How to Solve Inequalities on the GRE/GMAT: The Complete High-Score Guide

How to Solve Inequalities on the GRE/GMAT: The Complete High-Score Guide

Hook (10 seconds on camera): "Inequalities 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 mistakes and solving them in under 90 seconds."

What This Question Type Is Actually Testing

The GRE/GMAT isn’t testing your ability to solve inequalities—it’s testing your ability to: 1. Avoid sign errors (especially when multiplying/dividing by negatives). 2. Interpret compound inequalities (e.g., a < x < b vs. x < a or x > b). 3. Eliminate wrong answers efficiently (not just solve for x).

Anatomy of the Question

Every inequality question has: - Stem: A statement (e.g., "If 3x – 5 > 7, which of the following must be true?"). - Conditions: Constraints (e.g., "x is an integer" or "x > 0"). - Answer Choices: 3-5 options, often including: - Correct ranges (e.g., x > 4). - Distractors (e.g., x < 4, x ≥ 4). - "Cannot be determined" traps.

Example Question: If –2 ≤ 3x + 1 < 5, which of the following represents all possible values of x? (A) –1 ≤ x < 4/3 (B) –1 < x ≤ 4/3 (C) –3 ≤ x < 4/3 (D) –3 < x ≤ 4/3 (E) x < –1 or x ≥ 4/3

The Decision Framework (Step-by-Step)

Step 1: Isolate the variable. - Break compound inequalities into two parts (e.g., –2 ≤ 3x + 1 and 3x + 1 < 5). - Solve each part separately.

Step 2: Reverse the inequality when multiplying/dividing by a negative. - If you multiply/divide by a negative, flip the sign (e.g., –3x > 6 → x < –2).

Step 3: Combine the results. - For A < x < B, the solution is the overlap of both parts. - For x < A or x > B, the solution is two separate ranges.

Step 4: Match the solution to the answer choices. - Eliminate choices that don’t match the exact range. - Watch for strict (<, >) vs. inclusive (≤, ≥) inequalities.

Step 5: Check for traps. - Did you flip the sign when dividing by a negative? - Did you misinterpret "or" vs. "and"?

Worked Examples

Example 1: Straightforward

If 2x + 3 > 7, which of the following must be true? (A) x > 2 (B) x ≥ 2 (C) x < 2 (D) x ≤ 2 (E) x > 5

Solution: 1. Isolate x: 2x + 3 > 7 → 2x > 4 → x > 2. 2. Match to choices: Only (A) matches x > 2. 3. Eliminate others: (B) includes x = 2 (wrong), (C)-(E) are opposite or too extreme.

Answer: A

Example 2: Common Trap (Sign Error)

If –4x + 5 ≤ 13, which of the following must be true? (A) x ≥ –2 (B) x ≤ –2 (C) x ≥ 2 (D) x ≤ 2 (E) x > 2

Solution: 1. Isolate x: –4x + 5 ≤ 13 → –4x ≤ 8. 2. Flip the sign when dividing by –4: x ≥ –2. 3. Match to choices: Only (A) matches x ≥ –2. 4. Trap: (B) is the unflipped version (wrong).

Answer: A

Example 3: Hard Variant (Compound Inequality)

If –3 < 2x – 1 ≤ 5, which of the following represents all possible values of x? (A) –1 < x ≤ 3 (B) –1 ≤ x < 3 (C) –2 < x ≤ 3 (D) –2 ≤ x < 3 (E) x < –1 or x ≥ 3

Solution: 1. Break into two parts:
- –3 < 2x – 1 → –2 < 2x → –1 < x.
- 2x – 1 ≤ 5 → 2x ≤ 6 → x ≤ 3. 2. Combine: –1 < x ≤ 3. 3. Match to choices: Only (A) matches. 4. Trap: (B) includes x = –1 (wrong), (C)-(D) shift the range.

Answer: A

Wrong Answer Patterns

1. Unflipped Sign → Looks right if you forget to flip (e.g., x ≤ –2 instead of x ≥ –2).
2. Inclusive vs. Strict → x ≤ 3 vs. x < 3 (off by one value).
3. "Or" vs. "And" → x < –1 or x > 3 vs. –1 < x < 3.
4. Extreme Values → x > 5 when the solution is x > 2 (too restrictive).

Common Mistakes

1. Forgetting to flip the sign → Happens when dividing by a negative. Fix: Circle the sign before solving.
2. Misinterpreting "or" vs. "and" → Fix: Write the solution as two separate ranges if needed.
3. Ignoring inclusive vs. strict → Fix: Check if the inequality has ≤ or <.
4. Solving only one part of a compound inequality → Fix: Break into two parts first.
5. Overcomplicating → Fix: Solve step-by-step, don’t skip.

Time Strategy

• Target time: 60-90 seconds.
• Skip if: You’re stuck after 2 minutes (flag and return).
• Minimum work: Solve the inequality first, then match to choices.

Backsolving and Shortcuts

1. Plug in numbers → Test x = 0 or x = 1 to eliminate wrong choices.
2. Eliminate extremes → If x > 2, eliminate choices with x ≤ 2.
3. Check boundary values → For x ≤ 3, test x = 3 to confirm.

1-Minute Recap

"Here’s the exact process to solve inequalities in under 90 seconds: 1. Isolate x—break compound inequalities into two parts. 2. Flip the sign if you multiply/divide by a negative. 3. Combine the results—watch for ‘and’ vs. ‘or.’ 4. Match to the choices—eliminate anything that doesn’t fit. 5. Check for traps—did you flip the sign? Is it strict or inclusive?

Most mistakes happen in Step 2 or Step 4. Slow down, write it out, and you’ll get it right every time."

Final Note: Inequalities are easy points if you follow the framework. Practice 10-15 questions, and you’ll never miss one again.