# How do you solve and write the following in interval notation:−12 <3 − 3x ≤ 15?

Jun 29, 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:

$- \textcolor{red}{3} - 12 < - \textcolor{red}{3} + 3 - 3 x \le - \textcolor{red}{3} + 15$

$- 15 < 0 - 3 x \le 12$

$- 15 < - 3 x \le 12$

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 inequalities by a negative number we must reverse the inequality operators:

$\frac{- 15}{\textcolor{b l u e}{- 3}} \textcolor{red}{>} \frac{- 3 x}{\textcolor{b l u e}{- 3}} \textcolor{red}{\ge} \frac{12}{\textcolor{b l u e}{- 3}}$

$5 \textcolor{red}{>} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 3}}} x}{\cancel{\textcolor{b l u e}{- 3}}} \textcolor{red}{\ge} - 4$

$5 \textcolor{red}{>} x \textcolor{red}{\ge} - 4$

Or

$x < 5$ and $x \ge - 4$

Or, in interval notation:

$\left[- 4 , 5\right)$