When is an Absolute Value Not an Absolute Value?
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… when it’s a distance on a number line!
Okay, that doesn’t quite work as a joke. But it does work as a GMAT Quant strategy. Intimidated by absolute value GMAT problems? Read on to learn a quick and painless strategy.
Absolute values always come out as positive numbers. For instance, the absolute value of -7 is 7:
|-7| = 7
Distances are also always positive, both in the real world and on the GMAT. There’s no such thing as a negative distance. That means we can use distances—something we already understand intuitively—to think about absolute value.
The town of Greatport is at mile marker 5 on the highway, and the town of Fairmont is at mile marker 35.
If you want to travel from Greatport to Fairmont, you calculate the distance like this:
35 – 5 = 30
But if you want to travel from Fairmont back to Greatport, you don’t do this:
5 – 35 = -30
You know intuitively that the distance is still 30 miles, not negative 30, regardless of which direction you’re traveling. What you’re really doing, mathematically, is taking an absolute value.
|5 – 35| = |-30| = 30
|35 – 5| = |30| = 30
The distance between two towns is the absolute value of the difference between their mile markers.
Let’s add a third town: Veltria. But I’m not actually going to tell you where Veltria is. All I’m going to tell you is that it’s 5 miles away from Fairmont.
Here’s the equation that shows that:
|v – 35| = 5
By the way, this would be equally correct:
|35 – v| = 5
What this equation means is that the distance between Veltria and the 35-mile marker is 5 miles. Your intuition should tell you that Veltria can only be in two different locations: the 30-mile marker or the 40-mile marker. And in fact, those are the two values that fit the equation:
|30 – 35| = |-5| = 5
|40 – 35| = |5| = 5
So, when you see an equation like |10 – x| = 7, you can read it like this:
“The distance between 10 and x is 7.”
Without actually doing algebra, you can figure out that x can only equal 3 or 17.
Let’s bring a fourth town into the mix. It’s called Halfwayville, and here’s an equation that tells you where it’s located:
|h – 35| = |5 – h|
If you see this in an algebra problem on the GMAT, don’t start simplifying it with math. Instead, read it as if it’s telling you about the real world.
|h – 35| = “The distance between Halfwayville and the 35-mile marker”
|5 – h| = “The distance between Halfwayville and the 5-mile marker”
The equals sign between them means that those two distances are the same. In other words,
“Halfwayville is equally far from the 5-mile marker and the 35-mile marker.”
There’s only one place Halfwayville could be located: halfway between those two markers! That places it at the 20-mile marker.
What if they start bringing inequalities into the mix? Let’s locate the town of Easton on the highway. Here’s what you know:
|e – 35| > 7
The left side of the inequality is the distance between Easton and the 35-mile marker. So, read this inequality like this:
“The distance between Easton and the 35-mile marker is more than 7 miles.”
In other words,
“Easton is more than 7 miles from the 35-mile marker.”
Where could Easton be? Anywhere, as long as it’s at least 7 miles from Fairmont.
Let’s locate a new town: Middleburg. This time, all you know about it is this inequality, which has two absolute values:
|5 – m| > |35 – m|
Read it in plain English:
“The distance between Middleburg and the 5-mile marker is greater than the distance between Middleburg and the 35-mile marker.”
Or:
“Middleburg is closer to the 35-mile marker than to the 5-mile marker.”
Where could Middleburg be? It can’t be off to the left side of Greatport; if it was over there, it would be closer to Greatport. We want it to be closer to Fairmont. It could be between Greatport and Fairmont, as long as it’s closer to Fairmont. It could also be over to the right side of Fairmont. Here are all of the possibilities:
That inequality, |5 – m| > |35 – m|, is really just saying that Middleburg is to the right of the halfway point between Greatport and Fairmont.
What if they give you even less info? Let’s suppose that we’re now in a foreign country, where we don’t know where anything is at all. You see this equation:
|x – y| + |y – z| = |x – z|
Read it out piece by piece. We have three towns: Xandria, Yelby, and Zorb.
On the left side of the equation, we have this:
“The distance from Xandria to Yelby, plus the distance from Yelby to Zorb”
On the right side, we have this:
“The distance from Xandria to Zorb”
In other words, if you drive straight from Xandria to Yelby, then drive from Yelby to Zorb, you’ll cover the exact same distance as you would if you drove directly from Xandria to Zorb.
What does that mean?
Suppose that Xandria, Yelby, and Zorb were laid out like this. If you traveled from X to Y, then from Y to Z, you’d be going out of your way:
There’s actually only one situation where going through Yelby doesn’t add any distance to your trip. That’s the situation where Yelby is right on the line between Xandria and Zorb.
In that case, the distances are equal. If you go from X to Y, then from Y to Z, you’ve covered the same distance as going straight from X to Z.
In other words, this equation:
|x – y| + |y – z| = |x – z|
Means this:
“Y is located on the line that goes from X to Z.”
On a number line—which is where we do most absolute value problems—that just means that Y is in between X and Z, rather than being off to one side.
Absolute value problems are more intuitive than you might think, even when they’re combined with inequalities! The next time you see an absolute value of a difference, pause before you start applying algebra rules. Can you think about the problem in terms of distances, instead? If so, you may find that the solution is faster and simpler than you expected! ?
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Chelsey Cooley
is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GMAT prep offerings here.