$$\frac{x}{x-1} = 4$$

$$ \begin{aligned} \frac{x}{x-1} &= 4&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)\frac{x}{x-1} &= (x-1)\cdot4&& \text{cancel out the denominators} \\[1 em]x &= 4x-4&& \text{move the $ \color{blue}{ 4x } $ to the left side} \\[1 em]x-4x &= -4&& \text{simplify left side} \\[1 em]-3x &= -4&& \text{ divide both sides by $ -3 $ } \\[1 em]x &= \frac{-4}{-3}&& \\[1 em]x &= \frac{4}{3}&& \\[1 em] \end{aligned} $$

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