# Proof

(related to Corollary: 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 $$\left|\frac xy\right|=\frac{|x|}{|y|}$$.

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983