$$ \begin{aligned} \frac{1}{2x}-\frac{1}{2x^2} &= \frac{1}{x}&& \text{multiply ALL terms by } \color{blue}{ 2xx^2 }. \\[1 em]2xx^2\cdot\frac{1}{2x}-2xx^2\cdot\frac{1}{2x^2} &= 2xx^2\cdot\frac{1}{x}&& \text{cancel out the denominators} \\[1 em]1-x &= 2&& \text{ move 1 to the right } \\[1 em]-x &= 2-1&& \\[1 em]-x &= 1&& \text{switch signs on both sides} \\[1 em]x &= -1&& \\[1 em] \end{aligned} $$

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