Answer
$\left(-2,\frac{1}{7}\right]$
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.
$\displaystyle \frac{3x+1}{x+2} \leq \frac{2}{3}$
$\displaystyle \frac{3x+1}{x+2} -\frac{2}{3}\leq 0,$
$\displaystyle \frac{9x+3-2x-4}{3x+6} \leq0,$
$\displaystyle \frac{7x-1}{3x+6}\leq0,$
2. The cut points are: $\displaystyle \frac{7x-1}{3x+6}= 0$
$x=-2$ or $x=\frac{1}{7}$
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,
$\begin{array}{llll} Intervals: & a=test.v. & f(a),signs & f(a) \leq 0 ? \\ & & \displaystyle \frac{7a-1}{3a+6}& \\
(-\infty,-2) & -10 & \frac{(-)}{(-)}=(+) & F
\\ (-2,\frac{1}{7}) & 0 & \frac{(-)}{(+)}=(-) & T
\\ (\frac{1}{7},\infty) & 10 & \frac{(+)}{(+)}=(+) & F
\\ \end{array}$
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: $\left(-2,\frac{1}{7}\right]$
