How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$? 
How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?

In my title, I should be clear that the additive inverse should be unique. But how does this help?
I dont even get why uniqueness of a negative number matters... Is there a third class of numbers I should know about?

Hint: Use distributivity to show that $ax+(-(ax))=0$, then apply uniqueness.
