Answer
$ (2,\frac{13}{5}]$
Work Step by Step
1. Based on the given conditions, we have:
$\frac{3}{x-2}\ge5\Longrightarrow \frac{3}{x-2}-5\ge0\Longrightarrow \frac{13-5x}{x-2}\ge0$
2. Boundary points $x=2,\frac{13}{5}$, form intervals $(-\infty,2),(2,\frac{13}{5}],[\frac{13}{5},\infty)$
3. Choose test values $x=0,2.5,3$, test the inequality and get results $False,\ True,\ False$
4. Thus, we have the solution interval: $ (2,\frac{13}{5}]$
