Forming with absolute value - am I allowed to form it like that?

Question

So far I didn't face much with absolute value in maths that's why I'm not sure at all if I'm allowed to do the following.

Let's say we have $x,y, \tilde{x},\tilde{y} \in \mathbb{R} \setminus \left\{0\right\}$

$$\frac{\left |\frac{\tilde{x}}{\tilde{y}}-\frac{x}{y}\right |}{\left | \frac{x}{y} \right |}$$

And now I want form it to: $$\frac{\left | \frac{\tilde{x}}{\tilde{y}} \right |}{\left| \frac{x}{y} \right|}-\frac{\left|\frac{x}{y}\right|}{\left|\frac{x}{y}\right|} = \left|\frac{\tilde{x}y}{\tilde{y}x}\right|-1$$

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Is it correct like that?

Answer 1

Hint:

your first step is wrong because, for $a,b \in \mathbb{R}$ $$ |a-b|=\left|(|a|-|b|) \right| $$

the second step is correct because

$$ \left| \frac{a}{b} \right|=\frac{|a|}{|b|} $$

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$$\frac{\left |\frac{\tilde{x}}{\tilde{y}}-\frac{x}{y}\right |}{\left | \frac{x}{y} \right |}= \left|\frac{\left |\frac{\tilde{x}}{\tilde{y}}\right|-\left |\frac{x}{y}\right |}{\left | \frac{x}{y} \right |} \right|= \left|\frac{\left |\frac{\tilde{x}}{\tilde{y}}\right|}{\left | \frac{x}{y} \right |}-1 \right|=\left| \left|\frac{\tilde{x}y}{\tilde{y}x} \right|-1\right|$$

Answer 2

You are effectively asking whether $$\frac{|a-b|}{|b|}=\frac{|a|}{|b|}-\frac{|b|}{|b|}$$ or equivalently, whether $$\left|\frac ab-1\right|=\left|\frac ab\right|-1.$$ The answer is no; the right hand side may be negative.