Answer
Solution set: $ [-1,7]$
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.
$f(x)=(x+1)(x-7)\leq 0$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x+1)(x-7)=0$
$x=-1$ or $x=7$
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,
$\left[\begin{array}{llll}
Intervals: & (-\infty, -1) & (-1,7) & (7,\infty)\\
a=test.val. & -10 & 0 & 10\\
f(a) & (-9)(-17) & (1)(-7) & (11)(3)\\
f(a) \leq 0 ? & F & T & F
\end{array}\right]$
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: $ [-1,7]$
