Let \(x \in \mathbf{R}\), \(|x| \geq 0\).


For the absolute value function definition and other properties see here.

Proof:

By the definition of the absolute value function we have two cases:

  • if \(x \geq 0\), then \(|x| = x\) and so \(|x| \geq 0\).
  • if \(x < 0\), then \(|x| = -x\). Since \(x < 0\), then \(-x > 0\) so this means that \(|x| = -x > 0\). As required.

This second case definitely made me pause. By definition \(|x| = -x\). But \(x\) is negative. This means that \(-x\) is positive so \(|x|\) is positive! \(\blacksquare\)

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