Answer
$(-\infty,2)U(3,5) $
Work Step by Step
Step 1. Rewrite the inequality $\frac{1}{x-2}-\frac{2}{3x-9}\lt0$ as $\frac{x-5}{(x-2)(3x-9)}\lt0$. Identify boundary points (zeros and asymptotes) as $x=2,3,5$ which divide the x-axis into four intervals $(-\infty,2),(2,3),(3,5),(5,\infty)$
Step 2. Choose test values in each interval $x=0,2.5,4,6$ and test the inequality to get $True,\ False,\ True,\ False$
Step 3. Thus the solution intervals are $(-\infty,2)U(3,5) $
