$$ \begin{aligned} \frac{1}{4}(x+9) &= \frac{1}{3}(x-5)+1&& \text{multiply ALL terms by } \color{blue}{ 12 }. \\[1 em]12 \cdot \frac{1}{4}(x+9) &= 12 \cdot \frac{1}{3}(x-5)+12\cdot1&& \text{cancel out the denominators} \\[1 em]3x+27 &= 4x-20+12&& \text{simplify right side} \\[1 em]3x+27 &= 4x-8&& \text{move the $ \color{blue}{ 4x } $ to the left side and $ \color{blue}{ 27 }$ to the right} \\[1 em]3x-4x &= -8-27&& \text{simplify left and right hand side} \\[1 em]-x &= -35&& \text{switch signs on both sides} \\[1 em]x &= 35&& \\[1 em] \end{aligned} $$

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