Proving $\forall x, |x| \leq 2 => |x^3 - 7x + 3| \leq 25 $

Question

I'm trying to prove the following inequality:

$\forall x, |x| \leq 2 => |x^3 - 7x + 3| \leq 25 $

Suppose $ |x| \leq 2 $, then we can deduce that $ |x - 2| \leq 0$.

But even if I apply that deduced stuff in the the triangle inequality, I'm not sure how I will bring the $x^3$ part of the proof. Any idea on how to proceed ?

Answer 1

It it $$|x^3-7x+3|\le |x|^3+7|x|+3\le 8+14+3=25$$ if $|x|\le 2$

Answer 2

Starting from

$$|x^3 - 7x + 3|$$

applying the triangle inequality produces

$$|x^3 - 7x + 3|\le |x^3| + |7x| + |3|=|x^3| + 7|x| + 3$$ now since $|x|\le 2$ we have $$|x^3|\le 8$$ $$7|x|\le 14$$ so that $$|x^3| + 7|x| + 3\le 8 + 14 + 3 = 25$$ therefore we may conclude $$|x^3 - 7x + 3|\le 25$$