GMAT Inequalities: Why You’re Losing Points (And How to Stop)
GMAT inequalities trip up test-takers not because the concepts are hard, but because people apply equation logic to them. This guide covers every rule you need: the flip rule, the...
GMAT inequalities trip up test-takers not because the concepts are hard, but because people apply equation logic to them. This guide covers every rule you need: the flip rule, the Wavy Curve Method, squaring, reciprocals, min-max, and quadratics — with a quick-reference table and 10 practice questions with full solutions.
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. Test-takers 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.
A free GMAT diagnostic test gives you section scores and shows you exactly which Quant areas are costing you points right now.
GMAT Inequality Basics: Signs, Symbols, and the Number Line
An inequality compares two expressions using direction rather than equality. There are four signs, and each has a precise meaning on the number line.
| 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 circle means the endpoint is included — use this with ≤ or ≥. An open 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.
Drawing the number line is not optional on hard inequality questions. It takes five seconds and eliminates the most common range errors. The GMAT tests the edges of the range, not the middle. If you do not draw it, you will misread where the valid values start and stop.
Two Foundational 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.
Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Using 4 < 8: multiply both sides by −2 and you get −8 > −16. The sign flips.
This is the single most tested rule in GMAT Quant inequality questions. Miss it once under time pressure and it costs you the question.
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 internalise.
Quick-Reference Table: Every GMAT Inequality Operation
The complete reference for inequality operations. Return to this before test day.
| Operation | Condition | Effect on 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 |
| 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 |
| 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 |
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 and failure modes.
For all proper fractions (0 < x < 1), there is a consistent relationship between the number, its square root, and its square.
If x = 1/4: √x = 1/2 and x² = 1/16. So 1/2 > 1/4 > 1/16. The square root is always the largest; 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: −10 < x < 10. If x² > 100: x > 10 or x < −10.
If a < b, taking reciprocals depends on the signs:
Both positive: 1/a > 1/b (sign flips). 2 < 3, so 1/2 > 1/3.
Both negative: 1/a > 1/b (sign flips). −3 < −2, so 1/(−3) > 1/(−2).
Different signs: 1/a < 1/b (sign stays). −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 signs differ, use the flip rule to match them first. 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.
(a−3)(a−6) < 0 gives 3 < a < 6. (x−3)(x−2) > 0 gives x > 3 or x < 2.
Crackverbal’s GMAT online coaching covers inequalities, quadratics, and every other Quant area with timed practice built in from session one.
How to Solve Higher-Degree Inequalities: The Wavy Curve Method
The Wavy Curve Method is a structured approach for solving quadratic and higher-degree GMAT inequalities. 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 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. 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. The zero of the denominator is excluded from the solution set because the expression is undefined there.
Forgetting to alternate the signs correctly from right to left. Start right as positive, then alternate without exception. If a factor has an even power at a zero point, the wave bounces instead of crossing — the sign does not alternate at that point.
The Rule for Squaring Inequalities on the GMAT
You can only square both sides if you know the signs of both sides. This is one of the most nuanced areas of GMAT Quant inequalities.
| Situation | Can You Square? | Effect on 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. This comes up most often in Data Sufficiency questions where you have partial information about a variable.
Key Rules to Remember Before Test Day
A consolidated summary of every rule in this guide. Use this as a final check before timed practice.
| 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 or 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. Building a GMAT study schedule that includes timed inequality drills is one of the most efficient ways to close this gap.
10 GMAT Inequalities Practice Questions
Work through each question before checking the solution. 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
- A I only
- B II only
- C I and II only
- D I, II and III
- E None
Show Solution
Since 0 < a < 1, multiplying by a positive power preserves the inequality and makes the value smaller.
If 1 < a < b < c, which of the following has the greatest value?
- A c(a+1)
- B c(b+1)
- C a(b+c)
- D b(a+c)
- E c(a+b)
Show Solution
Jane was counting her numbers. How many integers x satisfy 1 < 5x + 5 < 25?
- A 1
- B 2
- C 3
- D 4
- E 5
Show Solution
If 5|5 − s| = 3, what is the sum of all the possible values of s?
- A 13
- B 10
- C 8
- D 7
- E 6
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?
- A −3.5
- B −2.5
- C 2.5
- D 3.5
- E 4.5
Show Solution
Rearrange: (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
If it is true that a > −2 and a < 7, which of the following must be true?
- A a > 2
- B a > −7
- C a < 2
- D −7 < a < 2
- E None of the above
Show Solution
On the number line, if m < n, p is halfway between m and n, and q is halfway between p and m, what is the value of (n − q)/(q − m)?
- A 1/4
- B 1/3
- C 4/3
- D 3
- E 4
Show Solution
A full GMAT diagnostic gives you section-level scores and shows exactly which Quant areas need the most work before test day.
Still have questions?
Crackverbal’s GMAT Quant mentors have helped 50,000+ students nail inequalities and every other Quant topic since 2006.
Talk to a Quant mentorThe Concepts Are Learnable. The Work Is Applying Them Fast.
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.
Crackverbal has helped 50,000+ students since 2006 prepare for the GMAT. If any of the concepts here feel unclear, our GMAT time management resources and Quant team are there to help.
Know your sign before you multiply or divide. Draw the number line before you analyse a range. Apply the Wavy Curve before you guess at a quadratic. Three habits, applied consistently, cover the vast majority of GMAT inequality errors.
A profile evaluation tells you exactly what score you need, which programmes are realistic, and how to build a prep timeline that fits your schedule.
Devmitra Sen is Head of Academics at Crackverbal and has trained over 4,000 students. Her scorers tell the story: GMAT 745, 725, 715, 705 alongside turnarounds like 575→715 and 375→675. She has produced multiple Q90 scores, including a perfect 100th percentile on GMAT Quant — a benchmark very few coaches can claim consistently. On Data Insights, her superpower is changing how students see, observe, and comprehend data: breaking it down, reasoning through it, and zeroing in on exactly what the question asks. The results follow: multiple 90+ percentile DI scores, including a 1st to 99th percentile turnaround in under two and a half months. She carries a quiet interest in the history of mathematical thought — particularly ideas rooted in India long before they were formalised elsewhere — a perspective that gives her an unusually grounded sense of why the subject matters.
TL;DR GMAT Multi-Source Reasoning (MSR) questions give you 2–3 tabs of data and ask you to cross-reference across all of...
TL;DR GMAT Graphics Interpretation (GI) questions appear in the Data Insights section and ask you to read a graph or...
TL;DR The GMAT is accepted by over 7,700 programs at more than 2,400 business schools globally — covering not just...
Your 705+ score starts with
one expert call.
30-minute strategy session. We'll audit your prep, find your gaps, and build a roadmap to your target score.