$$\frac{3}{x}+1 = \frac{2}{x}-3$$

$$ \begin{aligned} \frac{3}{x}+1 &= \frac{2}{x}-3&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{3}{x}+x\cdot1 &= x\cdot\frac{2}{x}-x\cdot3&& \text{cancel out the denominators} \\[1 em]3+x &= 2-3x&& \text{move the $ \color{blue}{ -3x } $ to the left side and $ \color{blue}{ 3 }$ to the right} \\[1 em]x+3x &= 2-3&& \text{simplify left and right hand side} \\[1 em]4x &= -1&& \text{ divide both sides by $ 4 $ } \\[1 em]x &= -\frac{1}{4}&& \\[1 em] \end{aligned} $$

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