# How do you solve |(x+1)/x| > 2, and represent the answer in interval notation?

May 20, 2018

The solution is $x \in \left(- \frac{1}{3} , 0\right) \cup \left(0 , 1\right)$

#### Explanation:

This is an inequality with absolute values

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

The solutions are

$\left\{\begin{matrix}\frac{x + 1}{x} > 2 \\ - \frac{x + 1}{x} > 2\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\frac{x + 1}{x} - 2 > 0 \\ - \frac{x + 1}{x} - 2 > 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\frac{x + 1 - 2 x}{x} > 0 \\ \frac{- x - 1 - 2 x}{x} > 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\frac{1 - x}{x} > 0 \\ \frac{- 3 x - 1}{x} > 0\end{matrix}\right.$

Solve the inequalities with a sign chart

$\iff$, $\left\{\begin{matrix}x \in \left(0 1\right) \\ x \in \left(- \frac{1}{3} 0\right)\end{matrix}\right.$

The solution is $x \in \left(- \frac{1}{3} , 0\right) \cup \left(0 , 1\right)$

graph{|(x+1)/x|-2 [-4.93, 4.934, -2.465, 2.465]}