# How do you solve the inequality x+abs(2x-3)<2 and write your answer in interval notation?

Mar 22, 2017

The inequality is not observed for any $x \in \mathbb{R}$

#### Explanation:

Consider

$\left\mid 2 x - 3 \right\mid < 2 - x$ Supposing $x \ne 2$ we have

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

$\frac{\left\mid 2 x - 3 \right\mid}{\left\mid 2 - x \right\mid} < \pm 1$

or

$\frac{\left\mid 2 x - 3 \right\mid}{\left\mid 2 - x \right\mid} < \min \left(- 1 , 1\right) = - 1$

arriving at

$\frac{\left\mid 2 x - 3 \right\mid}{\left\mid 2 - x \right\mid} < - 1$ which is impossible so does not exist any value for $x$ making feasible this inequality.