$$ \begin{aligned} \frac{1}{2}x+\frac{3}{2}(x+1)-\frac{1}{3} &= 5&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{1}{2}x+6\frac{3}{2}(x+1)-6\cdot\frac{1}{3} &= 6\cdot5&& \text{cancel out the denominators} \\[1 em]3x+9x+9-2 &= 30&& \text{simplify left side} \\[1 em]12x+9-2 &= 30&& \\[1 em]12x+7 &= 30&& \text{ move 7 to the right } \\[1 em]12x &= 30-7&& \\[1 em]12x &= 23&& \text{ divide both sides by $ 12 $ } \\[1 em]x &= \frac{23}{12}&& \\[1 em] \end{aligned} $$

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