$$ \begin{aligned} \frac{x}{2}+\frac{3}{2} &= x-1&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{x}{2}+2\cdot\frac{3}{2} &= 2x-2\cdot1&& \text{cancel out the denominators} \\[1 em]x+3 &= 2x-2&& \text{move the $ \color{blue}{ 2x } $ to the left side and $ \color{blue}{ 3 }$ to the right} \\[1 em]x-2x &= -2-3&& \text{simplify left and right hand side} \\[1 em]-x &= -5&& \text{switch signs on both sides} \\[1 em]x &= 5&& \\[1 em] \end{aligned} $$

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