Absolute Values: Is it a Mistake or Something I Don't Know?

[Jeff Ford]

An example in my textbook gives

$$\vert \frac{5-x}{5x} \vert \Leftrightarrow \frac {1}{5} (\frac{1}{\vert x \vert}) (\vert x-5 \vert)$$

Is there something I don't know about absolute values that allows $\vert 5-x \vert$ to become $\vert x-5 \vert$ or is this a mistake in the text?

 

[TD]

What would you say about |x| - |-x|?

 

[Jeff Ford]

It would be zero. So $$\vert 5 \vert + \vert -x \vert \Longleftrightarrow \vert -5 \vert + \vert x \vert$$

Thanks for the push.

 

[TD]

Well yes, but in your case |5-x| doesn't necessarily equal |5| + |-x|, but it does equal |x-5|, as you asked in the first place.

 

[mathman]

By definition |a|=|-a|. Let a=x-5 and the result is obvious.

 

[eNathan]

What would you say about |x| - |-x|?

$$\vert x \vert - \vert-x \vert = 0$$

The absolute value a number simply means that its positive, no matter what. So abs(x) - (abs(-x) would be the same as abs(x) - abs(x). This is actaully quite usfull in the field of programming.