#### Answer

Solution set: $(-\displaystyle \infty,5)\cup(5,\frac{13}{2})$

#### Work Step by Step

Follow the "Procedure for Solving Polynomial lnequalities",\ p.412:
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$(5-x)^{2}(x-\displaystyle \frac{13}{2}) < 0$
$f(x)=(5-x)^{2}(x-\displaystyle \frac{13}{2})$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(5-x)^{2}(x-\displaystyle \frac{13}{2})=0$
$x=5$ or $x=\displaystyle \frac{13}{2}$
3. Locate these boundary points on a number line, thereby dividing the number line into intervals.
4. Test each interval's sign of $f(x)$ with a test value, $a$, from that interval,
$\begin{array}{llll}
Interval & a & f(a),signs & f(a) < 0 ? \\
& & (5-a)^{2}(a-\frac{13}{2}) & \\
(-\infty,5) & 0 & (+)(-) & T\\
(5,\frac{13}{2}) & 6 & (+)(-) & T\\
(\frac{13}{2},\infty) & 10 & (+)(+) & F
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $(-\displaystyle \infty,5)\cup(5,\frac{13}{2})$