# How do you solve the inequality |1-x/3|>2/3?

Dec 1, 2015

Consider each case for the inequality being true to find the solution set $\left(- \infty , 1\right) \cup \left(5 , \infty\right)$

#### Explanation:

We can solve this if we recognize what the absolute value symbol is actually saying. It is equivalent to saying

$1 - \frac{x}{3} > \frac{2}{3}$
or
$1 - \frac{x}{3} < - \frac{2}{3}$

Then all we need to do is find what conditions cause each to be true.

For the first:
$1 - \frac{x}{3} > \frac{2}{3}$
$\implies 1 - \frac{2}{3} > \frac{x}{3}$
$\implies \frac{1}{3} > \frac{x}{3}$
$\implies 1 > x$

For the second:
$1 - \frac{x}{3} < - \frac{2}{3}$
$\implies 1 + \frac{2}{3} < \frac{x}{3}$
$\implies \frac{5}{3} < \frac{x}{3}$
$\implies 5 < x$

So one of the conditions, and thus the original inequality, will be true so long as either $x < 1$ or $x > 5$. Therefore we have the solution set
$\left(- \infty , 1\right) \cup \left(5 , \infty\right)$