Question

How do you solve the inequality `abs(2x-3)>=5`?

Answer 1

See a solution process below:

Explanation:

The absolute value function takes any negative or positive 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.

`-5 >= 2x - 3 >= 5`

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

`-5 + color(red)(3) >= 2x - 3 + color(red)(3) >= 5 + color(red)(3)`

`-2 >= 2x - 0 >= 8`

`-2 >= 2x >= 8`

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

`-2/color(red)(2) >= (2x)/color(red)(2) >= 8/color(red)(2)`

`-1 >= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) >= 4`

`-1 >= x >= 4`

Or

`x <= -1`; `x >= 4`

Or, in interval notation:

`(-oo, -1]; [4, +oo)`