# How do you solve and write the following in interval notation: 2< -3x + 2<=14?

Apr 16, 2017

See the entire solution process below:

#### Explanation:

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

$2 - \textcolor{red}{2} < - 3 x + 2 - \textcolor{red}{2} \le 14 - \textcolor{red}{2}$

$0 < - 3 x + 0 \le 12$

$0 < - 3 x \le 12$

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

$\frac{0}{\textcolor{b l u e}{- 3}} \textcolor{red}{>} \frac{- 3 x}{\textcolor{b l u e}{- 3}} \textcolor{red}{\ge} \frac{12}{\textcolor{b l u e}{- 3}}$

$0 \textcolor{red}{>} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 3}}} x}{\cancel{\textcolor{b l u e}{- 3}}} \textcolor{red}{\ge} - 4$

$0 > x \ge - 4$

Or

$x < 0$ and $x \ge - 4$

In interval notation [-4, 0)