#### Answer

$(-\infty,5)\cup(5,\frac{13}{2})$

#### Work Step by Step

Step 1. Rewrite the inequality as
$(x-5)^2(x-\frac{13}{2})\lt0$
The boundary points are
$x=5,\frac{13}{2}$
Step 2. Using the test points to examine signs across the boundary points, we have
$...(-)...(5)...(-)...(\frac{13}{2})...(+)...$
Thus, the solutions are
$x\lt\frac{13}{2}$ plus $x\ne5$
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
$(-\infty,5)\cup(5,\frac{13}{2})$