## College Algebra (6th Edition)

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