# How do you solve and write the following in interval notation: -1/6+|2−x/3|>1/2?

Apr 19, 2018

$x \in \left[- \infty , 4\right) \mathmr{and} x \in \left(8 , + \infty\right]$ or $x \notin \left(4 , 8\right)$

#### Explanation:

First we rearrange to get the $\left\mid f \left(x\right) \right\mid$ part on its own by adding $\frac{1}{6}$ to both sides.

$\left\mid 2 - \frac{x}{3} \right\mid > \frac{2}{3}$

Due to the nature of $\left\mid \right\mid$ we can take the inside to be positive or negative, since it turn either into a positive number.

$2 - \frac{x}{3} > \frac{2}{3}$ or $- 2 + \frac{x}{3} > \frac{2}{3}$
$\frac{x}{3} < 2 - \frac{2}{3}$ or $\frac{x}{3} > \frac{2}{3} + 2$
$\frac{x}{3} < \frac{4}{3}$ or $\frac{x}{3} > \frac{8}{3}$
$x < 4$ or $x > 8$

So, we have $x \in \left[- \infty , 4\right) \mathmr{and} x \in \left(8 , + \infty\right]$ or $x \notin \left(4 , 8\right)$