GMAT inequalities questions are not conceptually difficult. Most of the underlying rules are things you have seen before in school math. The challenge shows up in the specifics: knowing exactly when the sign flips, when you cannot multiply by a variable, and how to handle quadratic or fractional inequalities under time pressure.
A lot of GMAT Quant scores stall because of this topic. Not from lack of effort. But from applying equation logic to inequality problems. The two look similar. They behave differently in a few important ways, and those differences are exactly what the GMAT tests. This comes up in both Problem Solving and gmat data sufficiency questions, which makes it a high-value topic to nail.
Here is what tends to go wrong for most GMAT test-takers on inequality questions. They do not have a solid grasp of the basic rules. They treat inequalities like equations. They know the number line is useful but rarely draw it. And they multiply both sides by a variable without checking its sign first. This guide fixes all of that.
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Take the Free GMAT DiagnosticGMAT Inequality Basics: Signs, Symbols, and the Number Line
An inequality, like an equation, compares two expressions. The difference is that instead of saying they are equal, it says one is greater than, less than, or equal to the other. There are four GMAT inequality signs.
| Sign | Expression | Meaning |
|---|---|---|
| > | x > y | x is greater than y |
| < | x < y | x is less than y |
| ≥ | x ≥ y | x is greater than or equal to y |
| ≤ | x ≤ y | x is less than or equal to y |
The clearest way to make sense of an inequality is to represent it on a number line. Take x ≤ 2. Every value of x that is 2 or less satisfies this. On a number line, shade everything to the left of 2 and place a closed circle at 2 to show that 2 itself is included. For x > 5, shade everything to the right of 5 and place an open circle at 5, since 5 itself does not count.
The distinction between open and closed circles matters on the GMAT. A closed (filled) circle means the endpoint is included — use this with ≤ or ≥. An open (hollow) circle means the endpoint is not included — use this with < or >. For compound inequalities like -3 ≤ x ≤ 4, the same logic applies to both ends.
The Basic Rules for GMAT Inequality Manipulation
There are two foundational rules. The first is straightforward. The second is where most test-takers lose points.
Rule 1: Addition, subtraction, and multiplication or division by a positive number do not change the sign. Take a true inequality: 4 < 8. Adding 2 gives 6 < 10. Subtracting 2 gives 2 < 6. Multiplying by +2 gives 8 < 16. All of these hold. Nothing surprising here.
Can you multiply both sides by a variable? Adding or subtracting a variable on both sides works fine. Multiplying or dividing by a variable is a different matter. You need to know its sign first.
Here is a classic GMAT trap: if x/y > 1, many test-takers multiply both sides by y and conclude x > y. But that is only valid if y is positive. If x = -3 and y = -2, then x/y > 1 still holds, but x is not greater than y. The correct and complete inference from x/y > 1 is that x and y must share the same sign. That is the whole conclusion. If you are working on improving your gmat quant score, this variable-sign trap is one of the first things to internalize.
Quick-Reference Table: Every GMAT Inequality Operation
Before going into advanced rules, here is the complete reference table for inequality operations. Keep this in mind as you work through the sections below.
| Operation | Condition | Effect on Inequality Sign |
|---|---|---|
| Add or subtract any value | Always | Sign stays the same |
| Multiply or divide by a positive number | Always | Sign stays the same |
| Multiply or divide by a negative number | Always | Sign flips |
| Multiply or divide by a variable | Sign of variable unknown | Cannot perform this operation |
| Take reciprocal of both sides | Both sides same sign | Sign flips |
| Take reciprocal of both sides | Sides have different signs | Sign stays the same |
| Take reciprocal of both sides | Signs unknown | Cannot perform this operation |
| Square both sides | Both sides positive | Sign stays the same |
| Square both sides | Both sides negative | Sign flips |
| Square both sides | Signs unknown or mixed | Cannot perform this operation |
Advanced GMAT Inequality Rules: Fractions, Reciprocals, Min-Max, and Quadratics
Beyond the basic flip rule, GMAT Quant inequalities come in several specific forms. Each has its own rules. Here is what you need to know about each one.
Six Advanced Rule TypesFor all proper fractions (0 < x < 1), there is a consistent relationship between the number, its square root, and its square.
If x = 1/4, then √x = 1/2 and x² = 1/16. So 1/2 > 1/4 > 1/16. The square root is always the largest for proper fractions; the square is always the smallest.
A square root inequality has a square root on at least one side. The key rules:
If x² < 100, then −10 < x < 10. If x² > 100, then x > 10 or x < −10.
If a < b, taking reciprocals depends on the signs:
Both positive: 1/a > 1/b (sign flips). Example: 2 < 3, so 1/2 > 1/3.
Both negative: 1/a > 1/b (sign flips). Example: −3 < −2, so 1/(−3) > 1/(−2).
Different signs: 1/a < 1/b (sign stays). Example: −3 < 2, so 1/(−3) < 1/2.
Signs unknown: Cannot take reciprocals.
The only operation you can perform directly between two inequalities is addition, and only when the signs point the same direction.
If a > b and c > d, then a + c > b + d.
If the signs differ, use the flip rule to make them match first, then add.
Never subtract, multiply, or divide across two inequalities directly.
To find the maximum or minimum of an expression given two ranges, compute all four extreme combinations.
Example: −7 ≤ x ≤ 6 and −7 ≤ y ≤ 8. Max of xy?
(−7)(−7) = 49 | (−7)(8) = −56 | (6)(−7) = −42 | (6)(8) = 48
Factor the expression, find the critical points, and test regions on the number line.
Example: (a−3)(a−6) < 0 gives 3 < a < 6. Example: (x−3)(x−2) > 0 gives x > 3 or x < 2.
Knowing the rules is step one — applying them under time pressure is the real challenge
Our gmat online coaching covers inequalities, quadratics, and every other quant area with timed practice built in from session one.
Explore GMAT Online CoachingHow to Solve GMAT Inequalities with the Wavy Curve Method
The Wavy Curve Method is a structured approach for solving quadratic and higher-degree GMAT inequalities. It works across all of them: linear, quadratic, cubic, and beyond. It is worth learning properly because once you have it, it replaces guesswork entirely.
How to use it:
Draw a horizontal number line. Identify the zero points, the values at which each factor in the expression equals zero, and mark them on the line. Start drawing a wave from the top-right of the number line and move left through each zero point.
The key rule for the wave: If a factor has an odd power, the wave passes through the zero point. If it has an even power, the wave bounces off the zero point and stays on the same side.
Reading the wave: Regions above the number line mean the expression is positive. Points on the line mean the expression equals zero. Regions below mean the expression is negative.
Cubic example: (x−1)(x−2)(x−3) > 0. Critical points are 1, 2, 3. They divide the line into four regions. Starting from the rightmost region as positive and alternating: x > 3 (positive), 2 < x < 3 (negative), 1 < x < 2 (positive), x < 1 (negative). Since the inequality asks for > 0, the solution is 1 < x < 2 or x > 3.
Works for fractions too. Treat numerator and denominator factors the same way. Find the zero points for each, mark them, and draw the wave. The zero of the denominator is excluded from the solution set because the expression is undefined there.
The Rule for Squaring Inequalities on the GMAT
This is one of the more nuanced areas of GMAT Quant inequalities. The short answer: you can only square both sides if you know the signs of both sides. Here is exactly what that means in practice.
| Situation | Can You Square? | What Happens to the Sign | Example |
|---|---|---|---|
| Both sides known positive | Yes | Sign stays the same | a > 4 → a² > 16 |
| Both sides known negative | Yes | Sign flips | a < −4 → a² > 16 |
| One side positive, one negative | No | Result depends on values | x = −2, y = 1: x² > y². x = −2, y = 3: x² < y² |
| Signs unknown or mixed | No | Cannot determine | If x > −3, x can be negative, zero, or positive |
The reason you cannot square when signs are unknown is simple: squaring a negative number makes it positive, so the inequality’s direction becomes ambiguous. If the sign of at least one side is unclear, squaring produces unreliable results. This comes up most often in Data Sufficiency questions where you have partial information about a variable.
Key Points to Remember While Solving GMAT Inequality Questions
Before moving into practice questions, here is a consolidated summary of the rules to keep in mind on test day.
| Rule | What It Means in Practice |
|---|---|
| Add or subtract freely | Any quantity, both sides, sign unchanged |
| Multiply/divide by positive | Sign unchanged |
| Multiply/divide by negative | Sign always flips |
| Squaring rule | Only when both sides have known signs |
| Variable multiplication | Never multiply/divide by a variable of unknown sign |
| x² as a variable | x² is always non-negative — safe to divide by |
| Adding inequalities | Only addition works across two inequalities — never subtract, multiply, or divide |
A useful self-check before each step: do I know the sign of what I am multiplying or dividing by? If not, stop and find another way to manipulate the inequality. Planning your gmat study schedule to include timed inequality drills specifically is one of the most efficient ways to close this gap.
10 GMAT Inequalities Practice Questions
Here are 10 practice questions covering every type covered in this guide. Work through each before checking the solution. If you cannot reach the answer, use it as a signal of which concept to revisit, not as a measure of your ability.
Use gmat practice tests to put these under actual timed conditions once you have worked through them here.
Amy had a Maths test and found that a particular question read: “Which of the following inequalities must be true if 0 < a < 1?”
- I. a⁵ < a³
- II. a⁵ + a⁴ < a² + a³
- III. a⁴ − a⁵ < a² − a³
Show Solution
Since 0 < a < 1, multiplying by a positive power preserves the inequality and makes the value smaller.
All three statements hold for any value of a in (0,1).
If 1 < a < b < c, which of the following has the greatest value?
Show Solution
Since c is the largest number, maximizing what c multiplies gives the greatest value.
c(a+b) is the greatest among all options.
Jane was counting her numbers and there were x integers that she counted. How many integers x are there so that 1 < 5x + 5 < 25?
Show Solution
If it is observed that 5|5 − s| = 3, what is the sum of all the possible values of s?
Show Solution
If a and b are integers, is a > b?
(1) a + b > 0
(2) ba < 0
Show Solution
Is p > q?
(1) 6p > 5q
(2) pq < 0
Show Solution
If 6/a(a+1) > 1, which of the following could be the value of a?
Show Solution
Rearrange: 6/a(a+1) − 1 > 0 → [6 − a(a+1)] / a(a+1) > 0 → (6 − a² − a) / a(a+1) > 0
If x(x + y) ≠ 0 and x > 0, is 1/(x + y) < (1/x) + y?
(1) x + y > 0
(2) y > 0
Show Solution
Rearrange the question: is 1/(x+y) − 1/x < y? Since x > 0, multiply through by x(x+y) only when we know its sign.
If it is true that a > −2 and a < 7, which of the following must be true?
Show Solution
On the number line, if m < n, if p is halfway between m and n, and if q is halfway between p and m, then what is the value of (n − q)/(q − m)?
Show Solution
“I kept losing easy inequality questions in Data Sufficiency because I was multiplying by variables without checking signs. Once that clicked, my quant accuracy on that question type went from 50% to consistent. It is a conceptual fix, not a practice-more fix.”
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Take the Free GMAT DiagnosticFrequently Asked Questions on GMAT Inequalities
What are GMAT inequalities?
GMAT inequalities are mathematical expressions that compare two quantities using signs like >, <, ≥, or ≤, instead of an equals sign. Unlike equations, they represent a range of valid solutions. On the GMAT Focus Edition, they appear in both Problem Solving and Data Sufficiency questions in the Quantitative Reasoning section.
When does the inequality sign flip on the GMAT?
The GMAT inequality sign flips in two situations: when you multiply or divide both sides by a negative number, and when you take the reciprocal of both sides where both sides have the same sign. This is the flip rule and it is one of the most tested concepts in GMAT Quant. Missing it once under time pressure typically costs you the question.
Can you square both sides of a GMAT inequality?
Only if you know the signs of both sides. If both sides are positive, square without flipping the sign. If both sides are negative, square and flip the sign. If the signs are unknown or point in different directions, squaring is not valid because the result depends on the specific values and no general rule applies.
Can you multiply both sides of a GMAT inequality by a variable?
No, not unless you know the sign of that variable. If the sign is unknown, you do not know whether the inequality sign should stay or flip, which makes the operation unreliable. This is one of the most common traps in GMAT Data Sufficiency inequality questions, particularly in expressions like x/y > 1.
What is the Wavy Curve Method for GMAT inequalities?
The Wavy Curve Method is a technique for solving quadratic and higher-degree GMAT inequalities. You find the critical points where each factor equals zero, mark them on a number line, then draw a wave starting from the top-right. Regions above the line are positive; regions below are negative. It also works for cubic and fourth-degree inequalities, and for fractional inequalities.
What is the most common mistake in GMAT inequality questions?
Multiplying or dividing both sides by a variable of unknown sign. For example, from x/y > 1, many test-takers conclude x > y by multiplying both sides by y. But if y is negative, the sign flips and the conclusion reverses entirely. The correct inference from x/y > 1 is only that x and y share the same sign.
How do you solve quadratic inequalities on the GMAT?
Factor the expression, find the critical points, and mark them on a number line. For ax² + bx + c < 0, the solution is between the two roots. For ax² + bx + c > 0, the solution is outside the two roots: either less than the smaller root or greater than the larger root. The Wavy Curve Method handles this systematically without re-testing each region.
What types of inequality questions appear on the GMAT Focus Edition?
GMAT Focus Edition inequality questions cover linear inequalities, quadratic inequalities, absolute value inequalities, fractional inequalities, square root inequalities, reciprocal inequalities, and min-max optimization problems. Data Sufficiency questions frequently use inequalities to test whether given conditions are sufficient to determine a definite answer, making sign-awareness especially important.
What to Focus on Next with GMAT Inequalities
The concepts in this guide are not difficult once you have the rules in a clear structure. The challenge is applying them automatically under time pressure. That comes from deliberate practice on the specific question types where you currently make errors.
If you worked through all 10 practice questions above and found specific types harder than others, that is exactly the signal you need. Quadratics and reciprocal inequalities are where most test-takers lose points, not linear ones. Revisit the Wavy Curve Method on quadratic and fractional inequality questions specifically. Use the quick-reference table before timed practice sessions until it is fully internalized.
Crackverbal has helped over 30,000 students since prepare for the GMAT. If any of the concepts here feel unclear, our gmat time management resources and quant team are there to help. Drop a question in the comments and our Quant team will answer it directly.
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