For real numbers $x , y$ and $z$, show that $$| x | + | y | + | z | \leq | x + y - z | + | y + z - x | + | z + x - y |$$
Rewrite the given inequality in terms of the new variables $\alpha = x + y - z$, $\beta = y + z - x , \gamma = x + z - y$, and use the triangle inequality.
For real numbers $x , y$ and $z$, show that
$$| x | + | y | + | z | \leq | x + y - z | + | y + z - x | + | z + x - y |$$