# How do you solve and write the following in interval notation:  | 3x – 2 | ≥ 4?

Feb 19, 2017

$\left(- \infty , - \frac{2}{3}\right] \cup \left[2 , + \infty\right)$

#### Explanation:

The starting premise is.

$| x | \ge a$

$\Rightarrow x \ge a \text{ or } x \le - a$

$\text{Applying to } | 3 x - 2 | \ge 4$

$\Rightarrow 3 x - 2 \ge 4 \text{ or } 3 x - 2 \le - 4$

Solve each inequality.

$\textcolor{b l u e}{\text{First inequality}}$

$3 x - 2 \ge 4$

$3 x \cancel{- 2} \cancel{+ 2} \ge 4 + 2$

$\Rightarrow 3 x \ge 6$

divide both sides by 3

$\frac{\cancel{3} x}{\cancel{3}} \ge \frac{6}{3}$

$\Rightarrow \textcolor{red}{x \ge 2} \leftarrow \text{ solution}$

$\textcolor{b l u e}{\text{Second inequality}}$

$3 x - 2 \le - 4$

$\Rightarrow 3 x \le - 2$
$\Rightarrow \textcolor{red}{x \le - \frac{2}{3}} \leftarrow \text{ solution}$
The combined solution is $x \ge 2 \text{ or } x \le - \frac{2}{3}$
$\text{Expressed in interval notation as}$
$\left(- \infty , - \frac{2}{3}\right] \cup \left[2 , + \infty\right)$