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Lucas Fink

The Rules of Absolute Values for the SAT

Before we get into the rules, let’s see an example of an SAT question that uses absolute values:



1. In the equations above, x < 0 < y. What is the value of x + y?

(A) -8

(B) -5

(C) 2

(D) 16

(E) 26


Rule 1: Absolute values are grouping symbols

In the question above, there are two ways that we can deal with the first equation.

The first way is to treat the absolute value bars like parentheses. They serve a very similar purpose (the operations inside them have to be carried out before you can do any addition or subtraction), and you can distribute in the same way. So we can figure this:



Rule 2: Isolate absolute values

Alright, so maybe you saw the other way to work with that equation. Instead of distributing, you could just divide out the 2 from both sides. It works the same way, in the end.


In either case, you’ve done exactly what you need to: get that absolute value alone. Only then can you get rid of the bars.


Rule 3: Make two equations

Getting rid of the bars means splitting the equation into two: one positive equation and one negative. So that first equation ends up like this.

delim{|}{x-9}{|}=12 doubleright x-9=12 or x-9=-12

x=21 or x=-3

By following the instructions in the question, we know that only one of those is possible.


After we plug that in to the second equation, we’ll take away the absolute value bars and, again, split it into two equations.


y*-3=15 or y*-3=-15

y=-5 or y=5

The rest is pretty straight forward. Since y is greater than 0 according to the question, we know y=5, so the answer must be 2, which is (C).


Rule 4: Reverse inequalities for the negative value

If you have an inequality rather than an equation, you still have to split it into two possibilities. But you could also put those two possibilities into one expression. Let’s look at an example of that.

Without any absolute value, the inequality y<10 looks like this on a number line:


But if we put in some absolute value bars, it’ll look different.



That’s because while y can be -2 or -9, it can’t be -11. If it were -11, the absolute value would be 11, which is too high.

Splitting the inequalities, then, gives you this:


y<10 and y><noscript><img src=


Double check the rules

When you have absolute values on your SAT, always double check what information the question gives you. Does is define whether anything is positive or negative? Then, after you split the equation, be ready to scrap the value which doesn’t work.


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About Lucas Fink

Lucas is the teacher behind Magoosh TOEFL. He’s been teaching TOEFL preparation and more general English since 2009, and the SAT since 2008. Between his time at Bard College and teaching abroad, he has studied Japanese, Czech, and Korean. None of them come in handy, nowadays.

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