# How do you solve abs(1/3x - 3) = 9?

May 7, 2018

$x = - 18 \text{ or } x = 36$

#### Explanation:

$\text{the expression inside the absolute value bars can be}$
$\text{positive or negative leading to 2 possible solutions}$

$\frac{1}{3} x - 3 = 9 \leftarrow \textcolor{m a \ge n t a}{\text{positive value}}$

$\text{add 3 to both sides}$

$\Rightarrow \frac{1}{3} x = 9 + 3 = 12$

$\text{multiply both sides by 3}$

$\Rightarrow x = 3 \times 12 = 36$

$- \left(\frac{1}{3} x - 3\right) = 9 \leftarrow \textcolor{m a \ge n t a}{\text{negative value}}$

$\Rightarrow - \frac{1}{3} x + 3 = 9$

$\text{subtract 3 from both sides}$

$\Rightarrow - \frac{1}{3} x = 9 - 3 = 6$

$\text{multiply both sides by } - 3$

$\Rightarrow x = 6 \times - 3 = - 18$

$\textcolor{b l u e}{\text{As a check}}$

Substitute these values into the left side of the equation and if equal to the right side then they are the solutions.

$x = 36 \to | 12 - 3 | = | 9 | = 9$

$x = - 18 \to | - 6 - 3 | = | - 9 | = 9$

$\Rightarrow x = - 18 \text{ or "x=36" are the solutions}$