How does this work?

$\frac{x-1}{x-2}-\frac{2}{x}\,=\,\frac{1}{x-2}$

First let's get a common denominator.

$\frac{(x)(x-1)}{(x)(x-2)}-\frac{2(x-2)}{(x)(x-2)}\,=\,\frac{1x}{(x)(x-2)}$

We can multiply through by the denominator....however....you are right that x can't be 2 !

Also  x  can't be  0 . If we plug in  0  or  2  for  x  into the original equation, we will get an undefined result. Before we multiply through by the denominator, we must say that  x ≠ 0  and  x ≠ 2 .

$(x)(x-1)-2(x-2) \,=\,1x$         Distribute the  x  and the  -2 .

$x^2-x-2x+4 \,=\,x$           Subtract  x  from both sides and combine like terms.

$x^2-4x+4 \,=\,0$          Factor the left side and solve for  x  .

$(x-2)^2=0 \\~\\ x-2=0 \\~\\ x=2$

But, x cannot be 2 because it causes a zero in the denominator of the original problem.

So there is no solution.