Originally Posted by

**red_dog** Hi Patrick!

You must be more rigurous in such problems.

First of all we need a condition: $\displaystyle x+2>0\Leftrightarrow x>-2\Leftrightarrow x\in (-2,\infty)$, else the inequality has no solution.

Now:

If $\displaystyle 2x-1\geq 0\Leftrightarrow x\geq\frac{1}{2}$ (1) the inequality becomes

$\displaystyle 2x-1<x+2\Leftrightarrow x<3$ (2)

From (1) and (2) we have $\displaystyle x\in\left.\left[\frac{1}{2},3\right.\right)$ (3)

If $\displaystyle 2x-1<0\Leftrightarrow x\in\left(-2,\frac{1}{2}\right)$ (4) the inequality becomes

$\displaystyle -2x+1<x+2\Leftrightarrow x>-\frac{1}{3}$ (5)

From (4) and (5) we have $\displaystyle x\in\left(-\frac{1}{3},\frac{1}{2}\right)$ (6)

Now, from (3) and (6) yields $\displaystyle x\in\left(-\frac{1}{3},3\right)$