Answer
$(-\infty,-5)\cup\left(-\frac{5}{2},3\right)$
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.
$(x-3)(x+5)(2x+5) \lt 0$
$f(x)=(x-3)(x+5)(2x+5)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x-3)(x+5)(2x+5)=0$
$x=3$ or $x=-5$ or $x=-\frac{5}{2}$
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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& & (a-3)(a+5)(2a+5) & \\
(-\infty,-5) & -6 & (-)(-)(-) & T\\
(-5,-\frac{5}{2}) & -3 & (-)(+)(-) & F\\
(-\frac{5}{2},3) & 0 & (-)(+)(+) & T\\
(3,\infty) & 5 & (+)(+)(+) & F
\end{array}$
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: $(-\infty,-5)\cup\left(-\frac{5}{2},3\right)$
