$$ \begin{aligned} \frac{x}{3}-\frac{x+1}{2} &= 1&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{x}{3}-6\frac{x+1}{2} &= 6\cdot1&& \text{cancel out the denominators} \\[1 em]2x-(3x+3) &= 6&& \text{simplify left side} \\[1 em]2x-3x-3 &= 6&& \\[1 em]-x-3 &= 6&& \text{ move -3 to the right } \\[1 em]-x &= 6+3&& \\[1 em]-x &= 9&& \text{switch signs on both sides} \\[1 em]x &= -9&& \\[1 em] \end{aligned} $$
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