GMAT® Absolute Value Equations: A Complete Guide

Absolute value might be the friendliest-looking idea in GMAT® quant. Two straight bars, a familiar idea underneath, and the absolute value of 3 is 3. If the concept feels familiar, that's because it is: absolute value has probably been part of your math life since middle school, and the familiarity is earned.

The exam, though, rarely asks you for the absolute value of 3. It wraps the bars around an equation, and the moment a variable is inside them, a single equation can hold two solutions, exactly one, or none at all. So the skill worth building is bigger than computing $|3|$, and this guide covers it: what the bars mean, what they do to an equation, and the two-version method that solves them cleanly.

What absolute value means

The absolute value of a number is how far it is from zero. Some students memorize it as "distance from zero" and some as "how far from zero," and both phrasings point at the same idea, so use whichever one clicks for you.

Distance can't be negative. The distance from where you're sitting to where you're sitting is zero, and the distance to the door might be 20 feet. There's no version of that trip that's negative 20 feet. Since absolute value is just distance from zero, the output of an absolute value is always zero or positive.

The notation is two vertical bars with an expression between them:

$|x|$

• The two vertical bars are the absolute value notation.
• The expression between the bars is what gets measured.

A few rendered examples:

$|3| = 3$

$|-4| = 4$

$|0| = 0$

Notice the middle one: $-4$ sits four units from zero, so its absolute value is 4. And zero is zero units away from itself, which is why $|0| = 0$. That fact, absolute value output is always zero or positive, drives everything else in this guide.

Why the exam likes the bars

The GMAT® has a habit of taking a simple concept and building complications on top of it, and absolute value is one of its favorite wrappers. An operation that can take something positive or negative as its input and always outputs something zero or positive creates two openings for a question writer: it can hide a second solution inside an equation that looks like it should have one, and it can hand you a solution that looks right and then fails the check.

So if absolute value questions ever feel harder than the underlying material, the mismatch is by design. The material is basic, and the questions built on it don't have to be. The good news is that the complications follow a short list of patterns.

The bars behave like parentheses

In order of operations, absolute values occupy the same spot as parentheses: you do what's inside the bars before you do anything around them.

Take $3 + |4 - 6|$. You can't add the 3 to the 4 across the bars. Inside first: $4 - 6 = -2$, then $|-2| = 2$, and only then $3 + 2 = 5$.

If parentheses and bars end up nested inside each other, work from the innermost grouping out. This nesting is rare on the exam, but the rule is short: innermost first, whether that's parentheses or bars.

One more habit worth borrowing from parentheses, and then parting ways with: you can distribute a term across parentheses, but with absolute values we recommend you don't. Evaluating what's inside first and then taking the absolute value reaches the same place with less to track, and distributing a negative term across the bars wrecks the math in ways that are hard to spot under time pressure. Distribute across parentheses as much as you like. With bars, evaluate inside first.

The two-version method

Consider $|x + 5| = 3$. The bars are telling you that the expression inside, whatever it works out to, sits 3 units from zero. Two numbers do that, $+3$ and $-3$, so the equation is true in exactly two scenarios:

$x + 5 = 3$

$x + 5 = -3$

Solve each one. Subtracting 5 from both sides of the first equation gives $x = -2$. The same move on the second gives $x = -8$.

Both work, and you can confirm: $|-2 + 5| = |3| = 3$, and $|-8 + 5| = |-3| = 3$.

Two solutions from one equation. That's what the bars do: they turn one condition into two, because the expression inside can sit at either end of the given distance. Our worked solution on absolute value cases shows the same fork playing out in a real Official Guide problem. When a variable is inside them, two solutions is the usual outcome, not a sign that something went wrong. (Zero is the quiet exception: $|x| = 0$ has exactly one solution, since zero is the only number zero units from zero.)

The same method runs the same way on $|x - 3| = 11$: the inside must be $11$ or $-11$. The positive case gives $x = 14$. The negative case gives $x = -8$. Both check out, since $|14 - 3| = 11$ and $|-8 - 3| = |-11| = 11$.

Other approaches exist: some courses teach this with case notation, some with squaring both sides, etc. If you already have one that's working well for you, there's no requirement to switch. If you're starting from scratch, this is the version we've seen create the most success on the GMAT®: ask what two numbers sit at the given distance from zero, then build the two equations.

When a solution goes bad

Sometimes you'll run the two-version method, get two clean solutions, and have one of them fail in the original equation. This doesn't come up much, but GMAT® questions are tricky by nature, and it's worth knowing the pattern before test day rather than mid-question.

Consider $|x + 3| = 4x$. Two versions:

$x + 3 = 4x$

$x + 3 = -4x$

The first gives $3 = 3x$, so $x = 1$. Check it in the original equation: $|1 + 3| = 4$, and $4 \times 1 = 4$. It works.

The second gives $3 = -5x$, so $x = -\frac{3}{5}$. Check this one too: the left side becomes $\left|-\frac{3}{5} + 3\right| = \frac{12}{5}$, but the right side becomes $4 \times \left(-\frac{3}{5}\right) = -\frac{12}{5}$. A positive number can't equal a negative one, so $-\frac{3}{5}$ doesn't work.

The failure has a logic to it: the left side is an absolute value and can't be negative, while the right side, $4x$, came out negative for this particular solution. Any candidate that makes the non-absolute-value side negative can't survive the check.

So make it a habit: after solving an absolute value equation, plug both solutions back into the original equation, and write the check down like every other step. Most of the time both will work, but once in a while one won't, and the check is what can save the question.

If the bars are set equal to a negative number, there are no solutions at all. The equation $|x - 5| = -2$ has no answers, because no number's distance from zero is negative. A glance at the sign of what the bars are supposed to equal can save you the trouble of building two cases for an equation that has none.

Try this one

Try this one before reading on.

The inside must be $6$ or $-6$. The positive case: $2x - 4 = 6$, so $2x = 10$ and $x = 5$. The negative case: $2x - 4 = -6$, so $2x = -2$ and $x = -1$.

The sum is $5 + (-1) = 4$, which is choice (C), and both solutions check out in the original equation: $|2(5) - 4| = |6| = 6$, and $|2(-1) - 4| = |-6| = 6$.

Choice (D) is the trap: 5 is the answer to the positive case alone. A question asking for the sum of all possible values is asking whether you ran both versions, so the negative case isn't optional on questions like this one.

Where the test adds difficulty

Absolute value doesn't headline the quant section, but it shows up a reasonable amount inside equations and inequalities, and the exam's complications come from the packaging around the bars.

The Data Sufficiency format, which lives in the Data Insights section of the exam, leans on exactly this kind of ambiguity. An equation like $|x| = 3$ doesn't pin $x$ down, since $x$ could be $3$ or $-3$, and in Data Sufficiency that fork is the whole game: the question is whether the statements resolve it, and some of them pretend to. Remembering that a variable inside the bars usually has two possible values can help you spot the fork quickly.

Inequalities get a version of the same treatment. The two-version setup stays, with one addition: for the negative version, flip the inequality sign. So $|x + 5| < 3$ becomes $x + 5 < 3$ and $x + 5 > -3$, and both conditions have to hold, which solves to $x < -2$ and $x > -8$, or $-8 < x < -2$ written as one range. Our inequalities guide covers the flip and everything around it.

FAQ

What does absolute value mean on the GMAT®?

The absolute value of a number is how far it is from zero. Distance is never negative, so the output of an absolute value is always zero or positive.

How do you solve an absolute value equation on the GMAT®?

Create two versions of the equation, one where the expression inside the bars equals the positive value and one where it equals the negative value, then solve each. For $|x + 5| = 3$, the two versions are $x + 5 = 3$ and $x + 5 = -3$.

Why do absolute value equations usually have two solutions?

Two different numbers sit at every non-zero distance from zero, one positive and one negative. Either one can satisfy the equation, so a variable inside the bars usually produces two solutions.

Can an absolute value equal a negative number?

No. Absolute value is a distance from zero, and distance can't be negative. An equation like $|x - 5| = -2$ has no solutions.

Why do you need to check both solutions in an absolute value equation?

One solution can be extraneous: it solves the two versions but fails the original equation. In $|x + 3| = 4x$, the solution $x = -\frac{3}{5}$ makes the right side negative while the left side stays positive, so only $x = 1$ works.

What changes for absolute value inequalities?

The two-version setup stays, but the negative version flips the inequality sign. For $|x + 5| < 3$, the two versions are $x + 5 < 3$ and $x + 5 > -3$, and both must hold, giving $-8 < x < -2$.

Want to learn even more?

Related reading:

• GMAT® Equations: How to Simplify and Solve Them
• GMAT® Inequalities: A Complete Guide
• GMAT® Data Sufficiency: How to Approach DS Questions
• GMAT® Functions: A Complete Guide
• If the Absolute Value of x Equals the Absolute Value of y — GMAT® Worked Solution

The Math Basics series includes a full podcast lesson on absolute value, "GMAT® Focus Edition Math Basics: Absolute Value" (Lesson 22 in the series), and you can hear it on Spotify, Apple Podcasts, or YouTube.