$|x-2|-2x<3$
$\Leftrightarrow |x-2|<2x+3$
$\Leftrightarrow \left\{ \begin{array}{l}x-2<2x+3\\x-2>-2x-3\end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l}-x-5<0\\3x+1>0\end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l}x>-5\\x>-\frac{1}{3}\end{array} \right.$
$\Leftrightarrow x>-\frac{1}{3}$