Equation with x to the denominator: The value of x must not reduce to zero the denominator !

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$\LARGE \frac{2}{x-1}= 2$	$\LARGE x\neq 2 $$\qquad; \qquad $ $\LARGE \frac{2}{x-1}= 2$  ¦¦· x-2 $\LARGE \quad ; \qquad 2 = 2x - 2 \qquad ; \qquad S=\{2 \}$
$\LARGE \frac{3x-6}{x-3}= 2-\frac{3}{3-x}$	$ \LARGE x \neq 3 \qquad; \qquad 3x - 6 = 2(x - 3) + 3 \qquad; \qquad x = 3 \qquad : \qquad S=\{ \}$
$\LARGE \frac{x}{x+2}-\frac{1}{x-2}=\frac{2}{x+2}-\frac{x+2}{x^2-4}$	$\LARGE x \neq 2 \qquad; \qquad (x-2)^2=0 \qquad ; \qquad x = 2\qquad ;\qquad S =\{\} $
$\LARGE \frac{x^2}{x-1} = \frac{1}{x-1}$	$\LARGE x \neq 1 \qquad; \qquad x^2 - 1 = 0 \qquad ; \qquad (x - 1)(x + 1) = 0 \qquad ; \qquad x = 1 ; x = -1 \qquad ;\qquad S =\{ - 1\} $