$$ \begin{aligned} 2+\frac{1}{x-1} &= 0&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)\cdot2+(x-1)\cdot\frac{1}{x-1} &= (x-1)\cdot0&& \text{cancel out the denominators} \\[1 em]2x-2+1 &= 0&& \text{simplify left side} \\[1 em]2x-1 &= 0&& \text{ move the constants to the right } \\[1 em]2x &= 1&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= \frac{1}{2}&& \\[1 em] \end{aligned} $$

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