Answer
$x=1$ or $x=0$
Work Step by Step
$x+|3x-2|=2 $ or $x+|3x-2|=-2 $
Thus $x-2=-|3x-2| $ or $x+2=-|3x-2|$
Thus $x-2=-3x+2$ or $x-2=3x-2$ or $x+2=-3x+2$ or $x+2=3x-2$
Thus $x=1$ or $x=0$ or $x=0$ or $x=2$ (but this is not a solution after substituting back).