$$9 = \frac{x+4}{x+12}$$

$$ \begin{aligned} 9 &= \frac{x+4}{x+12}&& \text{multiply ALL terms by } \color{blue}{ x+12 }. \\[1 em](x+12)\cdot9 &= (x+12)\frac{x+4}{x+12}&& \text{cancel out the denominators} \\[1 em]9x+108 &= x+4&& \text{move the $ \color{blue}{ x } $ to the left side and $ \color{blue}{ 108 }$ to the right} \\[1 em]9x-x &= 4-108&& \text{simplify left and right hand side} \\[1 em]8x &= -104&& \text{ divide both sides by $ 8 $ } \\[1 em]x &= -\frac{104}{8}&& \\[1 em]x &= -13&& \\[1 em] \end{aligned} $$

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