1. Solve Absolute Value Equations
Section 6-5
Solve Absolute Value Equations

2. Example 1 Solve an absolute value equation: | x | = 9
• The distance between x and 0 is 9.
• There are always two answers for absolute value
• So, x = 9 and x = -9

3. Solving an Absolute Value Equation
The equation |ax + b| = c where c ≥ 0 is equivalent to the statement:
ax + b = c OR ax + b = -c

4. Example 2 Solve an absolute value equation: |x + 4| = 3 x + 4 = 3
• Rewrite the absolute value equation as two equations.
x + 4 = 3
x + 4 = -3
• Solve each equation separately.
x = -1
x = -7

5. Example 3 Rewrite an absolute value equation: 3|4x + 2| - 7 = 11
3|4x + 2| = 18
|4x + 2| = 6
•Rewrite the equation in the form |ax + b| = c.
Need to get absolute value BY ITSELF FIRST!
• Add 7 to both sides.
•Divide both sides by 3.
4x + 2 = 6
4x + 2 = -6
• Solve the absolute value equation.
4x = -8
4x = 4
x = 1
x = -2

6. Homework
Section 6.5 Pg
4 – 6, 9, 14, 15, 16, 20, 24 – 26

7. Solve Absolute Value Inequalities
Section 6-6
Solve Absolute Value Inequalities

8. Graphing Absolute Value Inequalities
If the original problem has:
> or ≥ Graphs goes AWAY from each other
< or ≤ Graphs goes TOWARDS each other.

9. Example 1 Solve an absolute value inequality: | x | > 2
• There are always two answers for absolute value inequalities
• Must make the number NEGATIVE and FLIP the symbol!
So x > 2 and
Graph:
x < -2
Original problem was >, so graph goes AWAY!

10. Example 1 - Continued Solve absolute value inequalities: b) |x| ≤ 1.5
x ≤ and
Graph:
All real numbers less than or equal to 1.5 and greater than or equal to -1.5.
x ≥ -1.5
Original problem was <, so graph goes towardS!

11. Example 2 Solve an absolute value inequality: |x + 2| > 1
x > -1
Graph:
Rewrite as a compound inequality.
x + 2 < -1
Solve each inequality.
x < -3
Original problem was >, so graph goes AWAY!

12. Example 3 Solve an absolute value inequality: |2x + 3| - 4 ≤ 5 + 4 +4
|2x + 3| ≤ 9
2x + 3 ≤ x + 3 ≥ -9
2x ≤ 6
x ≤ 3
Graph:
•Isolate the absolute value.
• Add 4 to both sides.
•Rewrite as a compound inequality.
•Solve each inequality.
2x ≥ -12
x ≥ -6
Original problem was ≤, so graph goes TOWARDS!

13. Homework
Section 6.6 Pg
6 – 13, 18