How do you solve and write the following in interval notation:  |1 + 5x| > 11?

Mar 17, 2017

$\left(- \infty , - \frac{12}{5}\right) \cup \left(2 , + \infty\right)$

Explanation:

To solve inequalities of this type, we have to solve 2 inequalities.

$| x | > a \text{ then " x< color(red)(-a)" or } x > \textcolor{red}{a}$

$\Rightarrow 1 + 5 x < \textcolor{red}{- 11} \text{ or } 1 + 5 x > \textcolor{red}{11}$

$\textcolor{b l u e}{\text{solve }} 1 + 5 x < - 11$

subtract 1 from both sides.

$\cancel{1} \cancel{- 1} + 5 x < - 11 - 1$

$\Rightarrow 5 x < - 12$

divide both sides by 5

$\frac{\cancel{5} x}{\cancel{5}} < \frac{- 12}{5}$

$\Rightarrow x < - \frac{12}{5} \leftarrow \textcolor{red}{\text{first solution}}$

$\textcolor{b l u e}{\text{solve }} 1 + 5 x > 11$

$\Rightarrow 5 x > 10$

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

$\text{Expressing the solution in interval notation}$

$\left(- \infty , - \frac{12}{5}\right) \cup \left(2 , + \infty\right)$