Title: Absolute Value Equations

The absolute value equation |ax + b| = c can be solved by rewriting as two linear equations

ax + b = c or ax + b = -c

and then solving each equation separately.

The solutions to this equation are or .

ABSOLUTE VALUES ALWAYS GIVE 2 EQUATIONS! One of which is the positive of what is in the absolute value signs, and the other is the negative of what is in the absolute value signs.

Example1: Solve |x| = 2

Solution:

x = 2 or x = -2

Example 2:

|x + 1| = 2

Solution:

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

x = 1 or x = -3

Example 3

|3x - 4| = 5

Solution: We write

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

3x = 9 or 3x = -1

x = 3 or x = -1/3

Example 4

|4x + 7| = -3

Solution:

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

Example 5 :

|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

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:

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

Solution:

We can write 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 or x = -1/5

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