Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.
Problem
Source: Kyiv mathematical festival 2012
Tags: Inequality, 3-variable inequality, algebra, inequalities