$$ \begin{aligned} x+\frac{1}{x} &= x-\frac{1}{x}&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]xx+x\cdot\frac{1}{x} &= xx-x\cdot\frac{1}{x}&& \text{cancel out the denominators} \\[1 em]x^2+1 &= x^2-1&& \text{move all terms to the left hand side } \\[1 em]x^2+1-x^2+1 &= 0&& \text{simplify left side} \\[1 em]x^2+1-x^2+1 &= 0&& \\[1 em]2 &= 0&& \\[1 em] \end{aligned} $$

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