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GMAT Question Patterns: Absolute Value
We reviewed questions from the GMAT Prep Tests and Official Guides to identify the most common absolute value patterns that appear on the exam.
This guide is focused on the different types of absolute value problems you may encounter on test day. The goal of this project is to uncover as many recurring question types as possible, for three key reasons:
Most GMAT absolute value problems now appear in the Problem Solving section. In the GMAT Focus Edition, Data Sufficiency no longer includes pure algebraic questions, as these have been removed from the syllabus. For details, refer to GMAT Syllabus for Focus Edition. That said, absolute value concepts may still show up in math-related Data Insights sections, such as Two-Part Analysis. For instance, check this question.
Both number plugging and algebraic methods remain effective strategies. On average, you can expect about 2–3 absolute value questions per test, based on GMAT Prep CATs.
Note:
We reviewed questions from the GMAT Prep Tests and Official Guides to identify the most common absolute value patterns that appear on the exam.
This guide is focused on the different types of absolute value problems you may encounter on test day. The goal of this project is to uncover as many recurring question types as possible, for three key reasons:
- Get a higher score – Master each pattern so you don’t miss out on easy points.
- Save time – No need to do hundreds of questions; just a few from each pattern will do.
- Peace of mind – Be confident that you've covered all known question types.
Most GMAT absolute value problems now appear in the Problem Solving section. In the GMAT Focus Edition, Data Sufficiency no longer includes pure algebraic questions, as these have been removed from the syllabus. For details, refer to GMAT Syllabus for Focus Edition. That said, absolute value concepts may still show up in math-related Data Insights sections, such as Two-Part Analysis. For instance, check this question.
Both number plugging and algebraic methods remain effective strategies. On average, you can expect about 2–3 absolute value questions per test, based on GMAT Prep CATs.
Note:
- The examples in this guide come from the Official Guides, GMAT Prep (Classic), and GMAT Focus.
- There are 11 absolute value questions in total across OG 2024–2025 and GMAT Focus.
Pattern #1: Absolute Value and Algebra - Focuses on solving equations that involve absolute value expressions and require algebraic manipulation.
If \(x\neq{0}\), then \(\frac{\sqrt{x^2}}{x}=\)
A. -1
B. 0
C. 1
D. x
E. \(\frac{|x|}{x}\)
Discussed HERE
If x<0, then \(\sqrt{-x|x|}\) is:
A. -x
B. -1
C. 1
D. x
E. \(\sqrt{x}\)
Discussed HERE
The equation |2x - 3| = x - 5 has how many solutions for x?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Discussed HERE. From GMAT Prep Focus
What is the sum of the different values of \(x\) such that \(|x+2| = 2|x-2|\)
A. \(4\)
B. \(\frac{16}{3}\)
C. \(6\)
D. \(\frac{20}{3}\)
E. \(8\)
Discussed HERE. From GMAT Prep Focus
What is the product of all the solutions of (x + 2)^2 = |x + 2|?
A. -6
B. -2
C. 2
D. 6
E. 12
Discussed HERE. From GMAT Prep Focus
Which of the following expressions equals 0 when x < 0 and equals x^2 when x ≥ 0?
A. \(x|x|\)
B. \(x + |x|\)
C. \(2x^2 - x|x|\)
D. \(\frac{1}{4}(x + |x|)^2\)
E. \((\frac{x - |x|}{2})^2\)
Discussed HERE. From GMAT Prep Focus
If \(x\neq{0}\), then \(\frac{\sqrt{x^2}}{x}=\)
A. -1
B. 0
C. 1
D. x
E. \(\frac{|x|}{x}\)
Discussed HERE
If x<0, then \(\sqrt{-x|x|}\) is:
A. -x
B. -1
C. 1
D. x
E. \(\sqrt{x}\)
Discussed HERE
The equation |2x - 3| = x - 5 has how many solutions for x?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Discussed HERE. From GMAT Prep Focus
What is the sum of the different values of \(x\) such that \(|x+2| = 2|x-2|\)
A. \(4\)
B. \(\frac{16}{3}\)
C. \(6\)
D. \(\frac{20}{3}\)
E. \(8\)
Discussed HERE. From GMAT Prep Focus
What is the product of all the solutions of (x + 2)^2 = |x + 2|?
A. -6
B. -2
C. 2
D. 6
E. 12
Discussed HERE. From GMAT Prep Focus
Which of the following expressions equals 0 when x < 0 and equals x^2 when x ≥ 0?
A. \(x|x|\)
B. \(x + |x|\)
C. \(2x^2 - x|x|\)
D. \(\frac{1}{4}(x + |x|)^2\)
E. \((\frac{x - |x|}{2})^2\)
Discussed HERE. From GMAT Prep Focus
Pattern #2: Must or Could be true questions with Absolute Value - Tests your ability to analyze all possible values of an expression within absolute value constraints to determine certainty or possibility.
If \(|y-\frac{1}{2}| < \frac{11}{2}\), which of the following could be a value of y?
(A) -11
(B) -11/2
(C) 11/2
(D) 11
(E) 22
Discussed HERE
If |x| > 3, which of the following must be true?
I. x > 3
II. X^2 > 9
III. |x-1|>2
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Discussed HERE
If \(w\neq{0}\), which of the following must be true?
I. \(\frac{|w|}{w} = 1\)
II. \(\frac{w^{2}}{|w|} = w\)
III. \(\sqrt{w^2} = |w|\)
(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
Discussed HERE. From GMAT Prep Focus
If f(x) = g(x - 1), where g(x) = |x| + 1, which of the following must be true?
(A) f(x) < 0
(B) f(x) = 0
(C) f(x) > 0
(D) f(x) = 2
(E) f(x) = g(x)
Discussed HERE. From GMAT Prep Focus
If \(|y-\frac{1}{2}| < \frac{11}{2}\), which of the following could be a value of y?
(A) -11
(B) -11/2
(C) 11/2
(D) 11
(E) 22
Discussed HERE
If |x| > 3, which of the following must be true?
I. x > 3
II. X^2 > 9
III. |x-1|>2
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Discussed HERE
If \(w\neq{0}\), which of the following must be true?
I. \(\frac{|w|}{w} = 1\)
II. \(\frac{w^{2}}{|w|} = w\)
III. \(\sqrt{w^2} = |w|\)
(A) II only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III
Discussed HERE. From GMAT Prep Focus
If f(x) = g(x - 1), where g(x) = |x| + 1, which of the following must be true?
(A) f(x) < 0
(B) f(x) = 0
(C) f(x) > 0
(D) f(x) = 2
(E) f(x) = g(x)
Discussed HERE. From GMAT Prep Focus
Pattern #3: Absolute Value and Distance - Uses the definition of absolute value as distance on the number line to interpret or solve problems.
Of the five coordinates associated with points A, B, C, D, and E on the number line above, which has the greatest absolute value?
(A) A
(B) B
(C) C
(D) D
(E) E
Discussed HERE
Which of the following inequalities is an algebraic expression for the shaded part of the number line above?
(A) |x| <= 3
(B) |x| <= 5
(C) |x - 2| <= 3
(D) |x - 1| <= 4
(E) |x +1| <= 4
Discussed HERE
Of the five coordinates associated with points A, B, C, D, and E on the number line above, which has the greatest absolute value?
(A) A
(B) B
(C) C
(D) D
(E) E
Discussed HERE
Which of the following inequalities is an algebraic expression for the shaded part of the number line above?
(A) |x| <= 3
(B) |x| <= 5
(C) |x - 2| <= 3
(D) |x - 1| <= 4
(E) |x +1| <= 4
Discussed HERE
Pattern #4: Min/Max and Absolute Value - Involves finding the minimum or maximum possible value of expressions that contain absolute values.
If y is an integer, then the least possible value of |23 - 5y| is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Discussed HERE
What is the greatest integer that is less than: \(-|\sqrt{3}-3|\) ?
A. -3
B. -2
C. -1
D. 0
E. 1
Discussed HERE. From GMAT Prep Focus
If y is an integer, then the least possible value of |23 - 5y| is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Discussed HERE
What is the greatest integer that is less than: \(-|\sqrt{3}-3|\) ?
A. -3
B. -2
C. -1
D. 0
E. 1
Discussed HERE. From GMAT Prep Focus
Pattern #5: Absolute Value and Inequalities - Covers how to interpret and solve inequalities that include absolute value expressions.
On the number line, which of the following specifies the set of all numbers x such that |x - 3| + |x - 4| < 2?
A. 1 < x <6
B. 1.5 < x < 5.5
C. 2 < x < 5
D. 2.5 < x < 4.5
E. 3 < x < 4
Discussed HERE. From GMAT Prep Focus
If a and b are, respectively, the least and greatest nonnegative real number solutions to \(|\frac{x - 6}{5}|≤ 5.5\), what is the value of b - a?
A. 11
B. 13.4
C. 27.5
D. 33.5
E. 55
Discussed HERE. From GMAT Prep Focus
Which of the following inequalities is equivalent to \( -3 \leq x\leq{1}\)?
A. \(|x|\leq{1}\)
B. \(|x|\leq{3}\)
C. \(1\leq{|x|}\leq{3}\)
D. \(|x+1|\leq{2}\)
E. \(|x-1|\leq{2}\)
Discussed HERE. From GMAT Prep Focus
On the number line, which of the following specifies the set of all numbers x such that |x - 3| + |x - 4| < 2?
A. 1 < x <6
B. 1.5 < x < 5.5
C. 2 < x < 5
D. 2.5 < x < 4.5
E. 3 < x < 4
Discussed HERE. From GMAT Prep Focus
If a and b are, respectively, the least and greatest nonnegative real number solutions to \(|\frac{x - 6}{5}|≤ 5.5\), what is the value of b - a?
A. 11
B. 13.4
C. 27.5
D. 33.5
E. 55
Discussed HERE. From GMAT Prep Focus
Which of the following inequalities is equivalent to \( -3 \leq x\leq{1}\)?
A. \(|x|\leq{1}\)
B. \(|x|\leq{3}\)
C. \(1\leq{|x|}\leq{3}\)
D. \(|x+1|\leq{2}\)
E. \(|x-1|\leq{2}\)
Discussed HERE. From GMAT Prep Focus
Pattern #6: Absolute Value and Word Problems - Applies absolute value concepts to real-world contexts involving distance or difference.
On a laboratory thermometer with equally spaced tick marks for the degrees, the distance between the tick marks for the temperatures P° and 32° is twice the distance between the tick marks for the temperatures P° and -10°. Which of the following equations does P satisfy?
A. 2|P - 32|= |P - 10|
B. P + 32 = |P - 10|
C. 2|P + 32= |P + 10|
D. |P - 32| = 2|P + 10|
E. |P - 32| = 2|P - 10|
Discussed HERE. From GMAT Prep Focus
On a laboratory thermometer with equally spaced tick marks for the degrees, the distance between the tick marks for the temperatures P° and 32° is twice the distance between the tick marks for the temperatures P° and -10°. Which of the following equations does P satisfy?
A. 2|P - 32|= |P - 10|
B. P + 32 = |P - 10|
C. 2|P + 32= |P + 10|
D. |P - 32| = 2|P + 10|
E. |P - 32| = 2|P - 10|
Discussed HERE. From GMAT Prep Focus
Theory on absolute value is HERE.
Hard Absolute Value PS questions are HERE.
Medium Absolute Value PS questions are HERE.
Easy Absolute Value PS questions are HERE.
Hard Absolute Value PS questions are HERE.
Medium Absolute Value PS questions are HERE.
Easy Absolute Value PS questions are HERE.
15 Apr 2025, 03:46
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22 Mar 2025, 19:53
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2025 Focus Update for the Absolute Value Patterns
We have gone through the latest set of questions and identified the latest patterns in the Absolute value questions. These are generally quite similar to their 10+year old cousins with exception that the DS section no longer contains absolute value questions, so some of the patterns are no longer used or have been changed.
📌 Major Patterns
Pattern 1: "Must Be True"
Questions testing logical implications of absolute value identities like \(|a| = |b|\).
Example 1.1 (Difficulty: Sub 505 Level)
If \(\frac{|z|}{w} = 1\), which of the following must be true?
A. \(z = -w\)
B. \(z = w\)
C. \(z^2 = w^2\)
D. \(z^2 = w^3\)
E. \(z^3 = w^3\)
Question Discussion Link
Example 1.2 (Difficulty: Sub 505 Level)
If \(k^2 = m^2\), which of the following must be true?
A. \(k = m\)
B. \(k = -m\)
C. \(k = |m|\)
D. \(k = -|m|\)
E. \(|k| = |m|\)
Question Discussion Link
Example 1.3 (Difficulty: Sub 505 Level)
If \(|x| = |y|\) and \(xy = 0\), which of the following must be true?
A. \(xy^2 > 0\)
B. \(x^2y > 0\)
C. \(x + y = 0\)
D. \(\frac{x}{(y + 1)} = 2\)
E. \(\frac{1}{x + 1/y = 1/2}\)
Question Discussion Link
Pattern 2: Evaluate/Calculate
Focus on sign changes, expression transformation, and conditional evaluation of absolute value.
Example 2.1 (Difficulty: 505–555 Level)
If \(x \ne 0\), then \(\frac{\sqrt{x^2}}{x}\) = ?
A. \(-1\)
B. \(0\)
C. \(1\)
D. \(x\)
E. \(\frac{|x|}{x}\)
Question Discussion Link
Example 2.2 (Difficulty: Sub 505 Level)
If \(x < 0\), which of the following represents a positive number?
A. \(\frac{x}{|x|}\)
B. \(\frac{|x|}{x}\)
C. \(x|x|\)
D. \(-x|x|\)
E. \(x|-x|\)
Question Discussion Link
Example 2.3 (Difficulty: 555–605 Level)
Which expression equals \(0\) when \(x < 0\) and \(x^2\) when \(x ≥ 0\)?
A. \(x|x|\)
B. \(x + |x|\)
C. \(2x^2 - x|x|\)
D. \(\frac{1}{4(x + |x|)^2}\)
E. \((\frac{(x - |x|)}{2})^2\)
Question Discussion Link
Pattern 3: Equations
Equations that require breaking into cases based on the definition of absolute value.
Example 3.1 (Difficulty: 705–805 Level)
What is the product of solutions for \(x^2 + 4x + 7 = |x + 2| + 3\)?
A. \(-6\)
B. \(-2\)
C. \(2\)
D. \(6\)
E. \(12\)
Question Discussion Link
Example 3.2 (Difficulty: 655–705 Level)
How many solutions does \(|2x - 3| = x - 5\) have?
A. \(0\)
B. \(1\)
C. \(2\)
D. \(3\)
E. \(4\)
Question Discussion Link
Example 3.3 (Difficulty: 605–655 Level)
What is the sum of different values of \(x\) such that \(|x + 2| = 2|x - 2|\)?
A. \(4\)
B. \(\frac{16}{3}\)
C. \(6\)
D. \(\frac{20}{3}\)
E. \(8\)
Question Discussion Link
📌 Minor Patterns
Pattern 4: Greatest/Least Integer
Estimating absolute value expressions and comparing them to nearest integers.
Example 4.1 (Difficulty: 555–605 Level)
What is the greatest integer less than \(-|\sqrt{3} - 3|\)?
A. \(-3\)
B. \(-2\)
C. \(-1\)
D. \(0\)
E. \(1\)
Question Discussion Link
🗂️ Bonus: Additional Absolute Value Questions (not purely Absolute Value)
Example A1 (Difficulty: 505–555 Level) — Sets
Number line points with absolute values \(r, s, t\). Average of coordinates?
Question Discussion Link
Example A2 (Difficulty: 555–605 Level) — Probability
X takes integer values 1–7. What’s the probability \(|X - 4| > 3/2\)?
Question Discussion Link
Example A3 (Difficulty: 555–605 Level) — Word Problem
An analyst defines “volatility” using absolute daily price differences...
Question Discussion Link
We have gone through the latest set of questions and identified the latest patterns in the Absolute value questions. These are generally quite similar to their 10+year old cousins with exception that the DS section no longer contains absolute value questions, so some of the patterns are no longer used or have been changed.
📌 Major Patterns
Pattern 1: "Must Be True"
Questions testing logical implications of absolute value identities like \(|a| = |b|\).
Example 1.1 (Difficulty: Sub 505 Level)
If \(\frac{|z|}{w} = 1\), which of the following must be true?
A. \(z = -w\)
B. \(z = w\)
C. \(z^2 = w^2\)
D. \(z^2 = w^3\)
E. \(z^3 = w^3\)
Question Discussion Link
Example 1.2 (Difficulty: Sub 505 Level)
If \(k^2 = m^2\), which of the following must be true?
A. \(k = m\)
B. \(k = -m\)
C. \(k = |m|\)
D. \(k = -|m|\)
E. \(|k| = |m|\)
Question Discussion Link
Example 1.3 (Difficulty: Sub 505 Level)
If \(|x| = |y|\) and \(xy = 0\), which of the following must be true?
A. \(xy^2 > 0\)
B. \(x^2y > 0\)
C. \(x + y = 0\)
D. \(\frac{x}{(y + 1)} = 2\)
E. \(\frac{1}{x + 1/y = 1/2}\)
Question Discussion Link
Pattern 2: Evaluate/Calculate
Focus on sign changes, expression transformation, and conditional evaluation of absolute value.
Example 2.1 (Difficulty: 505–555 Level)
If \(x \ne 0\), then \(\frac{\sqrt{x^2}}{x}\) = ?
A. \(-1\)
B. \(0\)
C. \(1\)
D. \(x\)
E. \(\frac{|x|}{x}\)
Question Discussion Link
Example 2.2 (Difficulty: Sub 505 Level)
If \(x < 0\), which of the following represents a positive number?
A. \(\frac{x}{|x|}\)
B. \(\frac{|x|}{x}\)
C. \(x|x|\)
D. \(-x|x|\)
E. \(x|-x|\)
Question Discussion Link
Example 2.3 (Difficulty: 555–605 Level)
Which expression equals \(0\) when \(x < 0\) and \(x^2\) when \(x ≥ 0\)?
A. \(x|x|\)
B. \(x + |x|\)
C. \(2x^2 - x|x|\)
D. \(\frac{1}{4(x + |x|)^2}\)
E. \((\frac{(x - |x|)}{2})^2\)
Question Discussion Link
Pattern 3: Equations
Equations that require breaking into cases based on the definition of absolute value.
Example 3.1 (Difficulty: 705–805 Level)
What is the product of solutions for \(x^2 + 4x + 7 = |x + 2| + 3\)?
A. \(-6\)
B. \(-2\)
C. \(2\)
D. \(6\)
E. \(12\)
Question Discussion Link
Example 3.2 (Difficulty: 655–705 Level)
How many solutions does \(|2x - 3| = x - 5\) have?
A. \(0\)
B. \(1\)
C. \(2\)
D. \(3\)
E. \(4\)
Question Discussion Link
Example 3.3 (Difficulty: 605–655 Level)
What is the sum of different values of \(x\) such that \(|x + 2| = 2|x - 2|\)?
A. \(4\)
B. \(\frac{16}{3}\)
C. \(6\)
D. \(\frac{20}{3}\)
E. \(8\)
Question Discussion Link
📌 Minor Patterns
Pattern 4: Greatest/Least Integer
Estimating absolute value expressions and comparing them to nearest integers.
Example 4.1 (Difficulty: 555–605 Level)
What is the greatest integer less than \(-|\sqrt{3} - 3|\)?
A. \(-3\)
B. \(-2\)
C. \(-1\)
D. \(0\)
E. \(1\)
Question Discussion Link
🗂️ Bonus: Additional Absolute Value Questions (not purely Absolute Value)
Example A1 (Difficulty: 505–555 Level) — Sets
Number line points with absolute values \(r, s, t\). Average of coordinates?
Question Discussion Link
Example A2 (Difficulty: 555–605 Level) — Probability
X takes integer values 1–7. What’s the probability \(|X - 4| > 3/2\)?
Question Discussion Link
Example A3 (Difficulty: 555–605 Level) — Word Problem
An analyst defines “volatility” using absolute daily price differences...
Question Discussion Link
