Algebra

What This Module Covers

Linear equations and systems, inequalities, quadratic equations, functions, and coordinate geometry. This module also introduces two of the most powerful GRE test-taking strategies: Picking Numbers and Backsolving.

Why It Matters on the GRE

Algebra shows up in roughly one-third of all GRE Quant questions — it is the single largest topic on the test. If you can solve systems quickly and factor quadratics on sight, you will save minutes you can spend on harder problems. The GRE also loves to hide algebra inside word problems and Quantitative Comparison questions, so recognizing the algebraic structure underneath the story is half the battle.

Core Concepts

1. Linear Equations

A linear equation has no exponents greater than 1. To solve for one variable, isolate it using inverse operations — undo addition with subtraction, undo multiplication with division.

Must Know: For a system of two equations, use elimination when the coefficients of one variable are already equal (or easily made equal). Use substitution when one equation already has a variable isolated. The GRE almost always rewards elimination because it's faster.

Example — System via Elimination:

1. Given: 3x + 2y = 16 and x + 2y = 8
2. Subtract the second equation from the first: (3x − x) + (2y − 2y) = 16 − 8
3. Simplify: 2x = 8, so x = 4
4. Substitute back: 4 + 2y = 8 → 2y = 4 → y = 2
5. Verify: 3(4) + 2(2) = 12 + 4 = 16 ✓

2. Inequalities

Inequalities behave almost exactly like equations — with one critical exception.

Must Know: Flip the inequality sign whenever you multiply or divide both sides by a negative number. If you forget this, you will pick the exact wrong answer. Also: you can add two inequalities that point in the same direction (< with <), but you can NEVER add inequalities that point in opposite directions.

Example:

1. Solve: −3x + 5 > 14
2. Subtract 5: −3x > 9
3. Divide by −3 — flip the sign: x < −3
4. Trap answer: If you forgot to flip, you'd write x > −3. The GRE will include this as a wrong answer choice.

3. Quadratic Equations

Standard form: ax² + bx + c = 0. Your first move should always be to try factoring — it's faster than the quadratic formula.

Must Know: Memorize these three identities cold:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• (a + b)(a − b) = a² − b² ← difference of squares

Also memorize the Zero Product Property: if (x − 3)(x + 2) = 0, then x = 3 or x = −2.

Example — Factoring:

1. Solve: x² − 5x + 6 = 0
2. Find two numbers that multiply to 6 and add to −5: those are −2 and −3
3. Factor: (x − 2)(x − 3) = 0
4. Apply Zero Product Property: x = 2 or x = 3
5. Key point: A quadratic can have 0, 1, or 2 real solutions. Don't assume there's only one answer — always check both roots.

4. Functions

Function notation f(x) just means "plug the value in for x everywhere you see x." For composite functions f(g(x)), work from the inside out.

Must Know: In f(g(x)), evaluate g(x) first, then plug that result into f. Getting the order backwards is the #1 function error on the GRE.

Example — Composite Function:

1. Given: f(x) = x² + 1 and g(x) = x − 3. Find f(g(5)).
2. Evaluate the inner function first: g(5) = 5 − 3 = 2
3. Now plug into f: f(2) = 2² + 1 = 5
4. Domain reminder: If a function has a denominator, exclude any x that makes it zero. If it has a square root, exclude any x that makes the radicand negative.

5. Coordinate Geometry

The slope and line equations are foundational — you'll see them in standalone questions and embedded in word problems about rates and motion.

Must Know:

• Slope: m = (y₂ − y₁) / (x₂ − x₁)
• Slope-intercept form: y = mx + b (m = slope, b = y-intercept)
• Parallel lines share the same slope.
• Perpendicular lines have slopes that are negative reciprocals: m₁ × m₂ = −1
• Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
• Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)

Example — Perpendicular Slope:

1. Line L has slope 2/3. What is the slope of a line perpendicular to L?
2. Negative reciprocal of 2/3: flip it to 3/2, then negate → −3/2
3. Check: (2/3) × (−3/2) = −1 ✓

6. Picking Numbers (GRE Strategy)

When answer choices contain variables instead of numbers — called VICs (Variables in Choices) — assign simple values to the variables, compute the target, and eliminate choices that don't match.

Must Know: Avoid 0 and 1 when picking numbers — they have special arithmetic properties (multiplying by 0 zeroes everything out; multiplying by 1 changes nothing). Use values like 2, 3, 5, or 10. Always test every answer choice, not just the one you think is right.

Example — VIC:

1. "If a store marks up a price by x%, then discounts the marked-up price by x%, which expression represents the final price in terms of original price P?" (answer choices involve P and x)
2. Pick x = 10 and P = 100. Marked-up price = 110. After 10% discount: 110 × 0.90 = 99.
3. Now evaluate each answer choice with x=10, P=100 and keep only the choice that gives 99.
4. This eliminates any "trap" expression like P (which gives 100 — the untouched original).

7. Backsolving (GRE Strategy)

When a Problem Solving question has five specific numerical answer choices, you can plug the answers directly back into the problem rather than solving algebraically.

Must Know: Start with choice (C) — the middle value. If C is too big, try B or A. If C is too small, try D or E. In the worst case you'll test three values and find the answer. This strategy is especially useful when setting up the algebra looks messy.

Example — Backsolving:

1. "A number increased by 15 equals three times the number minus 9. What is the number?" Choices: (A) 9 (B) 10 (C) 12 (D) 14 (E) 15
2. Start with (C): 12 + 15 = 27. Three times 12 minus 9 = 36 − 9 = 27. Equal! Done.
3. No algebra required — answer is (C) 12.

Common Traps

• Forgetting to flip the inequality sign: Dividing or multiplying by a negative flips the sign. Miss this once and you'll pick the answer that is exactly wrong.
• Squaring a binomial incorrectly: (a + b)² = a² + 2ab + b², NOT a² + b². The middle term 2ab is always there and the GRE always offers the incorrect version as a trap choice.
• Assuming one quadratic root: Quadratics have two roots. The GRE frequently asks for "the positive value" or "both values" — if you stop at one root, you'll miss the question.
• Reversing composite function order: In f(g(x)), g goes first. If you evaluate f first, you'll get a distractor that the test put there specifically for students who make this error.
• Slope sign errors: Going from point A to point B vs. point B to point A gives opposite signs. Always be consistent about which point is (x₁, y₁).

GRE Strategy

• On Quantitative Comparison questions with variables, always ask: "Could this variable be negative? Zero? A fraction?" The answer might be different for each case, leading to "D — Cannot be determined."
• When you see a quadratic, check for difference-of-squares or perfect-square patterns before reaching for the quadratic formula — factoring is 10× faster.
• If a system of equations has three or more variables but the question asks for a single combination like x + y, try to manipulate the equations to produce exactly that combination rather than solving for each variable individually.
• Picking Numbers and Backsolving are not "shortcuts for people who don't know algebra" — they are legitimate GRE strategies that often take less time than algebraic solutions. Know when to use them.

Worked Example

Question:

Quantity A: The positive value of x if x² = 16

Quantity B: The negative value of y if y² = 9

(A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given.

Solution:

1. Find Quantity A. Solve x² = 16. The two solutions are x = 4 and x = −4. The question specifies the positive value, so Quantity A = 4.

2. Find Quantity B. Solve y² = 9. The two solutions are y = 3 and y = −3. The question specifies the negative value, so Quantity B = −3.

3. Compare. 4 vs. −3. Since 4 is greater than −3, Quantity A is greater.

4. Answer: (A)

Why (C) is wrong: The trap here is reading too quickly — both quantities involve square roots, so a careless test-taker might process "4 vs. 3" and call them unequal, or confuse the signs entirely. The question is testing whether you correctly apply the constraint (positive vs. negative) to each root.

Why (D) is wrong: There are no variables left — both quantities resolve to fixed numbers (4 and −3). "Cannot be determined" only applies when you genuinely cannot tell which is larger, which requires at least one unresolved variable. Once both quantities are constants, D is never the answer.

Verification: Double-check by plugging back in. 4² = 16 ✓. (−3)² = 9 ✓. The constraints are satisfied and the comparison holds.