Question

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

Answer 1

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 > 4x - 3 > 13`

First, add `color(red)(3)` to each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`-13 + color(red)(3) > 4x - 3 + color(red)(3) > 13 + color(red)(3)`

`-10 > 4x - 0 > 16`

`-10 > 4x > 16`

Now, divide each segment by `color(red)(4)` to solve for `x` while keeping the system balanced:

`-10/color(red)(4) > (4x)/color(red)(4) > 16/color(red)(4)`

`-10/4 > (color(red)(cancel(color(black)(4)))x)/cancel(color(red)(4)) > 4`

`-5/2 > x > 4`

Or

`x < -5/2`; `x > 4`

Or, in interval notation:

`(-oo, -5/2)`; `(4, +oo)`

To graph this we will draw vertical lines at `-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: