Implication with absolute value

Task number: 2781

In the real domain show that \((x \le y) \land (-x\le y) \Longrightarrow |x|\le y\).

  • Resolution

    We will prove the contrapositive, i.e. \(|x|> y \Longrightarrow (x> y) \lor (-x> y)\).

    For \(x\ge 0\) we have \(x=|x|>y\), whereas for \(x\le 0\) we have \(|x|=-x>y\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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