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

$$ \begin{aligned} \frac{x}{x-1} &= \frac{1}{2}&& \text{multiply ALL terms by } \color{blue}{ (x-1)\cdot2 }. \\[1 em](x-1)\cdot2 \cdot \frac{x}{x-1} &= (x-1)\cdot2\cdot\frac{1}{2}&& \text{cancel out the denominators} \\[1 em]2x &= x-1&& \text{move the $ \color{blue}{ x } $ to the left side} \\[1 em]2x-x &= -1&& \text{simplify left side} \\[1 em]x &= -1&& \\[1 em]x &= -1&& \\[1 em] \end{aligned} $$

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