How to Solve Absolute Value Questions on the GRE/GMAT

How to Solve Absolute Value Questions on the GRE/GMAT

Introduction

"Absolute value questions appear 4-6 times on every GRE and 3-5 times on the GMAT—master them, and you’ll gain 20+ points by avoiding careless traps and solving them in under 60 seconds."

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GRE/GMAT doesn’t test your ability to compute absolute values—it tests: 1. Logical reasoning under constraints – Can you handle cases where a variable could be positive or negative? 2. Precision in algebra – Do you avoid sign errors when squaring or taking square roots? 3. Efficiency in elimination – Can you spot and discard wrong answers without solving every case?

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem – Introduces an equation or inequality with absolute value (e.g., |x + 3| = 5 or |2y - 1| ≤ 7).
2. Conditions (if any) – Additional constraints (e.g., "x is an integer" or "y > 0").
3. Answer Choices – Usually 5 options (GRE) or 4 (GMAT), with 1-2 traps per question.
4. What to Ignore – Overcomplicating the problem (e.g., assuming variables must be positive).

Representative Example

GRE-Style: If |3x - 4| = 8, which of the following could be the value of x? A) -4/3 B) 0 C) 4/3 D) 2 E) 4

GMAT-Style: If |2k + 5| > 7, which of the following must be true? A) k > 1 B) k < -6 C) k > 1 or k < -6 D) -6 < k < 1 E) k > -6

THE DECISION FRAMEWORK (Step-by-Step)

Step 1: Identify the Absolute Value Structure

• Is it an equation (|A| = B) or an inequality (|A| > B or |A| < B)?
• If it’s an equation, split into two cases (A = B and A = -B).
• If it’s an inequality, rewrite without absolute value using the rules:
• |A| > B → A > B or A < -B
• |A| < B → -B < A < B

Step 2: Solve Each Case Separately

• For equations: Solve both cases and check for extraneous solutions (e.g., if B is negative, no solution exists).
• For inequalities: Solve both cases and combine the results (use "or" for >/≥, "and" for </≤).

Step 3: Apply Additional Constraints

• If the problem gives conditions (e.g., "x is positive"), filter your solutions to match.
• If no constraints, consider all valid solutions.

Step 4: Match to Answer Choices

• For "which could be true" questions, test each option against your solutions.
• For "must be true" questions, eliminate options that don’t cover all cases.

Step 5: Eliminate Traps

• Watch for distractors that only satisfy one case (e.g., only A = B but not A = -B).
• Watch for sign errors (e.g., forgetting that |A| = B requires B ≥ 0).

Worked Examples

Example 1: Straightforward (GRE-Style)

Question: If |x - 2| = 5, what is the value of x? A) -7 B) -3 C) 3 D) 7 E) 3 or -3

Step-by-Step Solution: 1. Split into two cases:
- x - 2 = 5 → x = 7
- x - 2 = -5 → x = -3 2. Check for extraneous solutions: Both are valid (5 ≥ 0). 3. Match to answer choices: Only D (7) and B (-3) are options, but the question asks for "the value of x" (singular). Since both are possible, the correct answer is E (3 or -3).

Elimination Logic: - A (-7) and C (3) don’t match either solution. - D (7) is correct but incomplete (misses -3). - E is the only option that includes both solutions.

Example 2: Common Trap (GMAT-Style)

Question: If |3y + 1| ≤ 10, which of the following must be true? A) y ≤ 3 B) y ≥ -11/3 C) -11/3 ≤ y ≤ 3 D) y ≤ -11/3 or y ≥ 3 E) y ≥ 3

Step-by-Step Solution: 1. Rewrite the inequality:
-10 ≤ 3y + 1 ≤ 10 2. Solve for y:
- Subtract 1: -11 ≤ 3y ≤ 9
- Divide by 3: -11/3 ≤ y ≤ 3 3. Match to answer choices:
- A (y ≤ 3) is partially correct but misses the lower bound.
- B (y ≥ -11/3) is partially correct but misses the upper bound.
- C is fully correct.
- D and E are wrong (D is the "or" version of the inequality, which is incorrect here).

Elimination Logic: - A and B are half-truths (common traps). - D is the opposite of what we need (it’s for |A| ≥ B). - E is too restrictive.

Example 3: Hard Variant (Top Scoring Band)

Question: If |x² - 4| = 5, which of the following could be the value of x? A) -3 B) -1 C) 0 D) 1 E) 3

Step-by-Step Solution: 1. Split into two cases:
- x² - 4 = 5 → x² = 9 → x = 3 or x = -3
- x² - 4 = -5 → x² = -1 → No real solutions (discard this case). 2. Check answer choices:
- A (-3) and E (3) are valid.
- B (-1), C (0), and D (1) are invalid. 3. Since the question asks for "could be," either A or E is correct. But only A (-3) is an option here.

Elimination Logic: - The trap is ignoring the second case (which has no solution) and picking a wrong answer. - Another trap is assuming x² = -1 has solutions (it doesn’t in real numbers).

WRONG ANSWER PATTERNS

Common Mistakes

TIME STRATEGY

• Target time: 45–60 seconds per question.
• When to skip: If you’re stuck after 90 seconds, flag and move on.
• Minimum work needed:
• Split into cases (10 sec).
• Solve each case (20 sec).
• Eliminate wrong answers (15 sec).

BACKSOLVING AND SHORTCUTS

1. Plug in answer choices (for "could be" questions):
2. Test each option in the original equation.
3. Example: For |x - 2| = 5, test A (-3) → |-3 - 2| = 5 → valid.
4. Eliminate impossible cases early:
5. If |A| = B and B < 0, no solution exists (eliminate all options).
6. Use number lines for inequalities:
7. For |x - 1| < 3, draw -2 < x < 4 and visually eliminate wrong answers.

1-Minute Recap

"Here’s the exact process to solve any absolute value question in under 60 seconds:

1. Is it an equation or inequality?
2. Equation (|A| = B) → Split into A = B and A = -B.

3. Inequality (|A| > B) → Rewrite as A > B or A < -B.

4. Solve each case separately.

5. For equations, check if B is non-negative (if not, no solution).

6. For inequalities, combine the results (use "or" for >/≥).

7. Apply constraints (e.g., "x is positive") and eliminate wrong answers.

8. If the question says "could be," test the options.

9. If it says "must be," ensure the answer covers all cases.

10. Watch for traps:

11. Half-truth answers (only one case).
12. Sign errors (forgetting B ≥ 0).
13. Inequality direction mistakes.

That’s it. No extra steps, no overcomplicating. Split, solve, eliminate, and move on."

Final Notes

• Practice with a timer. Absolute value questions should take <1 minute.
• Memorize the rules for inequalities (> → or, < → and).
• Always split cases—this is the #1 way to avoid traps.

By following this framework, you’ll consistently solve absolute value questions correctly and quickly, boosting your score by 20+ points.