# Solving absolute value equations and inequalities

The absolute number of a number a is written as

$$\left | a \right |$$

And represents the distance between a and 0 on a number line.

An absolute value equation is an equation that contains an absolute value expression. The equation

$$\left | x \right |=a$$

Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

To solve an absolute value equation as

$$\left | x+7 \right |=14$$

You begin by making it into two separate equations and then solving them separately.

$$x+7 =14$$

$$x+7\, {\color{green} {-\, 7}}\, =14\, {\color{green} {-\, 7}}$$

$$x=7$$

or

$$x+7 =-14$$

$$x+7\, {\color{green} {-\, 7}}\, =-14\, {\color{green} {-\, 7}}$$

$$x=-21$$

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

The inequality

$$\left | x \right |<2$$

Represents the distance between x and 0 that is less than 2

Whereas the inequality

$$\left | x \right |>2$$

Represents the distance between x and 0 that is greater than 2

You can write an absolute value inequality as a compound inequality.

$$\left | x \right |<2\: or

$$-2<x<2$$

This holds true for all absolute value inequalities.

$$\left | ax+b \right |<c,\: where\: c>0$$

$$=-c<ax+b<c$$

$$\left | ax+b \right |>c,\: where\: c>0$$

$$=ax+b<-c\: or\: ax+b>c$$

You can replace > above with ≥ and < with ≤.

When solving an absolute value inequality it's necessary to first isolate the absolute value expression on one side of the inequality before solving the inequality.

**Example**

Solve the absolute value inequality

$$2\left |3x+9 \right |<36$$

$$\frac{2\left |3x+9 \right |}{2}<\frac{36}{2}$$

$$\left | 3x+9 \right |<18$$

$$-18<3x+9<18$$

$$-18\, {\color{green} {-\, 9}}<3x+9\, {\color{green} {-\, 9}}<18\, {\color{green} {-\, 9}}$$

$$-27<3x<9$$

$$\frac{-27}{{\color{green} 3}}<\frac{3x}{{\color{green} 3}}<\frac{9}{{\color{green} 3}}$$

$$-9<x<3$$

**Video lesson**

Solve the absolute value equation

$$4 \left |2x -1 \right | -2 = 10$$