$$ \begin{aligned} \frac{4x+1}{3} &= \frac{x-5}{2}&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{4x+1}{3} &= 6 \cdot \frac{x-5}{2}&& \text{cancel out the denominators} \\[1 em]8x+2 &= 3x-15&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ 2 }$ to the right} \\[1 em]8x-3x &= -15-2&& \text{simplify left and right hand side} \\[1 em]5x &= -17&& \text{ divide both sides by $ 5 $ } \\[1 em]x &= -\frac{17}{5}&& \\[1 em] \end{aligned} $$

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