Introduction to Absolute Value Equations

Introduction to Absolute Value

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          &nbsp
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 $\color{blue}{x = -28}$