Remember the first time you saw an absolute value like |x| on a GMAT question, probably many years removed from your most recent algebra class?
If you had a difficult time remembering what to do with it, don’t worry. A quick refresher will have you back on track in no time. I find that the easiest way to think about absolute value is spatially, in terms of distance and location. This helps me to remember that the values can go in either direction on a number line like the one below:
For example, if I say you are two feet away from me, without saying in which direction, you could be in one of several locations. You could be to my left, behind me, in front of me, and so on.
Now, normally when I give you the value of a variable, say x=5, you know exactly where it is on the number line. However, an absolute value tells you how far from zero the value is (so to speak), but removes the directional information.
So, |x| = 2 indicates that x = 2 or –2, but we don’t know which.
Adding information changes matters, but it still doesn’t give us a discrete value. For example, |–x| = 2 is no better. If we plug in our values from before, you’ll see that |–2| and |–(–2)| produce the same result.
However, we can add some information about location by adding or subtracting values within the bars. For example |x + 1| = 2 has the effect of shifting the possible values for x left one place (notice the counterintuitive direction) on the line, since the solutions for x are now -3 and 1.
|x| = 2 is indicated in red.
|x + 1| = 2 is indicated in blue.
In our next post, we’ll see an example problem and discuss how thinking about absolute value spatially can help us to solve it.