Question

How would I prove (-1)*x = -x? or is that an assumption I can make when proving other things?

Answer 1

First note that:\[1+(-1)=0\Longrightarrow x\cdot(1+(-1))=x\cdot0\Longrightarrow x\cdot1+x\cdot(-1)=0\]\[\Longrightarrow x+x\cdot(-1)=0\]

Answer 2

Then note that x+(-x)=0

Answer 3

So both x*(-1) and (-x) are acting like additive inverses of x. So by the uniqueness of the additive inverse (or you can subtract those two equations), we see that:\[-x=x\cdot(-1)\]

Answer 4

wow awesome, thats so clear, thank you!