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I'm in High School and my textbooks have like skipped a lesson of modulus,i.e. $|x|$ in previous classes. I know that $$|-4|=4 $$ but how does

$$|x|=\begin{cases} x &\text{ if } x\geq0\\-x&\text{ if } x<0\end{cases}$$

This negative value of $x$ is what I don't get. If modulus gives positive values, how can $f(x)$ give a negative value?

Ps: I have found these problems while learning sets.

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  • $\begingroup$ Wikipedia is a starting point. It has a nice explanation. $\endgroup$ – Dietrich Burde Aug 21 '17 at 14:34
  • $\begingroup$ I went there. I got it too. Its the -x in f(x) that i dont understand $\endgroup$ – Ashish Shukla Aug 21 '17 at 14:36
  • $\begingroup$ Look at the picture on the right there, with $-3$ in $f(3)$. So $|-3|=3$. $\endgroup$ – Dietrich Burde Aug 21 '17 at 14:37
  • $\begingroup$ I get the |-3|=3(Sorry to sound dumb:)) Its the f(x)= |x| i dont get that says f(x) is negative when x is negative where f(x) should be just x $\endgroup$ – Ashish Shukla Aug 21 '17 at 14:38
  • $\begingroup$ here is something for you sydney.edu.au/stuserv/documents/maths_learning_centre/… $\endgroup$ – Dr. Sonnhard Graubner Aug 21 '17 at 14:39
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(this negative value of x is what i dont get. If modulus gives positive values,how can f(x) give a negative value)

The definition might seem confusing at first, but it is in fact very logical: $$|x| = \begin{cases} x & x \ge 0 \\ -x & x< 0\end{cases}$$ You should read this as follows:

The absolute value of $x$, written as $|x|$, is equal to:

  • $x$ itself if $x$ is positive;
  • $-x$ if $x$ is negative.

Indeed: when $x$ is negative, $-x$ is positive!

Take an example, for $x = -3$ you would get: $$|\color{red}{-3}| = -(\color{red}{-3}) = 3$$


Or put differently: we probably think of $|-3|=3$ as "dropping the minus sign", but that's hard to put into symbols if we want to write down a symbolical definition. We can however easily add an extra minus and of course this achieves the exact same thing since $-(-x) = x$.

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  • $\begingroup$ Thank You very much......Its so clear now and you have explained it like the best way possible $\endgroup$ – Ashish Shukla Aug 21 '17 at 14:44
  • $\begingroup$ @AshishShukla You're welcome and thanks; I just added a bit to my answer. $\endgroup$ – StackTD Aug 21 '17 at 14:46
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Remember that $-x$ doesn't mean that the number is negative (even though there's a negative sign!) When $x$ is already negative, $-x$ is positive.

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