This document discusses solving absolute value equations. It provides 3 steps for solving equations containing absolute value: 1) isolate the absolute value bars, 2) make two equations setting the absolute value equal to the positive and negative answers, and 3) solve each equation. Examples are worked through demonstrating that absolute value equations can have more than one solution. The document also notes that if solving the equations results in the absolute value being equal to a negative number, then there is no solution or an empty set.

1. Absolute Value Equations
Chapter 6
2x −1 = 9

2. Absolute Value
 Absolute value – positive distance
from zero.
5 − 8 = -3 = 3

3. Solutions to Absolute Value
 Absolute value bars can have more
than one answer.
x =7
 What could be the values of x that
would make the sentence above true?
x = 7 or -7
therefore x = {7, -7}

4. Solving Equations Containing
Absolute Value
1. Isolate the absolute value bars.
2. Make two equations, one = to positive
answer and one = to negative answer.
3. Solve each equation.

5. Example #1
x − 7 = 13
x − 7 = 13 x − 7 = −13
x = 20 x = −6
x = {20,-6}

6. Example #2
x + 3 − 7 = 15
x + 3 = 22
x + 3 = 22 x + 3 = −22
x = 19 x = −25
x = {19,-25}

7. Example #3
2 3x − 4 − 1 = 5
2 3x − 4 = 6
3x − 4 = 3
3x − 4 = 3 3x − 4 = −3
7 1
x= x=
3 3
x = {7 , 1 }
3 3

8. No Solution or Empty Set
 If your absolute value equals negative,
then the answer is no solution or empty set
because absolute value is always positive.
x + 3 + 12 = 5
x + 3 = −7
x=∅