$$ \begin{aligned} \frac{1}{1000}x\cdot\frac{1}{10} &= 0&& \text{multiply ALL terms by } \color{blue}{ 1000 }. \\[1 em]1000 \cdot \frac{1}{1000}x\cdot\frac{1}{10} &= 1000\cdot0&& \text{cancel out the denominators} \\[1 em]\frac{x}{10} &= 0&& \text{multiply ALL terms by } \color{blue}{ 10 }. \\[1 em]10 \cdot \frac{x}{10} &= 10\cdot0&& \text{cancel out the denominators} \\[1 em]x &= 0&& \text{ divide both sides by $ 1 $ } \\[1 em]x &= \frac{0}{1}&& \\[1 em]x &= 0&& \\[1 em] \end{aligned} $$

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