# How do you solve and write the following in interval notation: | -3x | + 2 ≤ 8?

Apr 28, 2017

See the entire solution process below:

#### Explanation:

First, subtract $\textcolor{red}{2}$ from each side of the inequality to isolate the absolute value function while keeping the inequality balanced:

$\left\mid - 3 x \right\mid + 2 - \textcolor{red}{2} \le 8 - \textcolor{red}{2}$

$\left\mid - 3 x \right\mid + 0 \le 6$

$\left\mid - 3 x \right\mid \le 6$

The absolute value function takes any positive or negative term and transforms it to its positive form. Therefore, you must solve the term within the absolute value function for both its positive and negative equivalent:

$- 6 \le - 3 x \le 6$

We need to divide each segment of the system of inequalities 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:

$\frac{- 6}{\textcolor{b l u e}{- 3}} \textcolor{red}{\ge} \frac{- 3 x}{\textcolor{b l u e}{- 3}} \textcolor{red}{\ge} \frac{6}{\textcolor{b l u e}{- 3}}$

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

$2 \ge x \ge - 2$

Or

$x \le 2$ and $x \ge - 2$

Or, in interval notation:

$\left[- 2 , 2\right]$