Answer
Solution set: $[-3,-2]\cup[-1,\infty)$
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.
$(x+1)(x+2)(x+3) \geq 0$
$f(x)=(x+1)(x+2)(x+3)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x+1)(x+2)(x+3)=0$
$x=-1$ or $x=-2$ or $x=-3$
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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \geq 0 ? \\
& & (a+1)(a+2)(a+3) & \\
(-\infty,-3) & -5 & (-)(-)(-) & F\\
(-3,-2) & -2.5 & (-)(-)(+) & T\\
(-2,-1) & -1.5 & (-)(+)(+) & F\\
(-1,\infty) & 5 & (+)(+)(+) & T
\end{array}$
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: $[-3,-2]\cup[-1,\infty)$
