Answer
$(-\infty,3)\cup(4,\infty)$
Work Step by Step
Step 1. Rewrite the inequality as
$\frac{1}{x-3}-1\lt0$
which gives
$\frac{1-x+3}{x-3}\lt0$, $\frac{-x+4}{x-3}\lt0$, and $\frac{x-4}{x-3}\gt0$
Thus the boundary points are $x=3,4$
Step 2. Using test points to examine signs across the boundary points, we have
$...(+)...(3)...(-)...(4)...(+)...$
Thus the solutions are
$x\lt3$ plus $x\gt4$
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 $(-\infty,3)\cup(4,\infty)$
