Intermediate Algebra (11th edition)

116 CHAPTER 2 Linear Equations, Inequalities, and Applications

Solving Absolute Value Inequalities That Require Rewriting

Solve each inequality.

(a)

or

or

Solution set: 1 - q, - 104 ́ 3 4, q 2

xÚ 4 x...- 10

x+ 3 Ú 7 x+ 3 ...- 7

|x+ 3 |Ú 7

|x+ 3 |+ 5 Ú 12

NOW TRY EXAMPLE 5

EXERCISE 5
Solve each inequality.

(a)

(b)|x- 1 |- 4 Ú 2

|x- 1 |- 4 ... 2

NOW TRY
EXERCISE 6
Solve

| 3 x- 4 |=| 5 x+ 12 |.

Solving

To solve an absolute value equation of the form

solve the following compound equation.

axbcxd or axb 1 cxd 2

|axb||cxd|,

axbcxd

NOW TRY

OBJECTIVE 5 Solve equations of the form. If two

expressions have the same absolute value, they must either be equal or be negatives

of each other.

|axb||cxd|

Solving an Equation with Two Absolute Values

Solve

This equation is satisfied either if and are equal to each other or if

and are negatives of each other.

or

or

or

Check that the solution set is NOW TRY

OBJECTIVE 6 Solve special cases of absolute value equations and

inequalities.When an absolute value equation or inequality involves a negative

constant or 0alone on one side, use the properties of absolute value to solve the

equation or inequality.

5 - 1, 9 6.

x=- 1

9 = x 3 x=- 3

x + 9 = 2 x x+ 6 =- 2 x+ 3

x + 6 = 2 x- 3 x+ 6 = - 12 x- 32

x+ 6 2 x- 3

x+ 6 2 x- 3

|x+ 6 |=| 2 x- 3 |.

EXAMPLE 6

(b)

Solution set: 3 - 10, 4 4

- 10 ... x ... 4

- 7 ... x+ 3 ... 7

|x+ 3 |... 7

|x+ 3 |+ 5 ... 12

Special Cases of Absolute Value

Case 1 The absolute value of an expression can never be negative. That is,

for all real numbers a.

Case 2 The absolute value of an expression equals 0 only when the expression

is equal to 0.

|a|Ú 0

NOW TRY ANSWERS

5. (a)
   (b)


6. 5 - 8, - 16

1 - q, - 54 ́ 3 7, q 2

3 - 5, 7 4