Answer
$(-\infty, 3)\cup(4,\infty)$
Work Step by Step
Step 1. Move all the terms to the left side, we have $\frac{x+1}{x-3}-5\lt0\longrightarrow \frac{x+1-5x+15}{x-3}\lt0\longrightarrow \frac{-4x+16}{x-3}\lt0$
Step 2. Identify the boundary points as $x=3$ and $x=4$
Step 3. The boundary points separate the number line into three intervals $(-\infty, 3)$, $(3, 4)$ and $(4,\infty)$
Step 4. Use test points $x=0,3.5,5$ for each interval, we can determine the signs of the left side of the inequality as $-, +, -$
Step 5. Based on the signs, we can write the solution set as $(-\infty, 3)\cup(4,\infty)$
