$$ \begin{aligned} \frac{x}{1-x} &= \frac{2^1}{2}&& \text{multiply ALL terms by } \color{blue}{ (1-x)\cdot2 }. \\[1 em](1-x)\cdot2 \cdot \frac{x}{1-x} &= (1-x)\cdot2 \cdot \frac{2^1}{2}&& \text{cancel out the denominators} \\[1 em]2x &= -2x+2&& \text{move the $ \color{blue}{ -2x } $ to the left side} \\[1 em]2x+2x &= 2&& \text{simplify left side} \\[1 em]4x &= 2&& \text{ divide both sides by $ 4 $ } \\[1 em]x &= \frac{2}{4}&& \\[1 em]x &= \frac{1}{2}&& \\[1 em] \end{aligned} $$

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