If the absolute value is usually defined as a term used in mathematics to indicate the distance of a point or number from the origin (zero point) of a number line or coordinate system. How can this be possible:

For x < 0, | x | = - x

Thank you for the answer in advance.

  • 1
    $\begingroup$ What about it strikes you as impossible? If you plug in a particular value for $x$ (for instance, $-1$), does it still seem wrong? $\endgroup$ – Milo Brandt Aug 24 '16 at 2:00
  • $\begingroup$ It's simply a definition of a function. As Milo asks, what seems impossible about it? When $x<0$, the output of your function $|x|$ is given by $-x$. $\endgroup$ – Carser Aug 24 '16 at 2:01
  • $\begingroup$ My suspicion is that you're under the impression that $-x$ is negative because it begins with a minus sign. But if $x<0$, then $-x$ is positive. $\qquad$ $\endgroup$ – Michael Hardy Aug 24 '16 at 2:19
  • $\begingroup$ Note: -(-3)= 3. This is fine you can see the negative sign inside. But what if x =-3. Then -x = 3. Still okay. |x| =3. |-x| =3. But x <0 and -x >0. And -x =3. So |x| = 3 = -x. It looks wrong be if x is negative. Then -x is positive. $\endgroup$ – fleablood Aug 24 '16 at 8:00

Draw a number line. How far away is $-1$ from $0$? Well, it's exactly one unit away - that is, $\vert -1\vert=1$.

But $1=-(-1)$. Do you understand why?

In case you haven't seen this before, don't feel bad if it looks really strange. This is totally not obvious at first! Here's why it's true: the way we define negatives is $$\mbox{"$-a$ is the thing you need to add to $a$ to get $0$."}$$ That is, the defining property of (say) $-2$ is that $2+(-2)=0$. Now, what do you add to $-1$ to get $0$? Well, the answer is just $1$! So $-(-1)=1$. This sort of reasoning by algebraic definitions can seem really weird at first, and I strongly suggest you talk to your teacher(s) about it until it makes sense. Right now it might seem a little random, but it's actually super important; and down the road, it will be one of the key ideas behind abstract algebra.

In general, if $x$ is negative, then $\vert x\vert=-x$ because - despite how it may seem! - $-x$ is the "positive" version of $x$!

(What's really going on here: $-x$ flips the sign of $x$. If $x$ is positive, $-x$ is negative, and vice versa.)


One way to define |x| is that it is the larger of x and -x, with the understanding that the larger of 0 and 0 is 0. So |-3| is the larger of -3 and -(-3) and the larger of these two numbers is -(-3) =3.


|x| is always positive. (Well,non-negative-- it could be zero.)

x might be positive or it might be negative. (Or it might be zero... let's just ignore the zero options for this answer...)

|x| is whatever it takes to express the "size" of x. If x is positive, the positive expression is "x". But if x is negative, x is not positive. Instead -x is positive.

So |x| is either, x or -x-- whichever one of those two is not negative.

If x < 0, then x is the one that is negative and -x is positive(!). And |x| is the positive one of x and -x. So which one is |x|?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.