Answer
$(0,\frac{4}{11})U(\frac{1}{2},\infty)$
Work Step by Step
Step 1. Rewrite the inequality as $\frac{3}{2x-1}+\frac{4}{x}\gt0$ or $\frac{11x-4}{x(2x-1)}\gt0$
Step 2. Identify the boundary point as $x=0,\frac{4}{11},\frac{1}{2},$
Step 3. Use test points at $x=-1, \frac{1}{3},\frac{2}{5},1$, the signs of the left side rational are $-,+, -,+$
Step 4. The inequality requires positive (2nd and 4th regions), thus the solution intervals are $(0,\frac{4}{11})U(\frac{1}{2},\infty)$