Cyclic Absolute value equality

Question

Prove$$|x|+|y|+|z|\le|x+y-z|+|x-y+z|+|-x+y+z|$$

I'm not sure how to even start. Please provide some hints.

I know we can assume $x\ge y\ge z$ but to proceed further we need to assume their sign which can't be done without lose of generality

Answer 1

Hint: $\;|x+y-z|+|x-y+z| \ge 2|x|\,$ by the triangle inequality. Repeat.