$$3-\frac{9}{x} = 9$$

$$ \begin{aligned} 3-\frac{9}{x} &= 9&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot3-x\cdot\frac{9}{x} &= x\cdot9&& \text{cancel out the denominators} \\[1 em]3x-9 &= 9x&& \text{move the $ \color{blue}{ 9x } $ to the left side and $ \color{blue}{ -9 }$ to the right} \\[1 em]3x-9x &= 9&& \text{simplify left side} \\[1 em]-6x &= 9&& \text{ divide both sides by $ -6 $ } \\[1 em]x &= -\frac{9}{6}&& \\[1 em]x &= -\frac{3}{2}&& \\[1 em] \end{aligned} $$

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