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          
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 $\color{blue}{x = -7}$