$$ \begin{aligned} \frac{1}{x}+1+\frac{1}{x}+2 &= 0&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{1}{x}+x\cdot1+x\cdot\frac{1}{x}+x\cdot2 &= x\cdot0&& \text{cancel out the denominators} \\[1 em]1+x+1+2x &= 0&& \text{simplify left side} \\[1 em]3x+2 &= 0&& \text{ move the constants to the right } \\[1 em]3x &= -2&& \text{ divide both sides by $ 3 $ } \\[1 em]x &= -\frac{2}{3}&& \\[1 em] \end{aligned} $$

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