$$\frac{4}{5} = \frac{8}{x+7}-\frac{6}{5}$$

$$ \begin{aligned} \frac{4}{5} &= \frac{8}{x+7}-\frac{6}{5}&& \text{multiply ALL terms by } \color{blue}{ 5(x+7) }. \\[1 em]5(x+7)\cdot\frac{4}{5} &= 5(x+7)\cdot\frac{8}{x+7}-5(x+7)\cdot\frac{6}{5}&& \text{cancel out the denominators} \\[1 em]4x+28 &= 40-(6x+42)&& \text{simplify right side} \\[1 em]4x+28 &= 40-6x-42&& \\[1 em]4x+28 &= -6x-2&& \text{move the $ \color{blue}{ -6x } $ to the left side and $ \color{blue}{ 28 }$ to the right} \\[1 em]4x+6x &= -2-28&& \text{simplify left and right hand side} \\[1 em]10x &= -30&& \text{ divide both sides by $ 10 $ } \\[1 em]x &= -\frac{30}{10}&& \\[1 em]x &= -3&& \\[1 em] \end{aligned} $$

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