# Introduction to Absolute Value Equations

## What is Absolute Value?

The absolute value of a number is its distance from zero.
Absolute value and distance are always represented as positive values.

Notice that both $|8|$ and $|-8|$ equal $8$.

Example Problems:

 1.   $|-5\tfrac{1}{3}|$ 2.   $|5| + |-2|$ 3.   $3(|-9|$  $-$   $|-7|)$ $\color{blue}{= 5\tfrac{1}{3}}$ $5$   $+$   $2$ $=$   $3(5$   $-$   $7)$ since absolute value $\color{blue}{= 7}$ $=$  $3(-2)$ is always positive. $\color{blue}{= -6}$

## Solving Absolute Value equations?

Let’s look at the equation $|x| = 12$
We know that both $|12|$ and $|-12|$ equal $12$.
So there are 2 possible solutions $\color{blue}{x = 12}$ or $\color{blue}{x = -12}$

Let’s look at another problem.

 Original problem: $|x + 6| = 22$ To remove the absolute value signs we need to write the equation twice $x + 6 = 22$ once with the answer being positive $x = 16$ and once with the answer being negative $x + 6 = -22$ $x = -28$ So there are 2 possible solutions $\color{blue}{x = 16}$ or $\color{blue}{x = -28}$

One last example

 Original problem: $|2x-3| = 17$ To remove the absolute value signs we need to write the equation twice $2x – 3 = 17$ once with the answer being positive $2x = 20$ once with the answer being positive $x=10$ and once with the answer being negative $2x-3 = -17$ $2x = -14$ $x=-7$ So there are 2 possible solutions $\color{blue}{x = 10}$ or $\color{blue}{x = -7}$