DE-MO 2006 Problem 5

Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$ . Then prove $$|x| \geq \frac{1}{|a|+|b|+|c|-1}.$$