPT CHECK

Is the statement below true? Why or why not?

612a213b2 = 612a2 # 613b2

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