Prove: $\dfrac{|x|}{|y|}=\left|\dfrac{x}{y}\right|$

Question

Can you help me to prove that $$\dfrac{|x|}{|y|}=\left|\dfrac{x}{y}\right|$$, if $y\neq 0$.

Answer 1

Suppose $|ab|=|a||b|$. Then let $a=x$ and $b=1/x$ to get

$$1 = |x \cdot \frac{1}{x} | = |x| | \frac{1}{x}|$$

Rearrange to get $|\frac{1}{x}| = \frac{1}{|x|}$.

Now let $a=x$ and $b=1/y$. Then

$$\left| \frac{x}{y} \right| = \left| x \cdot \frac{1}{y} \right| = |x| \left| \frac{1}{y} \right| = |x| \frac{1}{|y|} = \frac{|x|}{|y|}.$$

It is up to you to show $|ab|=|a||b|$.

Answer 2

It follows directly from Prove: $\Big|\dfrac{1}{x}\Big|=\dfrac{1}{|x|}$ and Prove: $|xy|=|x|\cdot |y|$

$$\frac{|x|}{|y|}=\frac{1}{|y|}|x|=\left|\frac{1}{y}\right||x|=\left|\frac{x}{y}\right|,\quad y\neq 0$$