x=2x

doesn't have three solutions? One of them is clearly x = 0, but x = +∞ and -∞ should also be solutions - not in ℕ or ℝ, of course, given that +/-∞ does not belong to ℕ or ℝ, but I heard there's something called an extended-ℝ thingie that includes +/-∞, therefore in some domain x=2x should have three solutions, should it not?

If that's true, and x=2x does have three solutions in some domain... then that doesn't make sense at all! How a first-order equation would have multiple solutions? It's incredibly bizarre!

Where did I got out of the track in this reasoning? Once again sorry if the question sounds too idiotic...