$$ \begin{aligned} \frac{1}{x-1} &= \frac{3}{x+2}&& \text{multiply ALL terms by } \color{blue}{ (x-1)(x+2) }. \\[1 em](x-1)(x+2)\cdot\frac{1}{x-1} &= (x-1)(x+2)\cdot\frac{3}{x+2}&& \text{cancel out the denominators} \\[1 em]x+2 &= 3x-3&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ 2 }$ to the right} \\[1 em]x-3x &= -3-2&& \text{simplify left and right hand side} \\[1 em]-2x &= -5&& \text{ divide both sides by $ -2 $ } \\[1 em]x &= \frac{-5}{-2}&& \\[1 em]x &= \frac{5}{2}&& \\[1 em] \end{aligned} $$

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