Answer
$(-\infty,2)\cup(3,5) $.
Work Step by Step
Step 1. For $\frac{1}{x-2}-\frac{2}{3x-9}\lt0\Longrightarrow \frac{x-5}{3(x-2)(x-3)}\lt0$, identify boundary points as $x=2,3,5$.
Step 2. Form intervals $(-\infty,2),(2,3),(3,5),(5,\infty)$.
Step 3. Choose test values for each interval $x=0,2.5,4,6$.
Step 4. Test the inequality to get results: $True,\ False,\ True,\ False $
Step 5. We have the solution $(-\infty,2)\cup(3,5) $.
