$解:根据题意,得\frac{3x+2}{x-2}<0$
$则①$
$\begin{cases}{3x+2<0}\\{x-2>0}\end{cases}$
$或②$
$\begin{cases}{3x+2>0}\\{x-2<0}\end{cases}$
$解不等式组①,得不等式组无解$
$解不等式组②,得-\frac{2}{3}<x<2$
$∴-\frac{2}{3}<x<2,$
$∴-\frac{2}{3}<x<2时,分式\frac{3x+2}{x-2}的值为负数。$