ĐK: $x>1$.
Ta có:
$2\sqrt{x-1}-\sqrt{x+2}>x-2$
$\Leftrightarrow \dfrac{3x-6}{2\sqrt{x-1}+\sqrt{x+2}}>x-2$
$\Leftrightarrow (x-2)(3-2\sqrt{x-1}-\sqrt{x+2})>0$
$\Leftrightarrow \left[\begin{array}{l}\left\{\begin{array}{l}x-2>0\\3>2\sqrt{x-1}+\sqrt{x+2}\end{array}\right.\\\left\{\begin{array}{l}x-2<0\\3<2\sqrt{x-1}+\sqrt{x+2}\end{array}\right.\end{array}\right.$
$\Leftrightarrow 7-4\sqrt2<x<2$