Inequality proof for all $x,y,z \gt 0$

Question

$$\frac{xy}{z}+\frac{xz}{y}+\frac{zy}{x}\ge x+y+z$$

I want to prove this inequality for all $x,y,z \gt 0$

I started by multiply both sides by $2$, but the factorisation on the left-side was confusing.

Answer 1

Use Am-Gm $$\frac{xy}{z}+\frac{xz}{y}\geq 2x$$ $$\frac{yz}{x}+\frac{xz}{y}\geq 2z$$ $$\frac{xy}{z}+\frac{yz}{x}\geq 2y$$ Q.E.D