, 2017

Solution:

Note that reflecting for each choice of sign for $x$, $y$, $z$, we get new regions. Therefore, we can restrict to the case where $x, y, z > 0$. In this case, the sign of the expression only depends on $(x - y)(y - z)(z - x)$. It is easy to see that for this expression, every one of the $3! = 6$ orderings for $\{x, y, z\}$ contributes a region.

Therefore, our answer is $2^3 \cdot 3! = 48$.