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This must be true for all values of \(x\text{.}\) If \(x = 1\text{,}\) then the equation becomes \(1 = -a\) so \(a = -1\text{.}\) When \(x = \frac{1}{2}\) we get \(1 = b/2\) so \(b = 2\text{.}\) This tells us that we can decompose the fraction like this:

\begin{equation*} \frac{1}{(1-x)(1-2x)} = \frac{-1}{1-x} + \frac{2}{1-2x}\text{.} \end{equation*}

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