1. 6.5 Solving Absolute Value Equations and Inequalities
Algebra 1
6.5 Solving Absolute Value Equations and Inequalities

2. Steps to Solve an Absolute Value Equation
Isolate the absolute value expression
Drop the absolute value and create a positive and a negative equation
Solve each

3. Solving an Absolute Value Equation
Solve lx – 4l = 8
x – 4 = or x – 4 = -8
x = or x = -4
Mentally check your answers.

4. Solving an Absolute Value Equation
Solve l5x + 1l + 3 = 14
Isolate the absolute value expression.
l5x + 1l = 11
5x + 1 = or x + 1 = -11
5x = or x = -12
x = or x = - 2 and 2/5
Mentally check your answers. -3 -3 5 5

5. Investigating Absolute-Value Inequalities See if each number is a solution as it appears. If it is, fill it in.
1) lxl ≤ 2
(The distance from 0 is less than or equal to 2)
2) lxl > 3
(The distance from 0 is greater than 3)
3) lx + 2l ≥ 1
4) lx – 3l ≤ 2 . . . . . . . . . . . . . . . -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5
x ≤ 2 and x ≥ -2 or . . . . . . . . . . . . . -5 -4 -3 -2 -1 1 2 3 4 5 -4 -3 -2 -1 1 2 3 4 5 6
x > 3 or x < -3
WHY?
and
Absolute Value < or ≤ is an and inequality
Absolute Value > or ≥ is an or inequality

6. Steps to Solve an Absolute Value Inequality
Isolate the Absolute Value Expression
Drop the absolute value and create a positive inequality and a negative inequality (flip the sign)
Solve each inequality separately
<, ≤ is an and inequality
>, ≥ is an or inequality
5) Graph

7. Let’s Try! Solve lx + 6l < 8
x + 6 < and x + 6 > -8
x < and x > -14
-14 2

8. You Try! Solve l2x - 4l ≥ 2 2x - 4 ≥ 2 or 2x - 4 ≤ -2 +4 +4 +4 +4
2x ≥ or x ≤ 2
x ≥ or x ≤ 1 1 2 3

9. Let’s Try! Solve l2x + 1l - 3 ≥ 6 l2x + 1l – 3 ≥ 6 +3 +3 l2x + 1l ≥ 9
l2x + 1l ≥ 9
2x + 1 ≥ or x + 1 ≤ -9
2x ≥ or x ≤ -10
x ≥ or x ≤ -5 -5 4

10. You Try! Solve l3x - 3l + 4 < 10
l3x - 3l < 6
3x - 3 < and x - 3 > -6
3x < and x > -3
x < and x > -1 -1 3