Properties of the Absolute Value

Proof (related to "Properties of the Absolute Value")

\((1)\) We have to show that \(|-x|=|x|\) for all \(x\in\mathbb R\).

Case 1: If \(x \ge 0\), we have \(|x|=x\) and \(|-x|=-(-x)=x\).
Case 2: If \(x < 0\), then \(|x|=-x\). Since \(-x > 0\), we have \(|-x|=-x.\)

\((2)\)We have to prove that \(|xy|=|x||y|\) for all \(x,y\in\mathbb R\).

Case 1: If \(x\ge 0\) and \(y\ge 0\): We have \(|x|=x\) and \(|y|=y\). Thus \(|xy|=xy=|x||y|\).
Case 2: If \(x\ge 0\) and \(y < 0\): We have \(|x|=x\) and \(|y|=-y\). Thus \(|xy|=x\cdot(-y)=|x||y|\).
Case 3: If \(x < 0\) and \(y \ge 0\): We have \(|x|=-x\) and \(|y|=y\). Thus \(|xy|=(-x)\cdot y=|x||y|\).
Case 4: If \(x < 0\) and \(y < 0\): We have \(|x|=-x\) and \(|y|=-y\). Thus \(|xy|=(-x)\cdot(-y)=|x||y|\).

\((3)\) We will show that \(\left|\frac xy\right|=\frac{|x|}{|y|}\) for all \(x,y\in\mathbb R\), \(y\neq 0\).

Because \(\frac xy\cdot y=x\), it follows from \((2)\) that \(\left|\frac xy\right|\cdot |y|=|x|\). Thus \(\left|\frac xy\right|=\frac{|x|}{|y|}\).

References

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983