Answer
$[-5, 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.
$x^3+x^2-17x+15 \geq 0$,
$(x-1)(x+5)(x-3) \geq 0$
$f(x)=(x-1)(x+5)(x-3)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x-1)(x+5)(x-3)=0$
$x=1$ or $x=-5$ or $x=3$
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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \geq 0 ? \\
& &(a-1)(a+5)(a-3)& \\
(-\infty, -5) & -6 & (-)(-)(-) & F\\
(-5, 1) & 0 & (-)(+)(-) & T\\
(1, 3) & 2 & (+)(+)(-) & F\\
(3,\infty) & 4 & (+)(+)(+) & T
\end{array}$
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: $[-5, 1] \cup [3, \infty) $
