$$\frac{1+x}{2}+\frac{3-x}{4} = x\cdot6-4x\cdot5-18x\cdot4+52x\cdot3+101x\cdot2-144x-180$$

$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= x\cdot6-4x\cdot5-18x\cdot4+52x\cdot3+101x\cdot2-144x-180&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= 4x\cdot6-44x\cdot5-418x\cdot4+452x\cdot3+4101x\cdot2-4\cdot144x-4\cdot180&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= 24x-80x-288x+624x+808x-576x-720&& \text{simplify left and right hand side} \\[1 em]x+5 &= 512x-720&& \text{move the $ \color{blue}{ 512x } $ to the left side and $ \color{blue}{ 5 }$ to the right} \\[1 em]x-512x &= -720-5&& \text{simplify left and right hand side} \\[1 em]-511x &= -725&& \text{ divide both sides by $ -511 $ } \\[1 em]x &= \frac{-725}{-511}&& \\[1 em]x &= \frac{725}{511}&& \\[1 em] \end{aligned} $$

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