Given that |2x+1| > x-1 Find x?

Question

please solve

Answer 1

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

• First we consider the case where 2x + 1 ≥ 0, i.e. x ≥ - 1/2. In this case |2x + 1| = 2x + 1, so we can write the inequality as 2x + 1 > x - 1.

Solve the 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.

• Now consider the case 2x + 1 < 0, i.e. x < - 1/2. In this case |2x + 1| = - (2x + 1) = - 2x - 1, so we can write the inequality as
  - 2x - 1 > x - 1.

Add 1 to each side.

- 2x > x

Add 2x to each side.

- 2x + 2x > x + 2x

0 > 3x

Divide each side by 3.

x < 0.

So a real number x is a solution of the original inequality if x ≥ - 1/2 and x > - 2

                                                                                                  or if x < - 1/2 and x < 0.

Thus the set of solutions is ( - ∞, ∞ ) or all real numbers.