# How do you solve abs(x-2)<5x?

Jul 21, 2018

The solution is $x \in \left(\frac{1}{3} , + \infty\right)$

#### Explanation:

The inequality with absolute value is

$| x - 2 | < 5 x$

$x - 2 = 0$

$\implies$, $x = 2$

There are $2$ intervals to consider

${I}_{1} = \left(- \infty , 2\right)$ and ${I}_{2} = \left(2 , + \infty\right)$

Therefore,

In the first interval ${I}_{1}$

$- x + 2 < 5 x$

$6 x > 2$

$x > \frac{1}{3}$

This solution $\in {I}_{1}$

In the first interval ${I}_{2}$

$x - 2 < 5 x$

$4 x > 2$

$x > \frac{1}{2}$

This solution $\notin {I}_{2}$

The only solution is $x \in \left(\frac{1}{3} , + \infty\right)$

graph{|x-2|-5x [-5.55, 5.55, -2.773, 2.776]}