**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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