Integer axioms question

Question

I am stuck on a lecture problem.

Prove that if $x\in\mathbb{Z}$ and $x<0$, then $x^3<0$.

I believe that I am supposed to use the integer axioms but am having trouble making the first step or which ones to use. Any help would be appreciated. Thank You!

Answer 1

HINT If $x < 0$ then let $x = -y$ with $y > 0$ and now $$ x^3 = (-y)^3 = (-y) \times (-y) \times (-y) = y^2 \times(-y) = -y^3... $$