Answer
$(1,2)$
Work Step by Step
Step 1. Rewrite the inequality as $\frac{x}{x-1}-2\gt0$, which gives $\frac{x-2x+2}{x-1}\gt0$, $\frac{-x+2}{x-1}\gt0$, and $\frac{x-2}{x-1}\lt0$
Thus the boundary points are $x=1,2$
Step 2. Using the test points to examine signs across the boundary points, we have
$...(+)...(1)...(-)...(2)...(+)...$
Thus the solutions are $1\lt x\lt2$
Step 3. We can express the solutions on a real number line as shown in the figure.
Step 4. We can express the solutions in interval notation as $(1,2)$
