GMAT Inequalities: Why You’re Losing Points (And How to Stop)
Inequalities typically account for 3 to 5 questions on the GMAT, appearing in Quant and resurfacing inside Data Sufficiency in the Data Insights section. Master four things and most of...
Inequalities typically account for 3 to 5 questions on the GMAT, appearing in Quant and resurfacing inside Data Sufficiency in the Data Insights section. Master four things and most of them fall quickly: the flip rule, the number line, the wavy curve method, and knowing when you cannot multiply, divide, or square. Concepts first, number line second, plugging in last of the last resorts.
GMAT inequalities punch above their weight. The topic claims its fair share of the exam, roughly 3 to 5 questions, yet it sinks more Quant scores than that count suggests, and it sits squarely in the algebra portion of the GMAT syllabus.
The reason is simple. Test-takers treat inequalities like equations. They multiply both sides by a variable without knowing its sign, and the trap snaps shut. This guide fixes that habit for good.
Take a free diagnostic and see exactly where your Quant score is leaking before you spend weeks on the wrong topics.
How to Approach Inequalities on the GMAT
One strategic rule governs this entire topic, so it comes before the maths.
Plugging in is the last of the last of the last resorts. Inequality questions are dense with must-be-true scenarios, and those carry implied conditions the question never spells out. Under the timer, it is dangerously easy to plug in a value that quietly violates one of them, and one bad plug-in “proves” a wrong answer. Random plug-ins also burn time you do not have. Work with basic concepts and the number line instead. The one situation where plugging in is legitimate: when the answer choices themselves are concrete values that can be tested directly against the conditions. That is elimination, not guessing.
What Are GMAT Inequalities?
An inequality relates two expressions, just like an equation. The difference is the sign in the middle.
| 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 single best habit you can build is representing every inequality on a number line. An inequality is not a value. It is a range, and seeing the range is what lets you solve hard questions in under two minutes.
Take x ≤ 2. A closed (shaded) circle at 2 shows the endpoint is included, because the sign is ≤. Everything to its left is part of the solution.
Closed circle: endpoint included
Now take x > 5. An open (unshaded) circle at 5 shows the endpoint is excluded, because the sign is a strict >. The solution starts just after 5.
Open circle: endpoint excluded
Compound ranges combine both ideas. Here is −3 ≤ x ≤ 4: a shaded segment between two closed circles. Both endpoints are in, and so is everything between them.
Both endpoints included
The Two Basic Rules of GMAT Inequalities
Two rules govern every linear inequality. Get these right and the rest follows.
Rule 1: Adding or subtracting the same quantity on both sides changes nothing. The sign holds. Multiplying or dividing both sides by the same positive number also changes nothing.
Think of it this way. If A has more money than B and both receive 300 rupees, A still has more. Take 300 from each, A still has more. The relationship survives.
Rule 2: Multiplying or dividing both sides by a negative number reverses the sign. This is the flip rule, and it is the single most tested inequality concept on the GMAT.
Watch it with a true inequality, 4 < 8:
Multiply by +2: 8 < 16. Still true, sign unchanged.
Multiply by −2: −8 > −16. Sign flips.
Divide by +2: 2 < 4. Still true, sign unchanged.
Divide by −2: −2 > −4. Sign flips.
Can you multiply or divide both sides by a variable?
Not unless you know its sign. This is where most wrong answers are born.
If xy > 1, the tempting move is to multiply both sides by y and conclude x > y. But nobody told you y is positive. If x = 3 and y = 2, the inequality holds and x > y. If x = −3 and y = −2, the inequality still holds, but x is less than y.
The only safe deduction from xy > 1 is that x and y share the same sign.
We cannot divide by an unknown variable. Now, can we divide by z²? It feels safe, since z² is never negative. The answer is still no. You do not know whether z equals zero, and division by zero is undefined. Dividing by z² is legitimate only when the question tells you z ≠ 0.
How to solve a linear inequality
Three moves, always in this order: isolate the variable and keep it positive where possible, apply the properties above without cancelling or cross-multiplying blind, then represent the answer on a number line.
Solve: −6x + 4 ≤ −2
Show solution
Subtract 4 from both sides: −6x ≤ −6. Divide both sides by −6 and flip the sign, because the divisor is negative: x ≥ 1.
If a, b, c are non-zero integers and a > bc, which of the following must be true?
- I. ab > c
- II. ac > b
- III. abc > 1
- A. I only
- B. II only
- C. III only
- D. I, II, and III
- E. None of these
Show answer & solution
The trap answer is D. Dividing a > bc by b, or c, or bc looks harmless, but the signs of b and c are unknown. If either is negative, the sign flips and the statement fails. Since this is a must-be-true question, none of the three survives.
Advanced Rules for GMAT Inequalities
Beyond linear expressions, the GMAT keeps returning to six patterns. Each has one governing idea.
1. Signs of products and quotients
When the right-hand side of an inequality is zero, the inequality is really a statement about signs. If a > 0 and b > 0, multiplying gives ab > 0. Reading that logic backwards is where the marks are:
a and b have the same sign. Both positive or both negative, and neither is zero.
a and b have opposite signs, and neither is zero.
Same signs, or at least one of a and b equals zero. The ≥ opens the door to zero.
A positive product keeps a and b together. A negative product separates them across zero
Quotients behave identically, with one extra guardrail:
Same signs, both non-zero. Identical to ab > 0.
Same signs, or a = 0. But b can never be zero, because it sits in the denominator.
Opposite signs, both non-zero.
2. Fractions between 0 and 1
For any x where 0 < x < 1, this chain always holds: √x > x > x².
Check it with x = 14. The square root is 12, the square is 116, and 12 > 14 > 116. Squaring a positive fraction shrinks it. Rooting it grows it.
If x = 0.888, y = √0.888 and z = (0.888)², which of the following is true?
- A. x < y < z
- B. x < z < y
- C. y < x < z
- D. z < y < x
- E. z < x < y
Show answer & solution
0.888 sits between 0 and 1, so √x > x > x², which means y > x > z. Reading left to right: z < x < y.
3. Adding and subtracting inequalities
The core rule: two inequalities can be added when their signs point the same way. If a > b and c > d, then a + c > b + d. Bigger plus bigger beats smaller plus smaller.
What about subtraction? Subtraction is nothing but addition after aligning the signs. Say a > b and c < d. The signs point in opposite directions, so multiply the second inequality by −1 and flip it: −c > −d. Now both signs match, so add: a − c > b − d. That is a subtraction, done safely, through addition.
If 4a + 2b < n and 4b + 2a > m, then b − a must be:
- A. < m − n2
- B. ≤ m − n2
- C. > m − n2
- D. ≥ m − n2
- E. ≤ m + n2
Show answer & solution
Multiply the second inequality by −1 to align the signs: −4b − 2a < −m. Add it to the first: 2a − 2b < n − m, so a − b < n − m2. Multiply by −1 and flip: b − a > m − n2.
When both right-hand sides are zero, you can also multiply. If mn > 0 and pq > 0, then their product mnpq > 0. If one is positive and the other negative, say mn > 0 and pq < 0, the product is negative: mnpq < 0. This shortcut collapses some hard-looking questions into one line.
If mn > 0 and np < 0, which of the following must be negative?
- A. mnp
- B. mnp²
- C. mn²p
- D. mn²p²
- E. m²n²p²
Show answer & solution
Approach 1: the sign table. mn > 0 means m and n share a sign, neither zero. np < 0 means n and p have opposite signs, neither zero. Only two sign patterns exist:
| m | n | p | |
|---|---|---|---|
| Case 1 | + | + | − |
| Case 2 | − | − | + |
C. mn²p: Case 1 gives (+)(+)(−), negative. Case 2 gives (−)(+)(+), negative. Negative in both. This is the must.
Approach 2: multiply the inequalities. Both right-hand sides are zero, so multiplication is allowed:
One positive quantity times one negative quantity gives a negative product. Option C in three lines.
Approach 3: read the structure. Option C is simply the split (mn)(np), the two quantities whose signs you already know.
4. Reciprocal inequalities
Start from x < y and ask: what happens to 1x and 1y? Three cases:
Take x = 2 and y = 3. Then 1x = 0.5 and 1y ≈ 0.33. So 1x > 1y.
Take x = −3 and y = −2. Then 1x ≈ −0.33 and 1y = −0.5. So 1x > 1y.
Take x = −2 and y = 3. Then 1x = −0.5 and 1y ≈ 0.33. Still 1x < 1y.
Same sign: reciprocal can be taken, and the sign of the inequality flips.
Opposite signs: reciprocal can be taken, and the sign does not flip.
Signs unknown: the reciprocal cannot be taken at all.
Why it matters: knowing the sign is what enables the move. The GMAT hides the sign in the structure of the expression.
What is the largest integer x such that 12x > 0.01?
- A. 5
- B. 6
- C. 7
- D. 10
- E. 51
Show answer & solution
Both sides positive, same sign, so the reciprocal can be taken with a flip: 2x < 100. The largest power of 2 below 100: 2⁶ = 64 works, 2⁷ = 128 does not. So x = 6.
5. Square roots and squaring
Two templates cover nearly every square root question:
If x² < a²: then −a < x < a. Example: x² < 100 gives −10 < x < 10.
If x² > a²: then x > a or x < −a. Example: x² > 100 gives x > 10 or x < −10.
If (y − 5)² < 36, find the range of y.
Show solution
Template 1 with a = 6: −6 < y − 5 < 6. Add 5 throughout: −1 < y < 11.
And the reverse direction: when can you square an inequality? Only when you know the signs of both sides. Four cases:
If a > 4, both sides are positive, so a² > 16.
If a < −4, both sides are negative, so a² > 16.
No conclusion possible.
If x > −3, x could be negative, zero or positive, and x² ranges from 0 upward with no usable bound.
This is exactly the kind of pattern a private GMAT tutor catches in your practice sets long before the exam does.
6. Max-Min inequalities
When two ranges are given and you need the extremes of x + y, x − y, or xy, write the ranges one under the other, then combine the extreme values to get the candidates. The largest is the max, the smallest is the min.
If −13 < 7x + 1 < 29 and 19 < 2 − y < 23, what is the maximum possible integer value of x + y?
- A. −23
- B. −18
- C. −14
- D. −13
- E. −12
Show answer & solution
Clean up the first range: −2 < x < 4. Clean up the second: −21 < y < −17.
The maximum integer strictly below −13 is −14. Only option C sits inside the range.
That gap between knowing and applying is exactly what structured coaching closes. See how our live classes drill these patterns until they are automatic.
Quadratic Inequalities and the Wavy Curve Method
When the expression has an x² term, you cannot isolate x the way you would in a linear inequality.
Suppose the GMAT hands you this:
Solve: x² − 5x + 6 < 0
The x² term changes the game: the expression can dip below zero in one stretch of the number line and climb above it elsewhere. The answer is not a single cut-off. It is an interval, sometimes two.
The wavy curve method finds those intervals in six steps. Below, each step shows a small teaching example, and alongside it we check the same step on our main question.
Make the coefficient of the highest power positive
In ax² + bx + c, check whether a > 0. If not, multiply the whole inequality by −1 and flip the sign.
Make the right-hand side zero
Move everything to the left so the inequality reads: expression > 0, < 0, ≥ 0, or ≤ 0.
Find the critical points
Factorise the expression. The critical points are the values where the entire expression becomes zero.
Plot the zero points on the number line
With n distinct critical points, the line splits into n + 1 regions.
The rightmost region is positive
Anything to the right of the largest critical point makes the overall expression positive. Mark it +.
Alternate signs moving left, then read off your regions
Each region to the left takes the opposite sign of its neighbour: +, −, +, − and so on.
Here is the finished picture for x² − 5x + 6 < 0. The dotted wave shows the alternation. We need the negative stretch, so the solution is 2 < x < 3, with open circles because the inequality is strict.
(x − 2)(x − 3) < 0. Wave above the line means +, below means −
Same expression, different signs
The picture stays identical. Only the regions you read off change:
< 0: the middle, endpoints out. 2 < x < 3.
> 0: the outside, endpoints out. x < 2 or x > 3.
≤ 0: the middle, endpoints in. 2 ≤ x ≤ 3.
≥ 0: the outside, endpoints in. x ≤ 2 or x ≥ 3.
Your turn: solve 3x² − 7x + 4 ≤ 0
Solve: 3x² − 7x + 4 ≤ 0
Show solution
Factorise: (3x − 4)(x − 1) ≤ 0. Critical points: 1 and 43.
Rightmost region +, alternate leftward: +, −, +. We need ≤ 0, so take the negative region. Closed circles.
Answer: 1 ≤ x ≤ 43
(3x − 4)(x − 1) ≤ 0. Closed circles: endpoints included
Fractions inside the method
On the GMAT, do not work with a fractional inequality directly. Convert it into a non-fractional one first. Two steps, every time.
Example 1: x + 1x − 3 < 0
Multiply the numerator and the denominator by the denominator term
The overall sign of the fraction does not change:
(x + 1)(x − 3)(x − 3)² < 0
(x − 3)² is never negative and cannot be zero. Mental note: x ≠ 3.
Transfer the sign to the numerator, then run the wavy curve
The denominator is positive. So the negative sign comes from the numerator: (x + 1)(x − 3) < 0. The fraction is gone.
From here it is the usual wavy curve: critical points at −1 and 3. Solution: −1 < x < 3.
Seven Rules to Remember on Test Day
Pin these. Every inequality question traces back to one of them.
Add or subtract freely. Any quantity, both sides. The sign never changes.
Positive multipliers are safe. Multiply or divide by a positive value. The sign never changes.
Negative multipliers flip. Multiplying or dividing by a negative number always reverses the sign.
Square only when signs are known. Both positive: square, keep the sign. Both negative: square and flip. Mixed or unknown: no squaring.
Unknown sign, no operation. Never multiply or divide by a variable whose sign is unknown.
Even powers are not automatically safe. z² is never negative, but it can be zero. Divide by z² only when z ≠ 0 is given.
Combine inequalities through addition. Align the signs first; subtraction is just multiplying one by −1 and adding.
GMAT Inequalities Practice Questions
Seven questions in a graded ladder: two easy, two medium, three hard. Concepts and the number line first; use answer choices only to eliminate.
How many integers x satisfy 1 < 5x + 5 < 25?
- A. 1
- B. 2
- C. 3
- D. 4
- E. 5
Show answer & solution
Subtract 5: −4 < 5x < 20. Divide by 5: −45 < x < 4. The integers in range: 0, 1, 2 and 3. Four of them.
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 answer & solution
Expand and compare in pairs. E versus B: cancel cb, get ca versus c. Since a > 1, E wins. E versus D: cancel bc, get ca versus ab, i.e. c versus b. E wins. E beats every option.
If m < 12, then it must be true that:
- A. −m < −12
- B. −m − 2 < 14
- C. −m + 2 < −10
- D. m + 2 < 10
- E. m − 2 < 11
Show answer & solution
E: m − 2 < 11 rearranges to m < 13. Given m < 12, m is certainly less than 13. Must be true.
If 6a(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 answer & solution
The real condition is: 0 < a(a + 1) < 6. Test: B gives (−2.5)(−1.5) = 3.75. Positive and below 6. Works.
If 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 answer & solution
The given range is −2 < a < 7. Place −7 on the line. The entire given range sits to the right of −7. Every possible value of a is greater than −7. B is the must. Since B holds, E is out automatically.
If y > 0 and 0 < 1 − xy < 1, which of the following must be true?
- I. x > 0
- II. xy < 1
- III. x² + y² > 1
- A. I only
- B. II only
- C. I and II only
- D. II and III only
- E. I, II and III
Show answer & solution
Break the given down. Subtract 1: −1 < −xy < 0. Multiply by −1 and flip: 0 < xy < 1.
I: xy > 0, same sign as y > 0, so x > 0. True.
II: xy < 1 is directly in the breakdown. True.
III: x = 0.0002 and y = 0.0003 satisfy the given but x² + y² is nowhere near 1. Not always true.
If x, y, z and w are positive integers and xy < zw, which of the following must be true?
- I. x + zy + w < zw
- II. x + zy + w < xy
- III. x + zy + w = xy + zw
- A. None
- B. I only
- C. II only
- D. I and II
- E. I and III
Show answer & solution
Cross-multiply the given: xw < yz. Statement I cross-multiplies to xw < yz, exactly what we were given. Must be true. Statement II yields yz < xw, the opposite. Not true. Statement III has no chance. Only I holds.
Inequalities are one topic. If you are prepping on a tight timeline, a structured self-paced programme covers the full syllabus without guesswork.
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Talk to a GMAT ExpertThe Rules Fit on an Index Card. The Score Difference Comes from Applying Them.
Inequalities reward discipline more than brilliance. Work the number line, respect the flip rule, and never operate on a variable of unknown sign. If the broader section is the struggle, start with how to improve your GMAT Quant score systematically, then anchor it all inside a sensible GMAT study schedule.
Your score earns you a read. Your discipline on these rules earns you the marks. Sign checks, number lines, no blind operations. Take it into the rest of your Quant.
One-on-one tutoring means every practice set you solve gets reviewed for exactly these error patterns, by someone who has seen them thousands of times.
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.
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