GMAT® Inequalities: A Complete Guide

Inequalities are one of those topics that feels simple until you make a mistake under pressure. The rules aren't complicated. But the details — when to flip the sign, whether zero counts, how to read a compound inequality — are exactly the kind of details that cost people points on test day.

If you've been solving equations comfortably and inequalities still catch you off guard, that makes sense. Inequalities follow almost all the same rules as equations, with one major addition that changes everything. Once you internalize that one rule and build a system for avoiding the common traps, inequality questions become much more manageable.

This guide covers the symbols, the rules, the sign-flip principle, compound inequalities, number line graphing, common traps, and worked examples. What you need to handle inequality questions on the GMAT® Focus Edition.

Where Inequalities Appear on the GMAT® Focus Edition

Inequalities show up in two places on the GMAT® Focus Edition.

The Quantitative section includes inequalities in Problem Solving questions. You might need to solve an inequality, interpret one, or use one to answer a word problem.

The Data Insights section includes inequalities in Data Sufficiency questions. These often ask whether a given statement is sufficient to determine the relationship between variables — and that relationship is frequently expressed as an inequality.

The approach is the same in both sections. The difference is that DS questions ask whether you have enough information, while PS questions ask you to produce the answer. For a full breakdown of DS format and process, see our complete guide to GMAT® Data Sufficiency.

Inequality Symbols

Inequalities express relationships between values that aren't equal — or aren't always equal. Four symbols do the work:

• Greater than: $>$ means the left value is larger
• Less than: $<$ means the left value is smaller
• Greater than or equal to: $≥$ means the left value is larger or equal
• Less than or equal to: $≤$ means the left value is smaller or equal

The difference between "greater than" and "greater than or equal to" matters. If you see $x > 3$, then x could be 3.0001 but not 3. If you see $x ≥ 3$, then x could be exactly 3.

This distinction shows up on the GMAT® more than you might expect. A question that asks "how many integers satisfy $x > 3$ and $x < 7$?" has a different answer (4, 5, 6 — three integers) than "how many integers satisfy $x ≥ 3$ and $x ≤ 7$?" (3, 4, 5, 6, 7 — five integers).

The One Rule That Changes Everything

Here's the principle that makes inequalities different from equations.

All the same rules of equations apply to inequalities. The golden rule — whatever you do to one side, do to the other — still works. Combining like terms still works. Simplifying still works.

But there's one additional rule:

If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

One rule. But it's probably the source of more inequality errors on the GMAT® than anything else.

Why the Sign Flips

The rule can feel arbitrary if you've never thought about why it works. Here's the reasoning.

We can all agree that 5 is less than 6. So $5 < 6$ is true.

If we multiply both sides by negative 1, we get negative 5 and negative 6. But negative 5 is greater than negative 6. On the number line, negative 5 is to the right of negative 6, which means it's the larger value.

So $-5 > -6$ is the correct result. The less-than sign flipped to a greater-than sign because we multiplied by a negative.

If we didn't flip the sign, we'd write $-5 < -6$, which is false. That's why the flip is necessary.

Think of it like facing the opposite direction in a mirror. When you turn around, left becomes right and right becomes left. Multiplying by a negative does the same thing to an inequality — it reverses the relationship.

Worked Example 1: Sign Flip in Action

Solve: $-3x + 7 > 16$

Step 1: Subtract 7 from both sides.

$-3x > 9$

Step 2: Divide both sides by negative 3. Since we're dividing by a negative, flip the sign.

$x < -3$

The solution is any value of x that's less than negative 3. Not greater than — less than. The flip is what makes this work.

Worked Example 2: No Flip Needed

Solve: $3x + 5 < 2x - 7$

Step 1: Subtract $2x$ from both sides.

$x + 5 < -7$

Step 2: Subtract 5 from both sides.

$x < -12$

No flip needed. We never multiplied or divided by a negative, so the less-than sign stays the same.

Compound Inequalities

A compound inequality combines multiple inequalities into one expression. For example, instead of writing "$x > 5$ and $x < 10$," you might see:

$5 < x < 10$

This tells you the same thing: x is greater than 5 and less than 10. It's just a more compact way of expressing it.

The GMAT® likes compound inequalities. They appear in both Problem Solving and Data Sufficiency questions. If you haven't seen one before, it can look intimidating. But the rules are the same — you just apply operations to all parts of the inequality.

The Golden Rule for Compound Inequalities

Whatever you do to one part of a compound inequality, do to all parts.

If $a < b < c$, then $a + 1 < b + 1 < c + 1$. You can add the same value to all three parts. You can subtract. You can multiply or divide — and if you multiply or divide by a negative, you flip all the signs.

For example, if $a < b < c$ and you multiply everything by negative 2, you get $-2a > -2b > -2c$. Every sign flips.

Worked Example 3: Compound Inequality with Integer Count

This example comes from Episode 14 of our Real GMAT® Problems podcast series, which covers inequalities and word problem translation.

How many integers n satisfy $1 < 5n + 5 < 25$?

Step 1: Subtract 5 from all three parts.

$-4 < 5n < 20$

Step 2: Divide all three parts by 5.

$-\frac{4}{5} < n < 4$

Step 3: Count the integers in the range.

Negative four-fifths is approximately negative 0.8. So n must be greater than negative 0.8 and less than 4.

The integers that fit are 0, 1, 2, and 3. That's four integers.

The answer is 4.

The Zero Trap

That example contains one of the most common inequality traps on the GMAT®. Zero is an integer.

A lot of people miss zero when counting integers in a range. In the example above, zero is the smallest integer that satisfies the inequality. If you forget zero, you'd count 1, 2, 3 — three integers — and pick the wrong answer.

This trap shows up in any question that asks you to count integers within an inequality range. If the range includes zero, zero counts. Zero is an integer. It's neither positive nor negative, but it is an integer.

If you've ever missed this, you're in good company. It's one of the most common inequality errors on the test. Building a flashcard or adding a note to your memorization system can help. The definition of "integer" — no decimals, no fractions, zero included — is one of those basics worth reviewing regularly.

Number Line Graphing

Some GMAT® questions show inequalities graphed on a number line. You need to read these accurately.

The key is to look at the circle and the shading.

An open circle means the value isn't included. A filled circle means it is.

For $x > 3$, you'd see an open circle at 3 and shading to the right. The open circle tells you 3 isn't part of the solution. The shading to the right tells you all values greater than 3 are included.

For $x ≥ 3$, you'd see a filled circle at 3 and shading to the right. The filled circle tells you 3 is part of the solution.

The same logic applies for less-than inequalities. For $x < 3$, you'd see an open circle at 3 and shading to the left. For $x ≤ 3$, you'd see a filled circle at 3 and shading to the left.

When the inequality is bounded on both sides — like $3 ≤ x ≤ 5$ — you'd see filled circles at both 3 and 5 with shading in between. No arrows, because x can't extend to infinity in either direction.

Common Inequality Traps on the GMAT®

Trap 1: Forgetting to Flip the Sign

This is the most common inequality error. You're moving quickly, you divide by a negative, and you forget to flip the sign. The answer you get is the mirror image of the correct answer.

If this has happened to you, you're not alone. It happens to almost everyone at some point. The fix is to build a habit. Any time you multiply or divide by a negative, physically cross out the old sign on your scratch pad and write the new one. Don't try to do it in your head. Make it a physical action on paper.

If you build this habit during practice, it becomes automatic by test day. For more on building scratch work habits that hold up under time pressure, see our guide to GMAT® timing strategy.

Trap 2: Equation vs. Inequality Translation

When a word problem says "x is 10 greater than y," that's an equation: $x = y + 10$. A specific amount greater means an equals sign.

When a word problem says "x is greater than y," that's an inequality: $x > y$. No specific amount means a greater-than sign.

This distinction trips people up because the words "greater than" appear in both. But "10 greater than" is specific — it's an equation. "Greater than" without a number is general — it's an inequality.

If you see variables in the answer choices and the word problem uses "greater than" with a specific number, check whether it should be an equation. For more on translating word problems, see our complete guide to GMAT® word problems.

Trap 3: Forgetting Zero

As we saw in the worked example, zero is an integer. When a question asks you to count integers in a range, zero counts if it falls within that range.

This is a definition issue, not a math issue. The GMAT® defines an integer as a number with no decimals and no fractions. Zero fits that definition. It's neither positive nor negative, but it is an integer.

Trap 4: Sign Errors with Negative Coefficients

Sometimes an inequality has a negative coefficient on the variable, like $-x > 4$. To solve for x, you multiply or divide by negative 1 and flip the sign: $x < -4$.

People often solve this correctly in practice but miss it under time pressure because the negative coefficient doesn't trigger their "flip the sign" alarm. The trigger isn't just dividing by a negative — it's any multiplication or division by a negative, including when the negative is already attached to the variable.

Inequalities in Data Sufficiency

Inequalities appear frequently in Data Sufficiency questions. The typical format asks whether you can determine the relationship between two variables given certain statements.

For example, a question might ask: "Is $x > y$?" with two statements providing information about x and y.

The approach is the same as any DS question — evaluate each statement alone, then combine if needed. But inequalities add a layer because you often need to manipulate the statements to compare the variables.

A few tips for inequality DS questions:

Test numbers. Picking values for x and y and checking whether the statement holds is one of the most reliable approaches for inequality DS. Try positive numbers, negative numbers, zero, fractions, and integers. If different values give different answers, the statement isn't sufficient.

Watch for sign flips. When a statement gives you information like $-x > -y$, you can multiply both sides by negative 1 to get $x < y$. But you have to flip the sign. Missing that flip can lead you to the wrong conclusion.

Don't assume positivity. If a question involves variables and you don't know whether they're positive or negative, test both. A lot of inequality DS questions hinge on whether variables are positive or negative, and the answer often changes depending on the sign.

For the full DS process — the six-step method, AD/BCE elimination, and testing numbers — see our complete guide to GMAT® Data Sufficiency.

A System for Inequality Questions

Here's a simple process you can run on every inequality question on the GMAT®.

Step 1: Write down what's given and what's asked. This takes a few seconds and prevents translation errors.

Step 2: Isolate the variable using the golden rule. Add, subtract, multiply, or divide — just like an equation.

Step 3: Check for sign flips. Did you multiply or divide by a negative at any point? If yes, flip the sign. Cross out the old sign on your scratch pad and write the new one.

Step 4: If counting integers, include zero if it's in the range. Write out each integer individually so you don't miss any.

Step 5: If it's a DS question, test numbers. Try values that are positive, negative, zero, fractional, and integer. If different values give different answers, the statement isn't sufficient.

This system probably won't make inequality questions easy. But it can make them reliable. And reliability is what moves scores. For more on building reliable processes across all GMAT® question types, see our complete guide to studying for the GMAT®.

FAQ

Do you flip the inequality sign when subtracting?

No. You only flip the inequality sign when multiplying or dividing by a negative number. Subtraction — even subtracting a negative — doesn't require a sign flip. The same goes for addition.

Is zero an integer on the GMAT®?

Yes. Zero is an integer on the GMAT®. It's neither positive nor negative, but it is an integer. When a question asks you to count integers in a range that includes zero, zero counts.

What is a compound inequality?

A compound inequality is a single expression that combines multiple inequalities. For example, $5 < x < 10$ means x is greater than 5 and less than 10. You can perform operations on all parts of a compound inequality simultaneously, following the golden rule.

What's the difference between "greater than" and "greater than or equal to"?

"Greater than" (>) means the value can be any number larger than the comparison value, but not the value itself. "Greater than or equal to" (≥) means the value can be the comparison value or any number larger. On a number line, "greater than" uses an open circle, while "greater than or equal to" uses a filled circle.

How do inequalities appear in Data Sufficiency questions?

Inequality DS questions often ask whether you can determine the relationship between variables. You'll typically need to manipulate the statements, test numbers, or both. The key is to test multiple types of values — positive, negative, zero, fractions, and integers — because the relationship can change depending on the signs of the variables.

Can you multiply both sides of an inequality by a variable?

You can, but you need to know whether the variable is positive or negative. If the variable could be negative, you can't determine whether to flip the sign. This is why testing numbers is so valuable in inequality DS questions. If you don't know the sign of a variable, don't multiply or divide by it without considering both cases.

Want to learn even more?

Inequalities are one piece of the Quantitative puzzle. If you're building your quant foundations, our complete guide to GMAT® number properties covers primes, divisibility, odds and evens, and the testing-numbers method that handles many inequality questions too.

For the full DS process, our guide to GMAT® Data Sufficiency walks through the six-step method with worked examples.

If you're making avoidable mistakes on quant questions, our guide to the GMAT® error log shows a system for tracking and reviewing mistakes so they don't repeat on test day.

And if you're feeling stuck on quant more broadly, our guide to breaking through a GMAT® score plateau covers the three keys to getting unstuck.

For Isaac's full walkthrough of inequality basics, listen to Lesson 21 of our GMAT® Math Basics series, "Inequalities," on Spotify, Apple Podcasts, or YouTube. For worked inequality problems with common trap analysis, listen to Episode 14 of our Real GMAT® Problems series, "Inequalities and Translating Word Problems."