A False Answer

Solve $\frac{x+1}{x-1} = \frac{2}{x-1}+4$. Multiply both sides by $x-1$ to get $x+1 = 2+4\left(x-1\right)$. Expand: $x+1 = 4x-2$, so 3x is 3 and x is 1. Now put 1 back into the original: $x-1$ is 0, and you cannot divide by 0. x = 1 is not a solution at all.

Where did it come from? At x = 1, multiplying by $x-1$ meant multiplying both sides by 0, and that makes any two sides equal. So the new equation has a solution the original never had. Such an answer is called extraneous: a false answer that the working produced and the original equation refuses.

With x = 1 thrown out and nothing else left, this equation has no solution. That is a correct, finished answer, not a failure.