# How do you solve -6 ≤ 2 - 3m ≤ 7?

Apr 21, 2018

See a solution process below:

#### Explanation:

First, subtract $\textcolor{red}{2}$ from each segment of the system of inequalities to isolate the $m$ term while keeping the system balanced:

$- 6 - \textcolor{red}{2} \le 2 - \textcolor{red}{2} - 3 m \le 7 - \textcolor{red}{2}$

$- 8 \le 0 - 3 m \le 5$

$- 8 \le - 3 m \le 5$

Now, divide each segment by $\textcolor{b l u e}{- 3}$ to solve for $m$ while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

$\frac{- 8}{\textcolor{b l u e}{- 3}} \textcolor{red}{\ge} \frac{- 3 m}{\textcolor{b l u e}{- 3}} \textcolor{red}{\ge} \frac{5}{\textcolor{b l u e}{- 3}}$

$\frac{8}{3} \textcolor{red}{\ge} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 3}}} m}{\cancel{\textcolor{b l u e}{- 3}}} \textcolor{red}{\ge} - \frac{5}{3}$

$\frac{8}{3} \textcolor{red}{\ge} m \textcolor{red}{\ge} - \frac{5}{3}$

Or

$m \ge - \frac{5}{3}$ and $m \le \frac{8}{3}$

Or, in interval notation:

$\left[- \frac{5}{3} , \frac{8}{3}\right]$