$$\frac{3}{x+1} = \frac{4}{x-1}$$

$$ \begin{aligned} \frac{3}{x+1} &= \frac{4}{x-1}&& \text{multiply ALL terms by } \color{blue}{ (x+1)(x-1) }. \\[1 em](x+1)(x-1)\cdot\frac{3}{x+1} &= (x+1)(x-1)\cdot\frac{4}{x-1}&& \text{cancel out the denominators} \\[1 em]3x-3 &= 4x+4&& \text{move the $ \color{blue}{ 4x } $ to the left side and $ \color{blue}{ -3 }$ to the right} \\[1 em]3x-4x &= 4+3&& \text{simplify left and right hand side} \\[1 em]-x &= 7&& \text{switch signs on both sides} \\[1 em]x &= -7&& \\[1 em] \end{aligned} $$

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