Absolute Value Inequalities - reference question?

Question

How would I go about solving an inequality with two sets of absolute value on one side? For example: (just so you knoe, when i say >= i mean 'greater than or equal to')

|x+2| - |x| >= 2

I said that it holds true for all real values of x, but i got that answer vie guessing and checking. Is there an algebraic way to solve this?

NOTE: I am genuinely interested in learning the PRCEDURE for solving somehting like this. This problem is NOT on my homework, so I'd appreciate helpful/constructive answes only.

Answer 1

With absolutes, you would have to look at both equations:

Assuming x is positive, the the equation would be
x+2 - x >= 2 which would lead to x-x >= 2-2. Because 0=0, this would hold for all positive values of x.

The equation can also read as abs(x+2) >= 2 - abs(x). Now if x is negative, then:

-(-x)+2 >= 2 - (-x) which leads to 2+x >= 2 + x which is also true.