GMAT inequalities and absolute value questions reward a specific set of habits — thinking about distance on a number line, always carrying two cases, and never multiplying by a variable of unknown sign. This guide walks through every technique tested on the Focus Edition Quant section, with worked examples, practice questions, and a trap-by-trap playbook.
GMAT algebra inequalities and absolute value problems are some of the most reliable point-scorers on the test — if you can recognize the pattern, the work is almost mechanical. If you cannot, each question becomes a 3-minute rabbit hole in a section that only gives you 45 minutes total.
On the GMAT Focus Edition, inequality and absolute value questions primarily live in the Quantitative Reasoning section as Problem Solving items. But the same logic powers Data Sufficiency questions in the Data Insights section — sufficiency prompts lean heavily on questions like "is x positive?" or "is |a − b| > c?" and reward the same habits.
Arithmetic and algebra together account for more than half of GMAT Focus Quant content, and inequalities sit inside the algebra bucket. In practice you should expect one to three pure inequality or absolute value questions per Quant attempt, plus a handful more where the concept is woven into a word problem or number-properties question.
| Attribute | Value |
|---|---|
| Number of questions | 21 |
| Time allotted | 45 minutes |
| Average time per question | About 2 minutes 9 seconds |
| Question types | Problem Solving only (no Data Sufficiency) |
| Calculator allowed | No |
| Topics tested | Arithmetic and algebra (including inequalities and absolute value) |
Most students learn absolute value as "make it positive." That definition works for simple problems and then collapses the moment an absolute value appears inside an inequality with an unknown on both sides. The better mental model is distance.
The absolute value of x is the distance from x to zero on the number line. That has two consequences the GMAT tests constantly: |x| is never negative, and |x| = 5 always has two solutions because there are two points exactly 5 units from zero — one on each side.
The more general form, |x − c|, is the distance from x to the fixed point c on the number line. So |x − 4| < 5 is not a cryptic inequality — it literally says "the distance from x to 4 is less than 5." Every x within 5 units of 4 satisfies it, which means −1 < x < 9.
Worked Example — Distance Interpretation
Setup: Describe every x that satisfies |x − 4| < 5 using the distance definition.
Inequalities follow most of the same rules as equations — until they don't. One specific move changes the direction of the inequality, and missing that move is the single most common source of wrong answers on GMAT algebra inequalities.
Adding or subtracting the same quantity to both sides never flips the inequality. Multiplying or dividing both sides by a positive number also leaves the direction unchanged. These operations behave exactly as they would on an equation.
Multiplying or dividing by a negative number reverses the inequality. If you have −2x > 10 and you divide by −2, you must flip the sign to get x < −5. Forgetting that single flip turns a right answer into a wrong one nine times out of ten.
The same rule creates a trap in Data Sufficiency. If a question gives you xy > 0 and asks you to solve for something, you cannot divide both sides of an inequality by y unless you know whether y is positive or negative. The safe move is to split into cases: one branch assuming y > 0 and another assuming y < 0.
| Operation on both sides | Does it flip the sign? | GMAT example |
|---|---|---|
| Add or subtract any number | No | x + 3 > 7 becomes x > 4 |
| Multiply or divide by a positive number | No | 3x > 12 becomes x > 4 |
| Multiply or divide by a negative number | Yes — reverse the sign | −2x > 10 becomes x < −5 |
| Multiply or divide by a variable of unknown sign | Not allowed — split into cases | If y could be positive or negative, do not divide by y |
| Square both sides | Only safe when both sides are non-negative | a > y does NOT imply a² > y² in general |
The Case 1–Case 2 technique is the universal tool for GMAT absolute value equations. The pattern works whether the expression inside is linear, quadratic, or something messier — as long as you isolate the absolute value first and commit to writing out both branches.
If an equation looks like 5|x + 4| + 3 = 13, you cannot split into cases until |x + 4| is alone on one side. Subtract 3 and divide by 5 to get |x + 4| = 2, and now you are in position to apply the two-case rule cleanly.
Once isolated, the expression inside can equal either the positive or the negative of the right side. Continuing the example above: Case 1 is x + 4 = 2 and Case 2 is x + 4 = −2. That yields x = −2 or x = −6. Both cases have to be written out even when one looks obviously wrong — that is exactly where the GMAT hides the correct branch.
Some problems — especially those with a polynomial inside the bars — produce candidate solutions that fail the original equation. Always plug your answers back in before circling a choice. The check takes five seconds and catches every extraneous root.
Worked Example — Two-Case Method
Setup: Solve |4x − 3| = 7 and find the product of the two solutions.
This is the section that separates a 550-level solver from a 700-level one. GMAT absolute value inequalities come in two shapes, and the direction of the inequality controls the entire geometry of the solution. Confusing them is the most common wrong-answer trap on this topic.
When the absolute value is less than a positive number, the expression inside is trapped between the negative and the positive of that number. |X| < a rewrites as −a < X < a. Geometrically the solution is a single open interval — everything within distance a of zero.
When the absolute value is greater than a positive number, the expression inside has to be far from zero in either direction. |X| > a rewrites as X < −a OR X > a. The solution is two disjoint rays, with a gap between −a and a.
Never apply the less-than or greater-than rule until the absolute value is alone on one side. If a problem looks like 2|x − 1| + 3 < 11, subtract 3 and divide by 2 first to get |x − 1| < 4. Only then do you split.
| Inequality form | Rewrite as | Solution shape | Logical connective |
|---|---|---|---|
| |X| < a (where a > 0) | −a < X < a | A single interval between −a and a | AND |
| |X| ≤ a (where a > 0) | −a ≤ X ≤ a | Closed interval including endpoints | AND |
| |X| > a (where a > 0) | X < −a OR X > a | Two separated rays outside ±a | OR |
| |X| ≥ a (where a > 0) | X ≤ −a OR X ≥ a | Two rays including the endpoints | OR |
| |X| < 0 | No solution | Empty set — distance can never be negative | — |
Worked Example — Greater-Than Pattern
Setup: Solve |2x + 1| > 5 and graph the solution on a number line.
Use this converter the next time you see an absolute value inequality and need to remember which pattern produces AND versus OR:
Pick an absolute value inequality pattern to see its rewritten form and the shape of its solution.
Drawing the problem on a number line is not a beginner's crutch — it is the fastest way to avoid range-tracking errors at the 700+ level. The number line method GMAT tutors recommend extends smoothly into the wavy-line method for factored inequalities.
The type of endpoint reflects whether the inequality is strict. Use a filled (closed) circle for ≤ or ≥, meaning the endpoint itself is part of the solution. Use an open circle for < or >, meaning the endpoint is excluded. Mixing the two is an easy way to trade a right answer for a wrong one.
When an inequality comes pre-factored, like (x − 1)(x + 2) > 0, plot the zeros and draw a wavy line from right to left. The line sits above the axis where the expression is positive and below where it's negative, alternating as it crosses each zero. The solution is the set of intervals where the wavy line is on the correct side of zero.
If you ever forget which intervals are positive and which are negative, plug a single test value from each region into the original expression. That reality-check takes seconds and is far safer than trying to remember sign rules under time pressure.
Worked Example — Wavy Line Method
Setup: Find all x such that (x − 1)(x + 2) > 0 using the wavy-line method.
The most useful way to think about GMAT inequality traps is that the testmaker designs problems around predictable errors. If you know the five bait patterns below, you know where the exam is most likely to try to take a point from you.
| Trap | Why it works on students | How to avoid it |
|---|---|---|
| The negative-number trap | Students assume a variable is positive and divide by it. | Never multiply or divide by a variable unless you know its sign. Split into cases instead. |
| The missing-case trap | Students solve only the positive case of an absolute value. | Always write out both cases explicitly, even when one looks obviously wrong. |
| The squaring trap | Students square inequalities as if they were equations. | Only square when both sides are known to be non-negative, and always verify in the original. |
| The 'less than' / 'greater than' mix-up | Students apply the wrong compound form to absolute value inequalities. | Memorize: less than → AND between, greater than → OR outside. |
| The isolation trap | Students apply the two-case rule before the absolute value is alone. | First isolate |...| on one side, then split into cases. |
Before you commit to an answer on any inequality or absolute value question, run through this short list. It is designed to catch each of the five traps in under ten seconds.
Work through these four GMAT inequalities practice questions before moving on. Each one targets a different trap from the sections above — answer them under a 2-minute-per-question budget to mimic Quant section pacing.
Test your own inequalities of the form |ax + b| < c or |ax + b| > c. Enter the values and the solver returns the solution interval and its logical connective.
Enter the coefficient, constant, comparison value, and direction to see the solution interval for |ax + b| ? c (with a > 0 and c ≥ 0).
Inequalities appear as Problem Solving questions on the 21-question, 45-minute Quant section of the GMAT Focus Edition. They also show up embedded in Data Sufficiency questions on the Data Insights section. Expect one to three inequality or absolute value questions per Quant attempt, often combined with number properties or algebra.
Forgetting the second case. Every absolute value expression splits into a positive and a negative branch. Students who solve only the positive case quietly lose half of the solution set. Always write out both cases, even when one looks obviously wrong — the GMAT frequently baits you into abandoning the correct branch.
Only when you multiply or divide both sides by a negative number. Adding or subtracting any value — positive, negative, or variable — never flips the sign. This is why the GMAT avoids giving you the sign of a variable: if you do not know whether a variable is positive or negative, you cannot safely multiply or divide by it.
Not freely. Squaring is safe only when both sides are known to be non-negative. Otherwise, squaring can introduce extraneous solutions or reverse the inequality. For example, a > y does not imply a² > y² when either side could be negative. Treat squaring as a last resort and always test your answer against the original inequality.
No. The Quantitative section of the GMAT Focus Edition is non-calculator. Inequalities are designed to reward logical reasoning and number sense — drawing a number line, testing values, and applying sign rules — rather than arithmetic. If your scratch work is getting arithmetic-heavy, you likely missed a structural shortcut.
Look at the direction after the absolute value is isolated. |X| < a means distance from zero is less than a, so X is trapped between −a and a (an AND statement). |X| > a means distance from zero is more than a, so X is outside the interval — either less than −a OR greater than a. The direction of the sign controls everything.