$$\frac{1+x}{2}+\frac{3-x}{4} = 4x+6$$

$$ \begin{aligned} \frac{1+x}{2}+\frac{3-x}{4} &= 4x+6&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1+x}{2}+4\frac{3-x}{4} &= 4\cdot4x+4\cdot6&& \text{cancel out the denominators} \\[1 em]2x+2+3-x &= 16x+24&& \text{simplify left side} \\[1 em]x+5 &= 16x+24&& \text{move the $ \color{blue}{ 16x } $ to the left side and $ \color{blue}{ 5 }$ to the right} \\[1 em]x-16x &= 24-5&& \text{simplify left and right hand side} \\[1 em]-15x &= 19&& \text{ divide both sides by $ -15 $ } \\[1 em]x &= -\frac{19}{15}&& \\[1 em] \end{aligned} $$

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