Question

How do you graph and solve `|2x| ≥ 6`?

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.

`-6 >= 2x >= 6`

Now, divide each side of the system of inequalities by `color(red)(2)` to solve for `x` while keeping the system balanced:

`-6/color(red)(2) >= (2x)/color(red)(2) >= 6/color(red)(2)`

`-3 >= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) >= 3`

`-3 >= x >= 3`

Or

`x <= -3` and `x >= 3`

Or, in interval notation:

`(-oo, -3]` and `[3, +oo)`

To graph this we will draw two vertical lines at `-3` and `3` on the horizontal axis.

The lines will be solid lines because the inequality operators contain an "or equal to" clause.

We will shade to the left and right side of the lines respectively to show the intervals: