Help with an absolute value inequality?

Question

Can't figure this one out because I'm in school through a program and the book pretty much always adds one or two problems to the end without explaining them at all. It's quite annoying, but this is the last one I need to figure out for this section.

|x + 2| - x ≥ 0

The answer sheet asks for an answer and a graph, and I don't even know where to start. It's really the - x that throws me off. Thanks in advance (and later too I suppose) for any answers.

Answer 1

|x + 2| - x ≥ 0

With inequalities with one or more absolute values I divide the number line into at the point where the argument in the absolute value(s) is zero.

In your case the division is -2

CASE I

First consider to the left of the division.

x ≤ -2

in this section the absolute value of x +2 = -x-2

(This looks like a lot of negative signs for absolute values, but where the argument is negative minus a negative is POSITIVE.)

So we get:

-x-2-x≥ 0

-2x ≥ 2

x ≤ -1

Remembering our hypothesis that x ≤ -2 AND our solution x≤-1.

Which can be written as:

x ≤ -2 (Since that is the more restrictive.)

We are only half done.

Case II

Now consider x> -2 in this case the absolute value of x + 2 equals x + 2, so we get:

x + 2 - x ≥ 0

the x's cancel and we get the true statement: 2 ≥ 0

That tells us that all numbers in our the hypothesis x > -2 work.

Our solution is the union of the solutions in case I and case II;

x ≤ -2 ∪ x > -2,

Which is a lot of work to say what we could have noticed in the beginning ALL NUMBERS WORK.