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