lx - 2l - lx + 2l > x

Question

The absolute value of x - 2 minus the absolute value of x + 2 greater than x.

What is the Answer Please Help!

Answer 1

x-2 > x + │x+2│ OR -x+2 > x + │x+2│
-2 > │x+2│ or -2x + 2 > │x+2│

-2 > │x+2│ is the same as │x+2│ < -2 which is not true by definition of absolute value.
So -2x + 2 > │x+2│
then │x+2│ < -2x + 2
then x+2 < -2x+2 AND -x-2 < -2x + 2
3x < 0 AND x < 4

so x < 0

Answer 2

Ask yourself this:

1. If x>0, is |x - 2| - |x + 2| positive or negative or zero? (or think of it as "is |x - 2| > |x + 2| ?"

2. If x==0, then clearly |x - 2| - |x + 2| is zero.

3. Now, the interesting one: what if x is negative? Then

is |x - 2| > |x + 2| ?