Question

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

Answer 1

See a solution process below:

Explanation:

First, subtract `color(red)(6)` from each side of the inequality to isolate the absolution value function while keeping the inequality balanced:

`abs(2x) + 6 - color(red)(6) < 22 - color(red)(6)`

`abs(2x) + 0 < 16`

`abs(2x) < 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 < 2x < 16`

Divide each segment of the system of inequalities by `color(red)(2)` to solve for `x` while keeping the system balanced:

`-16/color(red)(2) < (2x)/color(red)(2) < 16/color(red)(2)`

`-8 < (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) < 8`

`-8 < x < 8`

Or

`x > -8`; `x < 8`

Or, in interval notation:

`(-8, 8)`