Answer
$\left[-8,-\frac{5}{2}\right)$
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 Rational function.
$\displaystyle \frac{(x-3)}{(2x+5)} \geq 1$
$f(x)=\displaystyle \frac{(x-3)}{(2x+5)} \geq 1$,
$f(x)=\displaystyle \frac{(x-3)}{(2x+5)} -1\geq 0$,
$f(x)=\displaystyle \frac{x-3-2x-5}{(2x+5)} \geq 0$,
$f(x)=\displaystyle \frac{(-x-8)}{(2x+5)} \geq 0$,
2. The cut points are:
$\displaystyle \frac{(-x-8)}{(2x+5)} =0$
$x=-8$ 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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \geq 0 ? \\
& & \displaystyle \displaystyle \frac{(-a-8)}{(2a+5)} & \\
(-\infty,-8) & -10 & \frac{(+)}{(-)}=(-) & F\\
(-8, -\frac{5}{2}) & -3 & \frac{(-)}{(-)}=(+) & T\\
(-\frac{5}{2},\infty) & 0 & \frac{(-)}{(+)}=(-) & F\\
\end{array}$
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: $\left[-8,-\frac{5}{2}\right)$
