Absolute Value $|1-(1/x)| = |(1/x)-1|$

Can someone please explain to me how: $$|1-(1/x)| = |(1/x)-1|$$ Im working on a limit problem in my calculus book and I cant seem to understand how they reversed this and it equals the same thing.

Thanks!

• Because $|a|=|-a|$ for all $a$. – vadim123 Sep 5 '15 at 15:38
• since $$|1-\frac{1}{x}|=|(-1)(\frac{1}{x}-1)|$$ – Dr. Sonnhard Graubner Sep 5 '15 at 15:40
• When faced with things like this, plug in some numbers and try to see the pattern. For example. If you let $\dfrac 1x = 5$, then you get $|1-5| = |-4| = 4$ and $|5 - 1| = |4| = 4$. Once you see how the numbers are working, you might get an idea how to attack the equation. – steven gregory Sep 5 '15 at 16:24

This is an identity, as has been mentioned already. In general, if $\vert x\vert =\vert y\vert$ then $x=\pm y$. You can see this geometrically or by considering cases, but perhaps the easiest way is to note that $\vert x\vert =\sqrt{x^{2}}$.