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A number \(x\) satisfies \(x = \dfrac{1}{1 + x}\).  Determine \(x - \dfrac{1}{x}\).

\(\begin{array}{|rcll|} \hline \mathbf{x} &=& \mathbf{\dfrac{1}{1 + x}} \quad &| \quad \times (1 + x)\\\\ x(1 + x) &=& 1 \quad &| \quad : x \\\\ 1 + x &=& \dfrac{1}{x} \quad &| \quad - \dfrac{1}{x} \\\\ 1 + x - \dfrac{1}{x} &=& 0 \quad &| \quad -1 \\\\ \mathbf{ x - \dfrac{1}{x} } &=& \mathbf{ -1 } \\ \hline \end{array}\)