Equations That have Absolute Value Sign on One Side

Example1:

Solve |x| = 2

Solution:

x = 2 or x = -2

Example 2:

Solve |x + 1| = 2

Solution:

x + 1 = 2 or x + 1 = -2

x = 1 or x = -3

Example 3

Solve |3x - 4| = 5

Solution:

3x - 4 = 5 or 3x - 4 = -5

3x = 9 or 3x = -1

x = 3 or x = -1/3

Example 4

Solve |4x + 7| = -3

Solution:

This equation has no solution, since an absolute value cannot be negative.

Example 5 :

Solve |2x - 6| = 0

Solution:

Since positive and negative 0 mean the same thing, we only need one equation

2x - 6 = 0

2x = 6

x = 3

Exercise 1: Solve absolute value equations

Equations That have Absolute Value Signs on Both Sides

If we have absolute value signs on both sides of the equation, we can play the same game with two choices as follows.

Example 6:

Solve |3x + 4| = | 2x - 3|

Solution:

3x + 4 = 2x - 3 or 3x + 4 = -(2x - 3)

3x + 4 = 2x - 3 or 3x + 4 = -2x + 3

3x = 2x - 7 or 3x = -2x - 1

x = -7 or 5x = -1

x = -7 or x = -1/5

Exercise 2: Solve absolute value equations