$$ \begin{aligned} \frac{1}{2}x+\frac{3}{2}(x+1)-\frac{1}{4} &= 0&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{1}{2}x+4\frac{3}{2}(x+1)-4\cdot\frac{1}{4} &= 4\cdot0&& \text{cancel out the denominators} \\[1 em]2x+6x+6-1 &= 0&& \text{simplify left side} \\[1 em]8x+6-1 &= 0&& \\[1 em]8x+5 &= 0&& \text{ move the constants to the right } \\[1 em]8x &= -5&& \text{ divide both sides by $ 8 $ } \\[1 em]x &= -\frac{5}{8}&& \\[1 em] \end{aligned} $$

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