**GRE Algebra: Inequalities & Absolute Value – Complete Study Guide**

GRE Algebra: Inequalities & Absolute Value – Complete Study Guide

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What This Is

Inequalities and absolute value are core GRE algebra topics, appearing in Quantitative Comparison (QC), Problem Solving (PS), and Data Interpretation (DI). The GRE tests your ability to: - Solve compound inequalities (e.g., |2x – 3| ≤ 5).
- Interpret absolute value as distance on a number line.
- Handle "cases" when absolute value expressions are split (e.g., |x + 1| = 3 → x + 1 = 3 or x + 1 = –3).
- Avoid traps like flipping inequality signs or ignoring extraneous solutions.

Real GRE-Style Example:
If |3 – 2x| > 7, which of the following could be the value of x? (A) –2 (B) 1 (C) 3 (D) 5 (E) 6 (Answer: A and D. Solution below.)

Mastering this topic boosts your score by 3–5 points because: ✅ QC questions often hinge on inequality/absolute value logic.
✅ PS questions test multi-step reasoning (e.g., |x – 1| + |x + 2| = 5).
✅ DI questions require interpreting ranges (e.g., –3 ≤ x ≤ 5).

Key Concepts & Techniques

1. Absolute Value as Distance

What it is: |A – B| = distance between A and B on the number line.
When to use it:
- When the problem mentions "distance" (e.g., "x is 3 units from 5" → |x – 5| = 3).
- To rewrite absolute value inequalities (e.g., |x – 2| < 4 → –4 < x – 2 < 4).

2. Splitting Absolute Value Equations (Cases)

What it is: |A| = B → A = B or A = –B (only if B ≥ 0).
When to use it:
- When the equation has one absolute value (e.g., |2x + 1| = 5).
- Never split inequalities (use the distance method instead).

3. Splitting Absolute Value Inequalities (Cases)

What it is:
- |A| < B → –B < A < B (if B > 0).
- |A| > B → A < –B or A > B (if B > 0).
When to use it:
- When the inequality has one absolute value (e.g., |x – 3| ≥ 2).
- Never split if B is negative (e.g., |x| < –2 has no solution).

4. Compound Inequalities (AND/OR)

What it is:
- AND (∩): a < x < b → intersection of x > a and x < b.
- OR (∪): x < a or x > b → union of two ranges.
When to use it:
- After splitting absolute value inequalities (e.g., |x + 1| > 3 → x < –4 or x > 2).
- For QC questions comparing ranges (e.g., "Is x > 5 or x < –1?").

5. Number Line Visualization

What it is: Plot critical points (e.g., x = –2, 3) and test intervals.
When to use it:
- For multi-absolute-value problems (e.g., |x – 1| + |x + 2| = 5).
- To eliminate wrong answers in QC/PS questions.

6. Inequality Sign Rules

What it is:
- Multiply/divide by a negative → flip the inequality sign.
- Never multiply/divide by a variable unless you know its sign (e.g., x > 5 → x is positive).
When to use it:
- When solving inequalities like –2x > 6 → x < –3.
- Avoid traps where ETS assumes x is positive (e.g., x² > 4 → x > 2 or x < –2).

7. Extraneous Solutions

What it is: Solutions that don’t satisfy the original equation (e.g., squaring both sides of √x = –2 gives x = 4, but √4 ≠ –2).
When to use it:
- After solving absolute value equations (always plug back in).
- For radical/quadratic inequalities (e.g., √(x + 3) > x).

Step-by-Step Strategy

Follow these steps for EVERY inequality/absolute value problem:

Step 1: Identify the Type

• Absolute value equation? → Split into two cases (e.g., |A| = B → A = B or A = –B).
• Absolute value inequality? → Rewrite using distance (e.g., |A| < B → –B < A < B).
• Compound inequality? → Solve each part separately, then combine with AND/OR.

Step 2: Isolate the Absolute Value

• Move all terms except the absolute value to the other side.
  Example: 3|2x – 1| + 4 ≤ 10 → 3|2x – 1| ≤ 6 → |2x – 1| ≤ 2.

Step 3: Solve the Inequality/Equation

• Equation: Split into cases (e.g., |2x – 1| = 2 → 2x – 1 = 2 or 2x – 1 = –2).
• Inequality: Rewrite as a compound inequality (e.g., |2x – 1| ≤ 2 → –2 ≤ 2x – 1 ≤ 2).

Step 4: Solve for x

• Simplify each case/inequality.
  Example (from Step 3):
• 2x – 1 = 2 → x = 1.5
• 2x – 1 = –2 → x = –0.5
• –2 ≤ 2x – 1 ≤ 2 → –1 ≤ 2x ≤ 3 → –0.5 ≤ x ≤ 1.5

Step 5: Check for Extraneous Solutions

• Plug solutions back into the original equation (especially for absolute value).
  Example: |x – 3| = –2 → No solution (absolute value is never negative).

Step 6: Graph on a Number Line (If Needed)

• For multi-absolute-value problems, plot critical points and test intervals.
  Example: |x – 1| + |x + 2| = 5
• Critical points: x = –2, 1
• Test intervals: x < –2, –2 ≤ x ≤ 1, x > 1

Fully Worked GRE-Style Example

Problem:
If |3 – 2x| > 7, which of the following could be the value of x? (A) –2 (B) 1 (C) 3 (D) 5 (E) 6

Solution (Using the Strategy):

Step 1: Absolute value inequality → Rewrite using distance.
|3 – 2x| > 7 → 3 – 2x < –7 or 3 – 2x > 7

Step 2: Solve each inequality separately.
1. 3 – 2x < –7 → –2x < –10 → x > 5 (flip sign when dividing by –2) 2. 3 – 2x > 7 → –2x > 4 → x < –2 (flip sign)

Step 3: Combine solutions.
x < –2 or x > 5

Step 4: Check answer choices.
- (A) –2 → No (x must be less than –2, not equal).
- (B) 1 → No (–2 < 1 < 5).
- (C) 3 → No (–2 < 3 < 5).
- (D) 5 → No (x must be greater than 5, not equal).
- (E) 6 → Yes (6 > 5).

Correct Answers: A and D (if the question allowed multiple answers; on the GRE, it would specify "which could be" and expect you to select all valid options).

Common Mistakes

1. Forgetting to Flip the Inequality Sign

Mistake: Solving –2x > 6 as x > –3.
Why it happens: Students forget to flip the sign when dividing by a negative.
Correct approach: –2x > 6 → x < –3.

2. Splitting Absolute Value Inequalities Incorrectly

Mistake: Solving |x + 1| > 3 as –3 < x + 1 < 3.
Why it happens: Confusing |A| > B with |A| < B.
Correct approach: |x + 1| > 3 → x + 1 < –3 or x + 1 > 3.

3. Ignoring Extraneous Solutions

Mistake: Solving √(x + 3) = x – 3 and getting x = 1 or x = 6, then picking both.
Why it happens: Not plugging solutions back into the original equation.
Correct approach: x = 1 → √4 = –2 (false). x = 6 → √9 = 3 (true). Only x = 6 is valid.

4. Misapplying AND/OR Logic

Mistake: Solving |x – 2| < 3 as x < –1 or x > 5.
Why it happens: Treating |A| < B like |A| > B.
Correct approach: |x – 2| < 3 → –3 < x – 2 < 3 → –1 < x < 5.

5. Assuming Variables Are Positive

Mistake: Solving x² > 4 as x > 2.
Why it happens: Forgetting that x could be negative.
Correct approach: x² > 4 → x > 2 or x < –2.

GRE Traps & Timing

1. The "Flip the Sign" Trap

Trap: ETS gives an inequality like –3x + 5 > 2 and expects you to forget to flip the sign.
How to avoid: Circle the negative sign before dividing/multiplying.

2. The "Absolute Value = Negative" Trap

Trap: Questions like |x – 1| = –2 appear to have solutions, but absolute value is never negative.
How to avoid: Always check if the right side is negative → no solution.

3. The "Multi-Absolute-Value" Trap

Trap: Problems like |x – 1| + |x + 2| = 3 require testing intervals, but students try to split into cases.
How to avoid: Plot critical points (x = –2, 1) and test each region.

4. Timing Budget

• QC questions: 60–90 seconds (focus on number line visualization).
• PS questions: 90–120 seconds (for multi-step problems).
• DI questions: 30–45 seconds (interpret ranges quickly).

Quick Practice

Question 1:

If |2x + 5| ≤ 9, what is the range of possible values for x? Answer: –7 ≤ x ≤ 2 Solution Path: Rewrite as –9 ≤ 2x + 5 ≤ 9 → –14 ≤ 2x ≤ 4 → –7 ≤ x ≤ 2.

Question 2:

Which of the following is a solution to |x – 3| > |2x + 1|? (A) –4 (B) –2 (C) 0 (D) 2 (E) 4 Answer: A (–4)
Solution Path: Test each option. For x = –4: |–7| > |–7| → 7 > 7 (false). Wait! This is a trick—solve algebraically: Square both sides: (x – 3)² > (2x + 1)² → x² – 6x + 9 > 4x² + 4x + 1 → –3x² – 10x + 8 > 0 → 3x² + 10x – 8 < 0 → (3x – 2)(x + 4) < 0 → –4 < x < 2/3. Only x = –4 is outside this range (but the inequality is strict, so no solution here).
Correction: The correct answer is none of the above (this is a hard question—ETS expects you to test options).

Last-Minute Cram Sheet

1. |A| = B → A = B or A = –B (only if B ≥ 0).
2. |A| < B → –B < A < B (if B > 0; no solution if B ≤ 0).
3. |A| > B → A < –B or A > B (if B > 0; all real numbers if B < 0).
4. Flip the inequality sign when multiplying/dividing by a negative.
5. Never multiply/divide by a variable unless you know its sign.
6. Extraneous solutions → Always plug back into the original equation.
7. Multi-absolute-value problems → Plot critical points and test intervals.
8. QC questions → Compare ranges, not exact values.
9. Trap: |A| = –B → no solution.
10. Trap: |A| < –B → no solution; |A| > –B → all real numbers.

Final Tip: On test day, draw a number line for every inequality/absolute value problem. It takes 5 seconds and eliminates 90% of careless errors.