$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= -2x\cdot3+12x\cdot3-8x&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= -42x\cdot3+412x\cdot3-4\cdot8x&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= -24x+144x-32x&& \text{simplify left and right hand side} \\[1 em]x+5 &= 88x&& \text{move the $ \color{blue}{ 88x } $ to the left side and $ \color{blue}{ 5 }$ to the right} \\[1 em]x-88x &= -5&& \text{simplify left side} \\[1 em]-87x &= -5&& \text{ divide both sides by $ -87 $ } \\[1 em]x &= \frac{-5}{-87}&& \\[1 em]x &= \frac{5}{87}&& \\[1 em] \end{aligned} $$

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