The Ultimate Guide to GRE Inequalities – 2026
GRE inequality questions test one thing above all else: whether you know when an operation flips the inequality sign. Multiplying or dividing by a negative number reverses it. Squaring, taking...
GRE inequality questions test one thing above all else: whether you know when an operation flips the inequality sign. Multiplying or dividing by a negative number reverses it. Squaring, taking reciprocals, or multiplying by a variable only works when you know the sign of both sides. Quadratic inequalities need a number line and critical points, not guesswork. Once these rules are automatic, inequality questions stop being a guessing game.
Inequality questions look like equations, but the rules change the moment you multiply, divide, square, or take a reciprocal. That mismatch is exactly what makes them a quiet score-killer: the algebra feels familiar, so test-takers apply equation rules on autopilot and pick a trap answer.
This guide walks through every rule you need for GRE Quant inequalities, from basic number line notation, through the properties that govern fractions, squares, and reciprocals, to quadratic inequalities and their role in Quantitative Comparison questions. Work through it once, properly, and you won’t need to relearn it under time pressure.
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What Is an Inequality? And how to read it
The symbols, the number line notation, and what an open versus a closed circle actually means.
An equation states that two expressions are equal, shown with the symbol =. An inequality states that two expressions are not necessarily equal, using one of four symbols:
- 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
On a number line, a closed (shaded) circle at an endpoint means the graph includes that value, used for ≤ or ≥. An open (unshaded) circle means the graph excludes that value, used for < or >.
The Two Rules every other rule builds on
What changes, and what doesn’t, when you add, subtract, multiply, or divide both sides.
Start with the true inequality 4 < 8, and test each operation on both sides.
Adding 2 to both sides: 6 < 10. Subtracting 2: 2 < 6. Multiplying by +2: 8 < 16. Dividing by +2: 2 < 4. All still true.
Adding, subtracting, multiplying, or dividing both sides by the same positive number never changes the direction of the inequality.
Multiplying 4 < 8 by -2: -8 > -16. Dividing by -2: -2 > -4. The sign reverses both times.
Multiplying or dividing both sides of an inequality by a negative number always reverses the inequality sign.
Yes. Adding or subtracting a variable works exactly like adding or subtracting a number. It never changes the direction of the inequality, regardless of the variable’s sign.
Not unless you know its sign. If x/y > 1, a common mistake is multiplying both sides by y to conclude x > y. But if x = 3 and y = 2, x/y > 1 holds and x > y is true. If x = -3 and y = -2, x/y > 1 still holds, but x is not greater than y. The only fact you can definitely conclude from x/y > 1 is that x and y have the same sign.
Question: If a, b, c are non-zero integers and a > bc, which of the following must be true?
- I. a/b > c
- II. a/c > b
- III. a/bc > 1
A. I only B. II only C. III only D. I, II, and III E. None of these
Solution: The tempting answer is D, dividing a > bc by b, by c, and by bc respectively. But b and c are integers of unknown sign. Dividing an inequality by an unknown-sign quantity is exactly what Property 2 warns against. Since the sign isn’t fixed, none of the three statements is guaranteed true in every case. The answer is E.
Our GRE Quant coaching drills exactly this instinct until it’s automatic under time pressure.
Advanced Properties you’ll actually get tested on
Fractions, squaring, square roots, reciprocals, adding inequalities, and max-min problems, each with its own trap.
All positive proper fractions fall in the range 0 < x < 1. For any value in this range, √x > x > x². For example, if x = 1/4, then √x = 1/2 and x² = 1/16, and clearly 1/2 > 1/4 > 1/16.
Example: If x = 0.888, y = √0.888, and z = (0.888)², which order is correct? Since 0.888 is a proper fraction, √0.888 > 0.888 > (0.888)², so y > x > z, which reverses to z < x < y.
You cannot square both sides of an inequality unless you know the sign of both sides.
Both sides negative: flip the sign when you square. If a < -4, the left side is negative, so squaring gives a² > 16.
Both sides positive: do not flip the sign. If a > 4, the left side is positive, so squaring gives a² > 16.
One side positive, one negative, or the sign unclear: you cannot square at all.
Square roots only exist for non-negative numbers, so this rule only applies when both sides are known to be zero or positive. If 0 ≤ x < y, then √x < √y. Taking the square root of both sides of a non-negative inequality preserves the direction. If either side could be negative, you cannot take the square root at all.
Taking the reciprocal of both sides can change the direction of the inequality, and the rule depends entirely on the signs of a and b when a < b:
Both positive: flip. If 2 < 3, then 1/2 > 1/3.
Both negative: flip. If -3 < -2, then 1/-3 > 1/-2.
Opposite signs: do not flip. If -3 < 2, then 1/-3 < 1/2.
Unknown signs: you cannot take reciprocals at all.
Example: If 1/3 ≥ 6/(x+1) ≥ 6, find the range of x. Taking the reciprocal and flipping (the entire inequality is positive): 1/3 ≥ (x+1)/6 ≥ 1/6. Multiplying throughout by 6: 2 ≥ x+1 ≥ 1. Subtracting 1: 1 ≥ x ≥ 0, so 0 ≤ x ≤ 1.
The only operation you can perform between two inequalities is addition, and only once both are pointing the same direction. If the signs differ, flip one first using Property 2, then add.
Example: If 4a + 2b < n and 4b + 2a > m, what must b – a be? Multiply the second inequality by -1 to flip it: -4b – 2a < -m. Now add the two like inequalities: (4a + 2b) + (-4b – 2a) < n + (-m), which gives 2a – 2b < n – m. Dividing by 2: a – b < (n-m)/2. Multiplying by -1 and flipping: b – a > (m-n)/2.
Optimization problems ask for the largest or smallest possible value of an expression built from two ranges, usually x + y, x – y, or xy. Some combination of the extreme values of each range will produce the max or min.
Method: place the two ranges one below the other with matching inequality signs, then combine both in a straight line and diagonally to get four candidate values. The greatest is the max; the smallest is the min.
Example: If -7 ≤ x ≤ 6 and -7 ≤ y ≤ 8, the four extreme values of xy are 49, 48, -56, and -42. So the maximum possible value of xy is 49 and the minimum is -56.
Solving Quadratic Inequalities step by step
The one method that works for quadratics, cubics, and beyond, no guesswork needed.
Take 3x² – 7x + 4 ≤ 0. Factor it: 3x(x – 1) – 4(x – 1) ≤ 0, which gives (3x – 4)(x – 1) ≤ 0. The critical points are x = 1 and x = 4/3.
Since the inequality needs the product to be ≤ 0, the middle region is the answer: for 1 ≤ x ≤ 4/3, one factor is negative or zero while the other is positive or zero, giving a non-positive product.
The range of x that satisfies 3x² – 7x + 4 ≤ 0 is 1 ≤ x ≤ 4/3.
Rearrange so the inequality reads ax² + bx + c > 0 or < 0.
Find the roots. These become your critical points.
They divide the line into regions, three for a quadratic.
Mark the rightmost region +, then alternate: -, +, and so on.
If the inequality is < 0, take the – region(s). If > 0, take the + region(s).
Yes. A cubic inequality has three critical points, dividing the number line into four regions instead of three. Mark the rightmost region + and alternate the same way. Whether the solution spans two of those four regions depends on whether the inequality reads < 0 or > 0.
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Using These Rules in Quantitative Comparison questions
A QC question is itself an inequality, so every rule above applies directly to it.
Take a basic comparison where Quantity B is clearly greater than Quantity A: A = 4, B = 6. Apply the same operations from Section 2 to both quantities and watch what holds.
| Operation | Quantity A | Quantity B | Comparison |
|---|---|---|---|
| Original | 4 | 6 | B greater |
| Add 2 | 6 | 8 | B still greater |
| Subtract 2 | 2 | 4 | B still greater |
| Multiply by +2 | 8 | 12 | B still greater |
| Divide by +2 | 2 | 3 | B still greater |
| Multiply by -2 | -8 | -12 | A now greater |
| Divide by -2 | -2 | -3 | A now greater |
Multiplying or dividing both quantities by a negative number never leaves the comparison consistent with the original. It always reverses it.
The Six Rules to lock in before test day
Everything in this guide reduces to these six checks.
Add or subtract any value to both quantities. The comparison holds.
Multiply or divide both quantities by a positive value. The comparison holds.
Square both quantities only when both are known to be positive (or both negative, then flip).
Never multiply or divide both quantities by a negative number without flipping the comparison.
Never multiply or divide both quantities by a variable whose sign you don’t know.
If the variable’s sign is always positive, like x² which is never negative, you can safely multiply or divide by it.
Frequently Asked Questions: GRE Inequalities
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Talk to a GRE ExpertEvery Mistake Traces Back to One Question
Every inequality mistake on the GRE traces back to one question: did this operation require knowing a sign I don’t actually know? Add and subtract freely. Multiply and divide by positive numbers freely. The moment a negative number, an unknown-sign variable, a square, a square root, or a reciprocal enters the picture, stop and check the sign first.
Inequalities are just one piece of GRE Quant. Our GRE Math coaching walks through every quant concept, from number properties to data interpretation, with live feedback.
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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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