Answer
Solution set: $(-\infty,-4]\cup[-2, 1]\cup[3, \infty)$
Work Step by Step
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.
$(x-1)(x+2)(x-3)(x+4) \geq 0$
$f(x)=(x-1)(x+2)(x-3)(x+4)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x-1)(x+2)(x-3)(x+4)=0$
$x=1$ or $x=-2$ or $x=3$ or $x=-4$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \geq 0 ? \\
& & (a-1)(a+2)(a-3)(a+4) & \\
(-\infty,-4) & -6 & (-)(-)(-)(-) & T\\
(-4,-2) & -3 & (-)(-)(-)(+) & F\\
(-2,1) & 0 & (-)(+)(-)(+) & T\\
(1,3) & 2 & (+)(+)(-)(+) & F\\
(3,\infty) & 5 & (+)(+)(+)(+) & T
\end{array}$
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: $(-\infty,-4]\cup[-2, 1]\cup[3, \infty)$
