Lesson 2: GRE Quantitative Reasoning - Algebra Fundamentals & Linear Equations
Master essential algebraic concepts including solving linear equations, understanding inequalities, working with absolute values, and manipulating algebraic expressions for GRE success.
Lesson 2: GRE Quantitative Reasoning - Algebra Fundamentals & Linear Equations 📐
Introduction
Welcome to the second lesson in your GRE preparation journey! After mastering basic arithmetic and number properties, we're now diving into algebra fundamentals—one of the most heavily tested areas on the GRE Quantitative section. 🎯
Algebra questions account for approximately 30-35% of all GRE Quantitative problems, making this topic absolutely essential for achieving a competitive score. Unlike basic arithmetic, algebra requires you to think abstractly about relationships between unknown quantities and to manipulate equations strategically.
💡 Why Algebra Matters for the GRE: The GRE doesn't just test whether you can solve equations mechanically—it tests your ability to recognize patterns, choose efficient solution strategies, and work with variables in novel contexts. Many word problems, data interpretation questions, and geometry problems also require algebraic thinking.
In this lesson, we'll cover:
- Linear equations and solution techniques
- Systems of equations (substitution and elimination methods)
- Inequalities and their properties
- Absolute value expressions and equations
- Algebraic manipulation strategies
Core Concepts: Building Your Algebraic Foundation 🏗️
1. Linear Equations: The Foundation
A linear equation is an equation where the highest power of the variable is 1. The standard form is:
ax + b = c
Where:
- a is the coefficient of x
- b and c are constants
- x is the variable (unknown)
Goal: Isolate the variable on one side of the equation to find its value.
Solution Strategy: PEMDAS in Reverse 🔄
When solving linear equations, perform operations in the opposite order of PEMDAS:
┌─────────────────────────────────────┐
│ Solving Linear Equations: Steps │
├─────────────────────────────────────┤
│ 1. Simplify both sides │
│ (distribute, combine like terms)│
│ 2. Eliminate addition/subtraction │
│ 3. Eliminate multiplication/division│
│ 4. Check your solution │
└─────────────────────────────────────┘
Example: Solve for x: 3(2x - 4) + 5 = 23
Step 1: Distribute → 6x - 12 + 5 = 23
Step 2: Combine terms → 6x - 7 = 23
Step 3: Add 7 → 6x = 30
Step 4: Divide by 6 → x = 5
💡 GRE Tip: Always check if you can simplify BEFORE solving. The GRE loves to include unnecessary complexity to waste your time!
2. Systems of Equations: Two Variables, Two Equations 🔗
A system of equations involves multiple equations with multiple variables. For the GRE, you'll primarily encounter two equations with two unknowns.
Method 1: Substitution
Solve one equation for one variable, then substitute into the other equation.
┌──────────────────────────────────┐
│ Substitution Method Flow │
├──────────────────────────────────┤
│ Equation 1 → Solve for x or y │
│ ↓ │
│ Substitute into Equation 2 │
│ ↓ │
│ Solve for remaining variable │
│ ↓ │
│ Back-substitute to find first │
└──────────────────────────────────┘
Method 2: Elimination (Addition/Subtraction)
Multiply equations by constants to make coefficients of one variable equal (or opposite), then add or subtract equations.
When to use each method:
- Substitution: When one equation already has a variable isolated (like y = 3x + 2)
- Elimination: When coefficients align nicely or variables appear with the same coefficient
🧠 Mnemonic: Substitution for Simple isolation, Elimination for Equal coefficients
3. Inequalities: Relationships Beyond Equality ⚖️
An inequality shows that one expression is larger or smaller than another using these symbols:
┌────────┬──────────────────────────┐
│ Symbol │ Meaning │
├────────┼──────────────────────────┤
│ > │ Greater than │
│ < │ Less than │
│ ≥ │ Greater than or equal to│
│ ≤ │ Less than or equal to │
└────────┴──────────────────────────┘
Critical Rules for Inequalities:
Rule 1: You can add or subtract any number from both sides without changing the direction.
If x + 3 < 7, then x < 4
Rule 2: You can multiply or divide by a positive number without changing the direction.
If 2x < 10, then x < 5
⚠️ CRITICAL RULE 3: When you multiply or divide by a negative number, you MUST flip the inequality sign!
If -2x < 10, then x > -5 (sign flipped!)
💡 Why the flip? Think about it: 3 < 5, but -3 > -5. Negatives reverse the order!
Rule 4: When squaring both sides, be careful! Only valid when both sides are positive.
If x > 3, then x² > 9 ✓
But if x < -3, then x² > 9 (not x² < 9!) ⚠️
4. Absolute Value: Distance from Zero 📏
The absolute value |x| represents the distance of x from zero on the number line, always non-negative.
-5 -4 -3 -2 -1 0 1 2 3 4 5
←───────────────────|───────────────────→
|-------- 5 --------|-------- 5 --------|
|-3| = 3 |3| = 3
Key Properties:
┌─────────────────────────────────────────┐
│ Absolute Value Properties │
├─────────────────────────────────────────┤
│ |x| ≥ 0 (always non-negative) │
│ |x| = |-x| (symmetry) │
│ |xy| = |x|·|y| (multiplicative) │
│ |x/y| = |x|/|y| (y ≠ 0) │
└─────────────────────────────────────────┘
Solving Absolute Value Equations
The equation |x| = a (where a > 0) has two solutions: x = a or x = -a
Why? Because both a and -a are distance a from zero!
Example: |x - 3| = 5
This means: "The distance from x to 3 is 5"
Solutions:
- x - 3 = 5 → x = 8
- x - 3 = -5 → x = -2
Both are 5 units away from 3! ✓
Solving Absolute Value Inequalities
Type 1: |x| < a (distance from zero is less than a)
Solution: -a < x < a
Visually: ←───(─────)───→
-a 0 a
Type 2: |x| > a (distance from zero is greater than a)
Solution: x < -a OR x > a
Visually: ←───)───────(───→
-a 0 a
🧠 Memory Device: "Less than" gives you an interval between (connected). "Greater than" gives you two separate regions (disconnected).
5. Strategic Algebraic Manipulation 🎯
The GRE often asks you to manipulate expressions rather than solve for specific values. Key strategies:
Factoring Common Terms
3x + 6y = 3(x + 2y)
ax + ay + az = a(x + y + z)
Combining Fractions
x/a + y/a = (x + y)/a
x/a + x/b = x(1/a + 1/b) = x(b + a)/(ab)
Difference of Squares (HIGH-YIELD for GRE!) 🌟
a² - b² = (a + b)(a - b)
This pattern appears constantly on the GRE!
Example: 37² - 36² = (37 + 36)(37 - 36) = 73 × 1 = 73
(Much faster than calculating 1369 - 1296!)
Squaring Binomials
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
⚠️ Common Error: (a + b)² ≠ a² + b² (You're missing the 2ab term!)
Detailed Examples with Step-by-Step Solutions 📝
Example 1: Multi-Step Linear Equation with Fractions
Problem: Solve for x: (2x - 3)/4 + (x + 1)/3 = 5
Solution:
Step 1: Find common denominator (LCD = 12)
3(2x - 3)/12 + 4(x + 1)/12 = 5
Step 2: Multiply entire equation by 12 to clear denominators
3(2x - 3) + 4(x + 1) = 60
Step 3: Distribute
6x - 9 + 4x + 4 = 60
Step 4: Combine like terms
10x - 5 = 60
Step 5: Add 5 to both sides
10x = 65
Step 6: Divide by 10
x = 6.5
Check: (2(6.5) - 3)/4 + (6.5 + 1)/3 = 10/4 + 7.5/3 = 2.5 + 2.5 = 5 ✓
💡 GRE Strategy: Multiply by the LCD immediately to eliminate fractions—it simplifies the work tremendously!
Example 2: System of Equations (Elimination Method)
Problem: Solve for x and y:
3x + 2y = 16
5x - 2y = 8
Solution:
Step 1: Notice that the y-coefficients are opposites (2 and -2)
Perfect for elimination by addition!
Step 2: Add the equations
3x + 2y = 16
+ 5x - 2y = 8
─────────────────
8x + 0y = 24
Step 3: Solve for x
8x = 24
x = 3
Step 4: Back-substitute into first equation
3(3) + 2y = 16
9 + 2y = 16
2y = 7
y = 3.5
Answer: x = 3, y = 3.5
Check: 3(3) + 2(3.5) = 9 + 7 = 16 ✓ and 5(3) - 2(3.5) = 15 - 7 = 8 ✓
🔧 Try This: What if the coefficients weren't opposites? For the system:
2x + 3y = 13
4x + 5y = 23
You'd multiply the first equation by -2 to get -4x, then add to eliminate x!
Example 3: Absolute Value Inequality
Problem: Solve for x: |2x - 5| ≤ 7
Solution:
Step 1: Recognize the "less than" pattern
|expression| ≤ a means -a ≤ expression ≤ a
Step 2: Apply the pattern
-7 ≤ 2x - 5 ≤ 7
Step 3: Solve the compound inequality (add 5 to all parts)
-7 + 5 ≤ 2x - 5 + 5 ≤ 7 + 5
-2 ≤ 2x ≤ 12
Step 4: Divide all parts by 2
-1 ≤ x ≤ 6
Answer: x is in the interval [-1, 6]
Interpretation: All values of x between -1 and 6 (inclusive) make |2x - 5| at most 7.
Visual Check:
At x = -1: |2(-1) - 5| = |-7| = 7 ✓
At x = 6: |2(6) - 5| = |7| = 7 ✓
At x = 2: |2(2) - 5| = |-1| = 1 < 7 ✓
Example 4: Word Problem Requiring System Setup
Problem: At a school fundraiser, adult tickets cost $12 and student tickets cost $7. If 150 tickets were sold for a total of $1,440, how many adult tickets were sold?
Solution:
Step 1: Define variables
Let a = number of adult tickets
Let s = number of student tickets
Step 2: Write equations based on given information
Total tickets: a + s = 150
Total revenue: 12a + 7s = 1440
Step 3: Use substitution (solve first equation for s)
s = 150 - a
Step 4: Substitute into second equation
12a + 7(150 - a) = 1440
12a + 1050 - 7a = 1440
5a + 1050 = 1440
5a = 390
a = 78
Step 5: Find s (if needed)
s = 150 - 78 = 72
Answer: 78 adult tickets were sold
Reality Check: Does this make sense?
- 78 adult tickets × $12 = $936
- 72 student tickets × $7 = $504
- Total: $936 + $504 = $1,440 ✓
- Total tickets: 78 + 72 = 150 ✓
💡 GRE Word Problem Strategy: Always define your variables clearly, write equations that directly translate the English into math, and verify your answer makes sense in context!
Common Mistakes to Avoid ⚠️
Mistake 1: Forgetting to Flip Inequality Signs
❌ Wrong: -3x < 12 → x < -4
✅ Correct: -3x < 12 → x > -4 (flipped because dividing by negative!)
Prevention: Circle or highlight negative coefficients when dividing/multiplying inequalities.
Mistake 2: Distributing Incorrectly
❌ Wrong: 3(x + 2) = 3x + 2
✅ Correct: 3(x + 2) = 3x + 6
Prevention: Multiply the coefficient by EVERY term inside the parentheses.
Mistake 3: Losing the Second Solution in Absolute Value Equations
❌ Incomplete: |x - 2| = 5 → x = 7 only
✅ Correct: |x - 2| = 5 → x = 7 or x = -3
Prevention: Always write both cases: expression = positive value AND expression = negative value.
Mistake 4: Squaring Both Sides Carelessly
❌ Wrong: If x + 3 = 5, then x² + 9 = 25
✅ Correct: If x + 3 = 5, then (x + 3)² = 25, which gives x² + 6x + 9 = 25
Prevention: Remember (a + b)² ≠ a² + b². Use the formula: (a + b)² = a² + 2ab + b²
Mistake 5: Confusing "And" vs "Or" in Compound Inequalities
For |x| < 3: ✅ Correct: -3 < x < 3 ("and" - x must satisfy both conditions)
For |x| > 3: ✅ Correct: x < -3 OR x > 3 ("or" - x satisfies either condition)
Prevention: Visualize on a number line. "Less than" creates a connected interval (and). "Greater than" creates disconnected regions (or).
Mistake 6: Canceling Terms Instead of Factors
❌ Wrong: (x + 3)/(x + 5) = 3/5 (attempting to "cancel" x)
✅ Correct: The expression cannot be simplified further. You can only cancel common factors, not terms!
Example of valid canceling:
✅ Correct: 3(x + 2)/(x + 2) = 3 (because (x + 2) is a common factor)
Key Takeaways 🎯
╔══════════════════════════════════════════════════════════╗
║ ALGEBRA FUNDAMENTALS SUMMARY ║
╠══════════════════════════════════════════════════════════╣
║ ║
║ 📌 LINEAR EQUATIONS ║
║ • Isolate variable using inverse operations ║
║ • Work in reverse order of PEMDAS ║
║ • Simplify before solving ║
║ ║
║ 📌 SYSTEMS OF EQUATIONS ║
║ • Substitution: When variable already isolated ║
║ • Elimination: When coefficients align nicely ║
║ • Two equations needed for two unknowns ║
║ ║
║ 📌 INEQUALITIES ║
║ • Add/subtract freely without changing direction ║
║ • Flip sign when multiplying/dividing by negative ║
║ • Solution often a range, not single value ║
║ ║
║ 📌 ABSOLUTE VALUE ║
║ • Represents distance from zero (always ≥ 0) ║
║ • |x| = a has TWO solutions: x = a or x = -a ║
║ • |x| < a gives -a < x < a (interval) ║
║ • |x| > a gives x < -a or x > a (two regions) ║
║ ║
║ 📌 ALGEBRAIC MANIPULATION ║
║ • a² - b² = (a+b)(a-b) ⭐HIGH-YIELD⭐ ║
║ • (a+b)² = a² + 2ab + b² ║
║ • Factor common terms to simplify ║
║ ║
╚══════════════════════════════════════════════════════════╝
Strategic Reminders for Test Day 🧠
Look for shortcuts: Can you plug in answer choices instead of solving algebraically? (Especially for complex word problems)
Estimate when possible: For Quantitative Comparison questions, you often don't need exact values
Check for special cases: Does x = 0 or x = 1 reveal something? What about negative values?
Don't over-solve: If the question asks "Which is greater?" you don't need to find exact values—just establish the relationship
Watch for restrictions: Can a variable be zero? Negative? These restrictions often reveal the answer
🤔 Did You Know?
The word "algebra" comes from the Arabic "al-jabr," meaning "reunion of broken parts." It was coined by the Persian mathematician Muhammad ibn Musa al-Khwarizmi in his 9th-century book. The term literally described the process of moving terms from one side of an equation to the other—reuniting the "broken" equation to solve it! 🌍
Interestingly, the term "algorithm" also comes from al-Khwarizmi's name, Latinized as "Algorithmi." So every time you solve an equation step-by-step, you're using techniques that have been refined for over 1,200 years! 📚
Quick Reference Card 📋
Photocopy this for quick review!
┌─────────────────────────────────────────────────────────┐
│ GRE ALGEBRA QUICK REFERENCE │
├─────────────────────────────────────────────────────────┤
│ │
│ SOLVING LINEAR EQUATIONS: │
│ ax + b = c → x = (c - b)/a │
│ │
│ SYSTEMS (2×2): │
│ Substitution: Solve for one, plug into other │
│ Elimination: Add/subtract to cancel variable │
│ │
│ INEQUALITY RULES: │
│ ✓ Add/subtract any number → sign stays same │
│ ✓ Multiply/divide by positive → sign stays same │
│ ⚠ Multiply/divide by NEGATIVE → FLIP SIGN! │
│ │
│ ABSOLUTE VALUE: │
│ |x| = a → x = a or x = -a │
│ |x| < a → -a < x < a │
│ |x| > a → x < -a or x > a │
│ │
│ KEY FORMULAS: │
│ Difference of squares: a² - b² = (a+b)(a-b) │
│ Perfect square: (a±b)² = a² ± 2ab + b² │
│ │
│ GRE STRATEGIES: │
│ • Plug in answer choices when stuck │
│ • Clear fractions by multiplying by LCD │
│ • Check for a² - b² pattern to avoid calculation │
│ • Always verify answer makes sense │
│ │
└─────────────────────────────────────────────────────────┘
📚 Further Study
To deepen your understanding of GRE algebra and practice with official-style questions:
Khan Academy GRE Prep - https://www.khanacademy.org/test-prep/gre - Free comprehensive algebra review with practice problems organized by difficulty
ETS Official GRE Quantitative Reasoning Practice Questions - https://www.ets.org/gre/test-takers/general-test/prepare.html - Official practice materials from the test maker, including free POWERPREP software
Manhattan Prep GRE Algebra Strategy Guide - https://www.manhattanprep.com/gre/store/online-resources/ - Detailed strategies specifically designed for GRE algebra question types
Ready to test your algebra skills? Let's move on to the practice questions! 💪
Remember: The GRE rewards strategic thinking, not just computational ability. Focus on understanding WHY each step works, and you'll be able to adapt to any algebraic challenge the test throws at you! 🎓