Answer
$\left(-7, -\frac{5}{2}\right] \cup (5, \infty)$
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 Rational function.
$\displaystyle \frac{2x+5}{x^2+2x-35} \geq 0$
$f(x)=\displaystyle \frac{2x+5}{x^2+2x-35}$
2. The cut points are:
$\displaystyle \frac{2x+5}{(x+7)(x-5)} = 0$
$x=-7$ or $x=\frac{-5}{2}$ or $x=5$
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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \geq 0 ? \\
& & \displaystyle \frac{2a+5}{(a+7)(a-5)} & \\
(-\infty,-7) & -10 & \frac{(-)}{(-)(-)}=(-) & F\\
(-7,-\frac{5}{2}) & -3 & \frac{(-)}{(+)(-)}=(+) & T\\
(-\frac{5}{2}, 5) & 0 & \frac{(+)}{(+)(-)}=(-) & F\\
(5,\infty) & 10 & \frac{(+)}{(+)(+)}=(+) & T\\
\end{array}$
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: $\left(-7, -\frac{5}{2}\right] \cup (5, \infty)$
