# Absolute values in GMAT Quant

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 we say you are two feet away from us, without saying in which direction, you could be in one of several locations. You could be to our left, behind us, in front of us, and so on.

Now, normally when we 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.

Here’s a practice question that expands upon the concept a bit. Our solution and explanation will follow in a later post. Remember, this is a Data Sufficiency question.

What is the value of x?

(1) |x-6| = 3

(2) |x| = 3(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

(B) Statement (2) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked

(D) EACH statement ALONE is sufficient to answer the question asked

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

## Solution:

First, remember that absolute value gives us distance without direction. So, a simple equation setting x at some distance from 0 is not enough to give us a specific value for x, since x could be to the left of (less than) or to the right of (greater than) 0. So, statement (1) is insufficient.

Moreover, adding or subtracting values within the bars only shifts the values on the number line. Thus, statement (2) is insufficient as well.

What about combining the statements? Well, thinking back to our spatial point, this is sort of like triangulation, in which we use fixed points to solve indirectly for an unknown. If I take my first two locations on the number line and shift them one direction or another, only one of those points (at most) will still line up with the original (see diagram).

Take statement (1) first. It gives us two possible values for x: 3 and –3 (indicated in red on the diagram).

Then, using statement 2, add 6 to each of those value to get 3 and 9. The only value that works for both statements is x = 3, so the statements are sufficient together.