$$ \begin{aligned} \frac{7}{x+1}+\frac{6}{x+6} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x+1)(x+6) }. \\[1 em](x+1)(x+6)\cdot\frac{7}{x+1}+(x+1)(x+6)\cdot\frac{6}{x+6} &= (x+1)(x+6)\cdot0&& \text{cancel out the denominators} \\[1 em]7x+42+6x+6 &= 0&& \text{simplify left side} \\[1 em]13x+48 &= 0&& \text{ move the constants to the right } \\[1 em]13x &= -48&& \text{ divide both sides by $ 13 $ } \\[1 em]x &= -\frac{48}{13}&& \\[1 em] \end{aligned} $$

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