$$3x+\frac{1}{2}+4x = x+14$$

$$ \begin{aligned} 3x+\frac{1}{2}+4x &= x+14&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2\cdot3x+2\cdot\frac{1}{2}+2\cdot4x &= 2x+2\cdot14&& \text{cancel out the denominators} \\[1 em]6x+1+8x &= 2x+28&& \text{simplify left side} \\[1 em]14x+1 &= 2x+28&& \text{move the $ \color{blue}{ 2x } $ to the left side and $ \color{blue}{ 1 }$ to the right} \\[1 em]14x-2x &= 28-1&& \text{simplify left and right hand side} \\[1 em]12x &= 27&& \text{ divide both sides by $ 12 $ } \\[1 em]x &= \frac{27}{12}&& \\[1 em]x &= \frac{9}{4}&& \\[1 em] \end{aligned} $$

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