1. Objectives Solve compound inequalities.
Write and solve absolute-value equations and inequalities.

2. Recall that the absolute value of a number x, written |x|, is the distance from x to zero on the number line. Because absolute value represents distance without regard to direction, the absolute value of any real number is nonnegative.

3. Absolute-value equations and inequalities can be represented by compound statements. Consider the equation |x| = 3.
The solutions of |x| = 3 are the two points that are 3 units from zero. The solution is a disjunction: x = –3 or x = 3.

4. Example 1: Solving Absolute-Value Equations
Solve the equation.
This can be read as “the distance from k to –3 is 10.”
|–3 + k| = 10
Rewrite the absolute value as a disjunction.
–3 + k = 10 or –3 + k = –10
Add 3 to both sides of each equation.
k = 13 or k = –7

5. Example 2: Solving Absolute-Value Equations
Solve the equation.
Isolate the absolute-value expression.
Rewrite the absolute value as a disjunction.
Multiply both sides of each equation
by 4.
x = 16 or x = –16

6. Example 3
Solve the equation.
This can be read as “the distance from x to +9 is 4.”
|x + 9| = 13
Rewrite the absolute value as a disjunction.
x + 9 = 13 or x + 9 = –13
Subtract 9 from both sides of each equation.
x = 4 or x = –22

7. Example 4
Solve the equation.
|6x| – 8 = 22
Isolate the absolute-value expression.
|6x| = 30
Rewrite the absolute value as a disjunction.
6x = 30 or 6x = –30
Divide both sides of each equation by 6.
x = 5 or x = –5

8. Lesson Quiz: Part I
Solve each equation. 1.
|2v + 5| = 9
2 or –7 2.
4. |5b| – 7 = 13
+ 4

9. Homework
Pg 154: 5-7, 16-19, 37, 38