I have been given the triangle inequality:

||x| - |y|| <= |x + y| <= |x| + |y|

where |x| is the absolute value of x.

Now I'm supposed to prove the following:

|3x + 2| + |3x - 2| >= 4 for all real x.

Now, I tried this:

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

This does not solve the problem. How do I do this?