$$ \begin{aligned} \frac{x+1}{5}+\frac{x+1}{7}+\frac{x+10}{9} &= \frac{x+12}{11}&& \text{multiply ALL terms by } \color{blue}{ 3465 }. \\[1 em]3465 \cdot \frac{x+1}{5}+3465\frac{x+1}{7}+3465\frac{x+10}{9} &= 3465 \cdot \frac{x+12}{11}&& \text{cancel out the denominators} \\[1 em]693x+693+495x+495+385x+3850 &= 315x+3780&& \text{simplify left side} \\[1 em]1188x+1188+385x+3850 &= 315x+3780&& \\[1 em]1573x+5038 &= 315x+3780&& \text{move the $ \color{blue}{ 315x } $ to the left side and $ \color{blue}{ 5038 }$ to the right} \\[1 em]1573x-315x &= 3780-5038&& \text{simplify left and right hand side} \\[1 em]1258x &= -1258&& \text{ divide both sides by $ 1258 $ } \\[1 em]x &= -\frac{1258}{1258}&& \\[1 em]x &= -1&& \\[1 em] \end{aligned} $$

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