# Absolute value equations

What number is x when | x | = 4?

According to the definition of absolute value, x is a number whose distance from zero is 4. So x = 4 or x = -4. An absolute value equation usually gives two solution paths.

Example 1

Solve x

| 2x | = 4

By definition

2x = 4 or 2x = -4, which gives x = 2 or x = -2.

Example 2

Solve x

| 3x-6 | = 9

By definition

3x-6 = 9 or 3x-6 = -9 to give x = 5 or x = -1

Example 3

Solve x

| 4x-7 | = -5

The absolute value is always greater than or equal to 0, so this equation has no solutions.

Example 4

Find x

By the definition we get 2 equations

Two quadratic equations are obtained. The solutions of the former are x = -2 or x = 3 and the solutions of the latter are x = -1 or x = 2. So the solutions of the original absolute value equation are x = -2, x = -1, x = 2 or x = 3

An equation in which the absolute values of two functions are equal is true when the values of the functions are equal or opposite.

Example 5

Solve the equation

|2x - 4| = |x - 5|

By definition we get two equations

2x - 4 = x - 5 or 2x - 4 = - (x - 5)

The first equation gives x = -1 and the second x = 3

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