Absolute Value Inequalities : For all real numbers a and b, b > 0, the following statements are true.

1. If | a | < b, then - b < a < b.

2. If | a | > b, then a > b or a < - b.

The Absolute Value Inequality is | 2x + 1 | > x - 1.

Here b = x - 1 > 0 ⇒ x > 1.

| 2x + 1 | > (x - 1) is equivalent to 2x + 1 > x - 1 or 2x + 1 < - (x - 1) and Solve the inequality.

Solve the first inequality : 2x + 1 > x - 1 for x.

Subtract 1 from each side.

2x + 1 - 1 > x - 1 - 1

2x > x - 2

Subtract x from each side.

2x - x > x - 2 - x

x > - 2.

Solve the second inequality : 2x + 1 < - (x - 1) for x.

2x + 1 < - x + 1

Subtract 1 from each side.

2x + 1 - 1 < - x + 1 - 1

2x < - x

Add x to each side.

2x + x < - x + x

3x < 0

Divide each side by 3.

x < 0.

The solution set is x > - 2 or x < 0.

But substitute the any real value for x in the original inequality, then satisfy the original inequality.