$$ \begin{aligned} \frac{3-2x}{4}-\frac{x+1}{2} &= \frac{x+5}{3}-\frac{1}{12}&& \text{multiply ALL terms by } \color{blue}{ 12 }. \\[1 em]12 \cdot \frac{3-2x}{4}-12\frac{x+1}{2} &= 12 \cdot \frac{x+5}{3}-12\cdot\frac{1}{12}&& \text{cancel out the denominators} \\[1 em]-6x+9-(6x+6) &= 4x+20-1&& \text{simplify left and right hand side} \\[1 em]-6x+9-6x-6 &= 4x+19&& \\[1 em]-12x+3 &= 4x+19&& \text{move the $ \color{blue}{ 4x } $ to the left side and $ \color{blue}{ 3 }$ to the right} \\[1 em]-12x-4x &= 19-3&& \text{simplify left and right hand side} \\[1 em]-16x &= 16&& \text{ divide both sides by $ -16 $ } \\[1 em]x &= -\frac{16}{16}&& \\[1 em]x &= -1&& \\[1 em] \end{aligned} $$

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