Worksheet 2-4 P. 1

NAME

DATE

2-4 Absolute Value in Open Sentences Objective:

To solve open sentences involving absolute value.

Vocabulary Rules for solving open sentences involving absolute value If a is a positive real number: 1. \x\ = a

is equivalent to the disjunction

x = — a orx = a.

2. \x\ > a

is equivalent to the disjunction

x < — aorx>a.

3. \x\ < a

is equivalent to the conjunction

x > —a and x < a (or —a < x < a).

CAUTION

Watch out for statements such as \x\ > -6 and \x < -6. Since the absolute value of every real number is nonnegative, \x is always greater than a negative number. Similarly, \x can never be less than a negative number.

Example 1 a. The solution set of x b. The solution set of

X

IS

f

{x: x —

> 2 is {x: x < -2 or x > 2}.

c. The solution set of

X

< 2 is{ x:

d. The solution set of

X

>

e. The solution set of

or x — 2 j .

A I -2 - 1

_ t

t

0

1

1

0

'-2 - 1

-3 -2 - 1

o

-3

t*t

•

. l -»

2

3

1

2

3

0

1

2

3

0

1

2

3

r\

2 < x < 2}.

Q

— 2 is (real numbers } .

No graph

< -2 is 0.

Example 2

Solve \2x - 6| = 10.

Solution

Use Rule 1 to write an equivalent disjunction. Then solve. \2x — 6| = 10 is equivalent to this disjunction:

or

2x - 6 = 10

2x — — 4

or

2x = 16

x = -2

or

x = 8

2x - 6 = -10

Check:

|2(-2) - 6| ^ 10 10 = 10 V

|2 • 8 - 6| * 10 10 = 10 V

.'. the solution set is { — 2, 8}. Solve.

1. JJCJ = 5 5. \3p - 4| = 5 9. 15 -2«| _ n

2. M = 8 6. \2t + 1| = 7 10. 13 + 4*| = 9

3. |2 + v| = 7 7. \lk + 14| = 0 11. U - 1| = -4

Study Guide, ALGEBRA AND TRIGONOMETRY, Structure and Method, Book 2 Copyright © by Houghton Mifflin Company. All rights reserved.

4. 3 - n| - 4 8. |5c - 20| = 5 12. I -•t = 0

25