$$ \begin{aligned} \frac{x}{3}-1 &= \frac{x}{5}+1&& \text{multiply ALL terms by } \color{blue}{ 15 }. \\[1 em]15 \cdot \frac{x}{3}-15\cdot1 &= 15 \cdot \frac{x}{5}+15\cdot1&& \text{cancel out the denominators} \\[1 em]5x-15 &= 3x+15&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ -15 }$ to the right} \\[1 em]5x-3x &= 15+15&& \text{simplify left and right hand side} \\[1 em]2x &= 30&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= \frac{30}{2}&& \\[1 em]x &= 15&& \\[1 em] \end{aligned} $$

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