$$ \begin{aligned} \frac{1}{3}(x+3)+\frac{1}{6}(x-6) &= x+3&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{1}{3}(x+3)+6\frac{1}{6}(x-6) &= 6x+6\cdot3&& \text{cancel out the denominators} \\[1 em]2x+6+x-6 &= 6x+18&& \text{simplify left side} \\[1 em]3x &= 6x+18&& \text{move the $ \color{blue}{ 6x } $ to the left side} \\[1 em]3x-6x &= 18&& \text{simplify left side} \\[1 em]-3x &= 18&& \text{ divide both sides by $ -3 $ } \\[1 em]x &= -\frac{18}{3}&& \\[1 em]x &= -6&& \\[1 em] \end{aligned} $$

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