Thread: Using triangle inequality to prove inequalities

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?

We can see that its minimum occurs at , i.e

We also find the same for

Conclusion follows

o_O: Sorry but that does not answer my question. I am supposed to use the triangle inequality to prove this.

Haha look at this ^^


by the property of absolute value, that is


Now consider
this is in the form , which is by triangle inequality.

Hence

Hahah, wow. How did I miss that? Thanks!