$$ \begin{aligned} \frac{x}{1+x} &= \frac{1}{3}&& \text{multiply ALL terms by } \color{blue}{ (1+x)\cdot3 }. \\[1 em](1+x)\cdot3 \cdot \frac{x}{1+x} &= (1+x)\cdot3\cdot\frac{1}{3}&& \text{cancel out the denominators} \\[1 em]3x &= 1+x&& \text{move the $ \color{blue}{ x } $ to the left side} \\[1 em]3x-x &= 1&& \text{simplify left side} \\[1 em]2x &= 1&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= \frac{1}{2}&& \\[1 em] \end{aligned} $$

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