$$ \begin{aligned} \frac{1}{x}-1-\frac{1}{x} &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{1}{x}-x\cdot1-x\cdot\frac{1}{x} &= x\cdot0&& \text{cancel out the denominators} \\[1 em]1-x-1 &= 0&& \text{simplify left side} \\[1 em]1-x-1 &= 0&& \\[1 em]-x &= 0&& \text{ divide both sides by $ -1 $ } \\[1 em]x &= \frac{0}{-1}&& \\[1 em]x &= 0&& \\[1 em] \end{aligned} $$

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