Proofs by Case (1)

Write clear and complete proofs by case for the following statement.

(1) If x is a real number, then |x + 3| - x > 2.

Proof. Assume that x is a real number.

- Case 1: Assume that x >= -3, so that x + 3 >= 0. In this case |x + 3| = x + 3. So we have

              |x + 3| - x = x + 3 - x = 3 > 2.

- Case 2: Now assume that x < - 3, so that x + 3 < 0. In this case |x + 3| = -(x + 3) = - 3 - x. So we have

|x + 3| - x = - 3 - x -x = - 2x - 3 > -2(-3) -3 = 6 - 3 = 3 > 2.

So in all cases |x+3| - x > 2.

QED