$$\frac{1}{x+5}+\frac{2}{x-3} = 0$$

$$ \begin{aligned} \frac{1}{x+5}+\frac{2}{x-3} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x+5)(x-3) }. \\[1 em](x+5)(x-3)\cdot\frac{1}{x+5}+(x+5)(x-3)\cdot\frac{2}{x-3} &= (x+5)(x-3)\cdot0&& \text{cancel out the denominators} \\[1 em]x-3+2x+10 &= 0&& \text{simplify left side} \\[1 em]3x+7 &= 0&& \text{ move the constants to the right } \\[1 em]3x &= -7&& \text{ divide both sides by $ 3 $ } \\[1 em]x &= -\frac{7}{3}&& \\[1 em] \end{aligned} $$

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