# How do you solve -15\leq - 4x + 5< - 3?

May 22, 2017

See a solution process below:

#### Explanation:

First, subtract $\textcolor{red}{5}$ from each segment of the system of inequalities to isolate the $x$ term while keeping the system balanced:

$- 15 - \textcolor{red}{5} \le - 4 x + 5 - \textcolor{red}{5} < - 3 - \textcolor{red}{5}$

$- 20 \le - 4 x + 0 < - 8$

$- 20 \le - 4 x < - 8$

Next, divide each segment by $\textcolor{b l u e}{- 4}$ to solve for $x$ while keeping the system balanced. However, because we are multiplying or dividing an inequality by a negative number we must reverse the inequality operators:

$\frac{- 20}{\textcolor{b l u e}{- 4}} \textcolor{red}{\ge} \frac{- 4 x}{\textcolor{b l u e}{- 4}} \textcolor{red}{>} \frac{- 8}{\textcolor{b l u e}{- 4}}$

$5 \textcolor{red}{\ge} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 4}}} x}{\cancel{\textcolor{b l u e}{- 4}}} \textcolor{red}{>} 2$

$5 \textcolor{red}{\ge} x \textcolor{red}{>} 2$

Or

$x \le 5$; $x > 2$

Or, in interval notation:

$\left(2 , 5\right]$