How would you prove $\frac{1}{xz+x}+\frac{1}{xy+y}+\frac{1}{zy+z} \geq \frac{3}{2}$?

Question

If $x$, $y$, $z$ are integers with $xyz=1$, $\frac{1}{xz+x}+\frac{1}{xy+y}+\frac{1}{zy+z} \geq \frac{3}{2}$ any suggestions to proceed?

I think would be fine to work them out separately. Thanks

Answer 1

You can disprove it for example of they all = 3.