GMAT Math will ask you about absolute values. Mastering what the GMAT asks about them requires sophisticated understanding.
Somewhere along the line, perhaps in middle school, you probably learned:
|positive| = positive and |negative| = positive
In other words, the equation |x| = 5 has the solution: x = 5 or x = −5. (Notice: the word “or” is not a garnish there; it’s actually an essential piece of mathematical equipment.)
Expanding the Pattern
That’s great, but the GMAT is simply not going to ask you to solve the equation |x| = 5. When the GMAT asks about absolute value, it’s going to be something more in the vein of |3x – 7| = 5. The basic idea is (as is often the case in more advanced algebra) is to replace “x” in the simpler equation above with whatever “thing” is between the absolute value. If q is a positive constant, then
|thing| = q
has the solution:
thing = q or thing = -q
In the given example,
|3x – 7| = 5
3x – 7 = 5 or 3x – 7 = -5
3x = 12 or 3x = 2
x = 4 or x = 2/3
That’s an example of an absolute value equation, which the GMAT could ask. The GMAT is even more likely to ask about an absolute value inequality.
Rethinking Absolute Value
OK, let’s face it. The definition of absolute value that says “keeps a positive positive, and makes a negative positive” – the utility of that definition peaks in middle school. We need to have a more sophisticated understanding of absolute value to handle everything the GMAT will ask of it.
Here is the more sophisticated definition of absolute value. The absolute value of x, |x| is the distance of x from zero on the number line. Of course, it’s always positive, because distance is away positive.
To extend that further: |x – 4| is the distance of x from 4; |x – 7| is the distance of x from 7;|x + 3| is the distance of x from -3 (this is because x + 3 = x – (-3) when written as subtraction).
That is profoundly important in solving the absolute value inequalities that the GMAT will ask of you. Suppose a GMAT Math question asks you: represent the region -1 < x < 9 as an absolute value inequality.
The first step is to find the midpoint of the region: 4 is exactly halfway between -1 and 9. Now the distances: 9 is a distance of 5 from 4, and so is -1. So the distance from 4 (viz. |x – 4|) can’t equal 5, but it can be anything up to 5. Thus
|x – 4|< 5
is the absolute value inequality representation of the region -1 < x < 9. Integrate this understanding, and you will be able to handle anything the GMAT asks you about absolute value.
Here’s a free Magoosh practice problem, a challenging absolute value practice question, with a video explanation of the answer.