# How do you solve and graph  -3 ≤ 3 - 2x< 11?

Jul 5, 2017

See a solution process below:

#### Explanation:

First, subtract $\textcolor{red}{3}$ from each segment of the system of inequalities to isolate the $x$ term while keeping the system balanced:

$- \textcolor{red}{3} - 3 \le - \textcolor{red}{3} + 3 - 2 x < - \textcolor{red}{3} + 11$

$- 6 \le 0 - 2 x < 8$

$- 6 \le - 2 x < 8$

Now, divide each segment by $\textcolor{b l u e}{- 2}$ to solve for $x$ while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

$\frac{- 6}{\textcolor{b l u e}{- 2}} \textcolor{red}{\ge} \frac{- 2 x}{\textcolor{b l u e}{- 2}} \textcolor{red}{>} \frac{8}{\textcolor{b l u e}{- 2}}$

$3 \textcolor{red}{\ge} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 2}}} x}{\cancel{\textcolor{b l u e}{- 2}}} \textcolor{red}{>} - 4$

$3 \textcolor{red}{\ge} x \textcolor{red}{>} - 4$

Or

$x > - 4$ and $x \le 3$

Or, in interval notation:

$\left(- 4 , 3\right]$