$$ \begin{aligned} \frac{12}{x}+8 &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{12}{x}+x\cdot8 &= x\cdot0&& \text{cancel out the denominators} \\[1 em]12+8x &= 0&& \text{ move the constants to the right } \\[1 em]8x &= -12&& \text{ divide both sides by $ 8 $ } \\[1 em]x &= -\frac{12}{8}&& \\[1 em]x &= -\frac{3}{2}&& \\[1 em] \end{aligned} $$

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