How do you solve the inequality abs(2x) + 6 < 22?

Jun 16, 2018

See a solution process below:

Explanation:

First, subtract $\textcolor{red}{6}$ from each side of the inequality to isolate the absolution value function while keeping the inequality balanced:

$\left\mid 2 x \right\mid + 6 - \textcolor{red}{6} < 22 - \textcolor{red}{6}$

$\left\mid 2 x \right\mid + 0 < 16$

$\left\mid 2 x \right\mid < 16$

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

$- 16 < 2 x < 16$

Divide each segment of the system of inequalities by $\textcolor{red}{2}$ to solve for $x$ while keeping the system balanced:

$- \frac{16}{\textcolor{red}{2}} < \frac{2 x}{\textcolor{red}{2}} < \frac{16}{\textcolor{red}{2}}$

$- 8 < \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} x}{\cancel{\textcolor{red}{2}}} < 8$

$- 8 < x < 8$

Or

$x > - 8$; $x < 8$

Or, in interval notation:

$\left(- 8 , 8\right)$