How do you graph and solve abs{4x -3} > 13?

Jan 19, 2018

See a solution process below:

Explanation:

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.

$- 13 > 4 x - 3 > 13$

First, add $\textcolor{red}{3}$ to each segment of the system of inequalities to isolate the $x$ term while keeping the system balanced:

$- 13 + \textcolor{red}{3} > 4 x - 3 + \textcolor{red}{3} > 13 + \textcolor{red}{3}$

$- 10 > 4 x - 0 > 16$

$- 10 > 4 x > 16$

Now, divide each segment by $\textcolor{red}{4}$ to solve for $x$ while keeping the system balanced:

$- \frac{10}{\textcolor{red}{4}} > \frac{4 x}{\textcolor{red}{4}} > \frac{16}{\textcolor{red}{4}}$

$- \frac{10}{4} > \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} x}{\cancel{\textcolor{red}{4}}} > 4$

$- \frac{5}{2} > x > 4$

Or

$x < - \frac{5}{2}$; $x > 4$

Or, in interval notation:

$\left(- \infty , - \frac{5}{2}\right)$; $\left(4 , + \infty\right)$

To graph this we will draw vertical lines at $- \frac{5}{2}$ and $4$ on the horizontal axis.

The lines will be a dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade to the left and right side of the lines respecitively: