(x/(x-1))/((x/(x-1))-1)

How would you solve this problem algebraically? I'm not interested in just the answer, I know that already. I try setting it equal to y, and then solving for that. What happens:

>y=(x/(x-1))/((x/(x-1))-1)
>y((x/(x-1))-1)=(x/(x-1))
>y(x/(x-1))-y=(x/(x-1))
>y(x/(x-1))=(x/(x-1))+y

wat... multiply the whole thing by (x-1) and you end up with x.

say z=x/(x-1).
Then we have
y = z/(z-1)
so y = z(z(x))

Solving, set q=x-1
y=(x/q)/((x/q)-1)
 =(x/q)/((x-q)/q)
 =(x/q)*(q/(x-q))
but x-q = x-(x-1) = x-x+1 = 1, so
 =(x/q)*(q/1)
 = x

Using both these results, we know that z(z(x)) = x, so z(x) is self-inverse, like 1/x (1/(1/x) = 1*(x/1) = x).