$$ \begin{aligned} 1+\frac{1}{x} &= -1&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot1+x\cdot\frac{1}{x} &= x\cdot(-1)&& \text{cancel out the denominators} \\[1 em]x+1 &= -x&& \text{move the $ \color{blue}{ -x } $ to the left side and $ \color{blue}{ 1 }$ to the right} \\[1 em]x+x &= -1&& \text{simplify left side} \\[1 em]2x &= -1&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= -\frac{1}{2}&& \\[1 em] \end{aligned} $$

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