$$ \begin{aligned} x+\frac{1}{2}+x+\frac{1}{2} &= 0&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2x+2\cdot\frac{1}{2}+2x+2\cdot\frac{1}{2} &= 2\cdot0&& \text{cancel out the denominators} \\[1 em]2x+1+2x+1 &= 0&& \text{simplify left side} \\[1 em]4x+2 &= 0&& \text{ move the constants to the right } \\[1 em]4x &= -2&& \text{ divide both sides by $ 4 $ } \\[1 em]x &= -\frac{2}{4}&& \\[1 em]x &= -\frac{1}{2}&& \\[1 em] \end{aligned} $$

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