$$ \begin{aligned} \frac{1}{10}x &= \frac{1}{11}x+\frac{1}{1100}&& \text{multiply ALL terms by } \color{blue}{ 1100 }. \\[1 em]1100 \cdot \frac{1}{10}x &= 1100 \cdot \frac{1}{11}x+1100\cdot\frac{1}{1100}&& \text{cancel out the denominators} \\[1 em]110x &= 100x+1&& \text{move the $ \color{blue}{ 100x } $ to the left side} \\[1 em]110x-100x &= 1&& \text{simplify left side} \\[1 em]10x &= 1&& \text{ divide both sides by $ 10 $ } \\[1 em]x &= \frac{1}{10}&& \\[1 em] \end{aligned} $$

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