Triangle Inequality and Pseudometric

**The problem statement, all variables and given/known data**

where a>0, b>0, c>0 and

Show whether exhibits Triangle inequality?

__Relevant equations:__

(M4) (for all x,y and z in X)

__The attempt at a solution__

I started my solution by solving by squaring the both sides of the equation separately.

I am tending to think it does not satisfy the triangle inequality any other simple way to prove it? Also is this a pseudometric? if it does not satisfy the triangle inequality?