Question

How do you solve `-2| 4x + 3| < - 8`?

Answer 1

Please see the explanation.

Explanation:

Divide by -2 (this changes the inequality from < to >):

`|4x + 3| > 4`

The absolute value function splits into two equations based on the sign within the absolute value; here is how it is defined `|a| = a; a >= 0 and |a| = -a; a < 0`:

`4x + 3 > 4; 4x + 3>= 0` and `-(4x + 3) > 4; 4x + 3 < 0`

Simplify the restrictions:

`4x + 3 > 4; x >= -3/4` and `-(4x + 3) > 4; x < -3/4`

Multiply the second inequality by -1:

`4x + 3 > 4; x >= -3/4` and `4x + 3 < -4; x < -3/4`

Subtract 3 from both inequalities:

`4x > 1; x >= -3/4` and `4x < -7; x < -3/4`

Divide both inequalities by 4:

`x > 1/4; x >= -3/4` and `x < -7/4; x < -3/4`

Because both inequalities are contained within the restrictions, drop the restrictions:

`x > 1/4` and `x < -7/4`