# How do you solve and write the following in interval notation: 10<=-2(x-2)<=20?

Oct 8, 2017

See a solution process below:

#### Explanation:

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

$\frac{10}{\textcolor{b l u e}{- 2}} \textcolor{red}{\ge} \frac{- 2 \left(x - 2\right)}{\textcolor{b l u e}{- 2}} \textcolor{red}{\ge} \frac{20}{\textcolor{b l u e}{- 2}}$

$- 5 \textcolor{red}{\ge} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 2}}} \left(x - 2\right)}{\cancel{\textcolor{b l u e}{- 2}}} \textcolor{red}{\ge} - 10$

$- 5 \textcolor{red}{\ge} x - 2 \textcolor{red}{\ge} - 10$

Now, add $\textcolor{red}{2}$ to each segment to solve for $x$ while keeping the system balanced:

$- 5 + \textcolor{red}{2} \ge x - 2 + \textcolor{red}{2} \ge - 10 + \textcolor{red}{2}$

$- 3 \ge x - 0 \ge - 8$

$- 3 \ge x \ge - 8$

Or

$x \le - 3$ and $x \ge - 8$

Or

$x \ge - 8$ and $x \le - 3$

Or, in interval notation:

$\left[- 8 , - 3\right]$