multiplication of two inequalities where they have a range

Question

I have these two inequalities $$-3\le a\le 1$$ and $$-2\le b\le 2$$

How can we combine the above two inequalities to say $-6\le ab\le 6$?

Answer 1

Because $$|a+1|\leq2$$ and $$|b|\leq2,$$ which gives $$|ab+b|\leq4$$ or $$-4-b\leq ab\leq4-b, $$ which gives $$-6\leq ab\leq6.$$

Answer 2

Your inequalities imply $|a|\le 3$ and $|b|\le 2$, which gives $|ab| = |a||b| \le 2 \cdot 3 = 6$ (where the inequalitites can be multiplied since they involve only nonnegative numbers).