> Sure it does, because if that is the case, you restrict your domain when you divide by a variable expression. If you divide both sides by x-1, then you effectively rule out 1 from the domain. > That's the problem.

That's the problem that a mathematician must handle. The solution to the equation

  x * f(x) = x

is very simple: x is either 0 or such that f(x) = 1.