How to prove the symmetrical relation for the following statement?

Question

I would like to know how to prove the symmetrical relation for $\sim$ according to the following definition:

Suppose $\sim$ is defined on the set of the integers as follows : $a\sim b$ iff $ab ≤ a|b|$

Please explain to me. Thank you.

Answer 1

HINT : Simplest approach is to make cases. There are just 4 cases a,b>0, a>0 b<0, a,b<0 and a<0 b>0.

You will see that not all cases are possible if the condition is true, and for the remaining cases, the symmetric relationship b~a is also true.

Let me know if this is not sufficient and I will write it out! Hope it helps!