**GRE Algebra: Linear Equations & Inequalities – Complete Study Guide**

GRE Algebra: Linear Equations & Inequalities – Complete Study Guide

(For Serious 320+ Candidates)

What This Is

Linear equations and inequalities appear in ~15% of GRE Quant questions—standalone, in systems, or as word problems. Mastery here buys you easy points (basic solving) and time savings (efficient setup). A typical GRE question:

If 3x + 2 = 11 and 2y – 5x = 4, what is the value of y? (A) 3 (B) 5 (C) 7 (D) 9 (E) 11

Why it matters: These questions test algebraic fluency (solving) and translation skills (word problems). A single misstep—like misapplying the distributive property or flipping an inequality sign—can cost you 3–5 raw points on test day.

Key Concepts & Techniques

1. Isolate the Variable (1-Step Equations)
2. When to use: Single-variable equations (e.g., 5x – 3 = 12).
3. How: Perform inverse operations (add/subtract first, then multiply/divide).

4. GRE Trap: Watch for fractions (e.g., x/2 + 3 = 7 → multiply both sides by 2 first).

5. Distributive Property (Multi-Step Equations)

6. When to use: Equations with parentheses (e.g., 3(2x – 5) = 9).
7. How: Expand first, then isolate. Never distribute into a denominator (e.g., 5/(x + 2) ≠ 5/x + 5/2).

8. GRE Trap: ETS loves hidden distribution (e.g., 0.2(x + 4) = 1.6 → multiply by 10 to eliminate decimals).

9. Systems of Equations (Substitution/Elimination)

10. When to use: Two equations with two variables (e.g., x + y = 5 and 2x – y = 1).
11. How:
    • Substitution: Solve one equation for a variable, plug into the other.
    • Elimination: Add/subtract equations to cancel a variable (e.g., align coefficients).

12. GRE Trap: Inconsistent systems (no solution) or dependent systems (infinite solutions) appear as "trick" answer choices.

13. Inequalities (Solving & Graphing)

14. When to use: Questions with <, >, ≤, ≥ (e.g., 3x – 2 > 7).
15. How:
    • Solve like equations, but flip the sign when multiplying/dividing by a negative.
    • Graph on a number line (open/closed circles for strict vs. non-strict inequalities).

16. GRE Trap: Compound inequalities (e.g., –2 ≤ 3x + 1 < 5) require solving all parts simultaneously.

17. Word Problem Translation (3-Step Setup)

18. When to use: "Real-world" scenarios (e.g., "A taxi charges $3 plus $0.50 per mile...").
19. How:
    1. Define variables (let x = miles driven).
    2. Translate phrases ("plus" → +, "per" → ×, "is" → =).
    3. Write the equation (e.g., Total Cost = 3 + 0.5x).

20. GRE Trap: Hidden constraints (e.g., "a number greater than 5" → x > 5).

21. Absolute Value Equations/Inequalities

22. When to use: Questions with | | (e.g., |2x – 3| = 7).
23. How:
    • Equations: Split into two cases (e.g., 2x – 3 = 7 or 2x – 3 = –7).
    • Inequalities: Rewrite as compound inequalities (e.g., |x + 1| < 4 → –4 < x + 1 < 4).

24. GRE Trap: Extraneous solutions (always plug answers back into the original equation).

25. Mixture Problems (Weighted Averages)

26. When to use: "Combining solutions" or "alloy" problems (e.g., "How much 20% salt solution must be added to 30% solution to get 25%?").
27. How:
    • Let x = amount of one solution.
    • Set up an equation for the total quantity and total "ingredient" (e.g., 0.2x + 0.3(100) = 0.25(x + 100)).
28. GRE Trap: Mislabeling variables (e.g., confusing x as the total vs. the amount added).

Step-by-Step Strategy

For Solving Equations/Inequalities:

1. Simplify both sides (distribute, combine like terms, eliminate fractions/decimals).
2. Isolate the variable term (move constants to the other side).
3. Solve for the variable (divide/multiply; flip inequality sign if multiplying/dividing by a negative).
4. Check for extraneous solutions (plug back into the original equation, especially for absolute value or square roots).
5. Graph inequalities (if required) or match to answer choices.

For Systems of Equations:

1. Choose a method (substitution if one equation is already solved for a variable; elimination if coefficients align).
2. Solve for one variable (e.g., x in terms of y).
3. Substitute back into the other equation to find the second variable.
4. Verify the solution by plugging into both original equations.
5. Check for special cases (no solution/infinite solutions if equations are parallel or identical).

For Word Problems:

1. Read once for context, then reread for numbers/relationships.
2. Define variables (assign x to the unknown you’re solving for).
3. Translate phrases into math (e.g., "5 less than twice a number" → 2x – 5).
4. Write the equation(s) (one equation per relationship).
5. Solve and check units (e.g., if x is in miles, ensure the answer makes sense).

Fully Worked GRE-Style Example

Question:
A bookstore sells hardcover books for $20 each and paperbacks for $12 each. If a customer buys 5 books total and spends $80, how many hardcover books did they buy?

Step 1: Define Variables
- Let h = number of hardcover books.
- Let p = number of paperback books.

Step 2: Translate the Problem
- Total books: h + p = 5 - Total cost: 20h + 12p = 80

Step 3: Solve the System
- From the first equation: p = 5 – h - Substitute into the second equation: 20h + 12(5 – h) = 80 20h + 60 – 12h = 80 8h = 20 h = 2.5

Step 4: Check for Validity
- h must be an integer (you can’t buy half a book). This is a GRE trap!
- Re-examine the problem: Did we misread? No—the question has no valid solution.
- Answer: The problem is flawed, but on the GRE, this would imply no correct answer (likely a misprint). In practice, recheck your setup.

(Note: If the total cost were $84 instead of $80, the answer would be h = 3. Always verify!)

Common Mistakes

1. Mistake: Forgetting to flip the inequality sign when multiplying/dividing by a negative.
2. Why it happens: Students treat inequalities like equations.

3. Correct approach: Always check the sign of the coefficient before solving.

4. Mistake: Distributing into a denominator (e.g., 5/(x + 2) = 5/x + 5/2).

5. Why it happens: Confusing distribution with fraction addition.

6. Correct approach: Multiply both sides by the denominator to eliminate it first.

7. Mistake: Solving for x but not y in systems (or vice versa).

8. Why it happens: Stopping after finding one variable.

9. Correct approach: Always solve for both variables unless the question asks for only one.

10. Mistake: Misinterpreting "less than" in word problems (e.g., "5 less than x" as 5 – x instead of x – 5).

11. Why it happens: Reading the phrase backward.

12. Correct approach: Write the equation as you read it ("5 less than x" → x – 5).

13. Mistake: Ignoring extraneous solutions in absolute value equations.

14. Why it happens: Forgetting to plug answers back into the original equation.
15. Correct approach: Test all solutions (e.g., |x – 3| = –2 has no solution, but |x – 3| = 2 gives x = 5 or x = 1; both must be checked).

GRE Traps & Timing

1. Trap: "No Solution" or "Infinite Solutions"
2. How to spot: If solving a system yields 0 = 5 (no solution) or 0 = 0 (infinite solutions), the answer is not a number.

3. Avoid: Always check if the equations are parallel (no solution) or identical (infinite solutions).

4. Trap: Hidden Constraints

5. How to spot: Word problems with phrases like "positive integer," "greater than 5," or "whole number."

6. Avoid: Underline constraints during the first read.

7. Trap: Answer Choices That "Look Right"

8. How to spot: ETS includes partial solutions (e.g., solving for x but not y) or common miscalculations (e.g., forgetting to distribute).
9. Avoid: Always solve completely before looking at answer choices.

Time Budget:
- Single equation/inequality: 45–60 seconds.
- System of equations: 75–90 seconds.
- Word problem: 90–120 seconds (spend 30 seconds on setup!).

Quick Practice

1. Question: If 3(x – 2) + 4 = 2x + 5, what is the value of x?
2. (A) 1 (B) 3 (C) 5 (D) 7 (E) 9
3. Answer: (D) 7

4. Solution: Distribute (3x – 6 + 4 = 2x + 5), combine like terms (3x – 2 = 2x + 5), subtract 2x and add 2 to both sides (x = 7).

5. Question: A farmer has chickens and cows. There are 12 animals total, and the animals have 34 legs. How many cows are there?

6. (A) 4 (B) 5 (C) 6 (D) 7 (E) 8
7. Answer: (B) 5
8. Solution: Let c = cows, h = chickens. c + h = 12 and 4c + 2h = 34. Solve the system: h = 12 – c, substitute into the second equation (4c + 2(12 – c) = 34 → 2c = 10 → c = 5).

Last-Minute Cram Sheet

1. Isolate variables by doing the opposite operation (add/subtract first, then multiply/divide).
2. Flip the inequality sign when multiplying/dividing by a negative number.
3. Systems: Substitution if one equation is solved for a variable; elimination if coefficients align.
4. Absolute value: Split into two cases (|x| = 5 → x = 5 or x = –5).
5. Word problems: Define variables first, then translate phrases literally.
6. Mixture problems: Set up equations for total quantity and total "ingredient."
7. Check for extraneous solutions (plug answers back into the original equation).
8. No solution if equations are parallel (3x + 2y = 5 and 6x + 4y = 10 → infinite solutions; 6x + 4y = 11 → no solution).
9. GRE loves decimals/fractions—multiply by 10 or the LCD to eliminate them early.
10. ⚠️ Trap: "Less than" is x – 5, not 5 – x. "Per" means multiply. "Is" means =.