$$ \begin{aligned} 1+\frac{1}{x}+1 &= -\frac{x}{x}+1&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot1+x\cdot\frac{1}{x}+x\cdot1 &= -x \cdot \frac{x}{x}+x\cdot1&& \text{cancel out the denominators} \\[1 em]x+1+x &= -x+x&& \text{simplify left and right hand side} \\[1 em]2x+1 &= -x+x&& \\[1 em]2x+1 &= 0&& \text{ move the constants to the right } \\[1 em]2x &= -1&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= -\frac{1}{2}&& \\[1 em] \end{aligned} $$

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