Algebra questions test your ability to set up equations from word descriptions, solve them efficiently, and manipulate algebraic expressions. Knowing the key patterns cold is faster than deriving from scratch under time pressure.

A) Translating Words to Math

B) Linear Equations

• One unknown: isolate the variable with inverse operations. Both sides can be added to, subtracted from, multiplied by, or divided by the same nonzero number without changing equality.
• Two unknowns, two equations: use substitution or elimination. With two independent equations, exactly one solution. With equivalent equations (same line), infinitely many solutions. With contradictory equations, no solution.
• Substitution: solve one equation for one variable, substitute into the other.
• Elimination: multiply equations to make one variable's coefficient the same magnitude, then add or subtract to cancel it.

Example (one unknown): 3x − 4 = 11. Add 4 to both sides: 3x = 15. Divide both sides by 3: x = 5. Check: 3(5) − 4 = 11 ✓

Example (substitution): 3x + y = 7 and x − y = 1. Solve the second for y: y = x − 1. Substitute into the first: 3x + (x − 1) = 7 → 4x = 8 → x = 2. Back-substitute: y = 2 − 1 = 1. Solution: x = 2, y = 1.

Example (elimination): 3x + 2y = 12 and 5x − 2y = 4. The y-coefficients are already opposite in sign, so add the equations: 8x = 16 → x = 2. Back-substitute into the first: 6 + 2y = 12 → y = 3. Solution: x = 2, y = 3.

Special cases: 2x + 4 = 2(x + 3) simplifies to 4 = 6 — a contradiction, so no solution. 2x + 4 = 2(x + 2) simplifies to 4 = 4 — always true, so infinitely many solutions (any x works).

C) Quadratic Equations

Standard form: ax² + bx + c = 0 (a ≠ 0). A quadratic has at most two real roots.

• Factoring: find two numbers that multiply to c (when a=1) and add to b. x² − 5x + 6 = (x−2)(x−3) → x = 2 or x = 3.
• Difference of squares: a² − b² = (a−b)(a+b). Example: 9x² − 25 = (3x−5)(3x+5).
• Perfect square trinomials: (x+y)² = x² + 2xy + y², (x−y)² = x² − 2xy + y².
• Quadratic formula: x = (−b ± √(b²−4ac)) / (2a). Use when factoring fails.
• Discriminant b²−4ac: >0 → two distinct real roots (parabola crosses x-axis twice); =0 → one repeated root (parabola touches x-axis at its vertex); <0 → no real roots (parabola entirely above or below x-axis). The discriminant is derived from the quadratic formula — it's what's under the radical.
• Sum of roots = −b/a. Product of roots = c/a. Useful shortcut on GMAT — you don't need to solve for each root individually.

Example (sum/product shortcut): x² − 7x + 10 = 0. Sum = 7, Product = 10 → roots must be 2 and 5 (2+5=7, 2×5=10). No need to use the quadratic formula.

D) Extended Algebraic Identities

• (a+b+c)² = a² + b² + c² + 2(ab + bc + ca)
• (a+b)³ = a³ + 3a²b + 3ab² + b³
• (a−b)³ = a³ − 3a²b + 3ab² − b³
• a³ + b³ = (a+b)(a² − ab + b²)
• a³ − b³ = (a−b)(a² + ab + b²)

E) Exponent Divisibility Identities

• aⁿ − bⁿ is always divisible by (a − b), for any positive integer n.
• aⁿ − bⁿ is divisible by (a + b) if n is even.
• aⁿ + bⁿ is divisible by (a + b) if n is odd.
• Examples: 7⁴ − 3⁴ divisible by both (7−3)=4 and (7+3)=10. 5³ + 2³ divisible by 7.

F) Functions

A function is a rule that takes a number in and produces exactly one number out. Think of it as a machine: you drop a number in, the machine does something to it, and spits one result out. Every time you use the same input, you get the same output.

Notation: f(x) = 3x − 1 means "take any number x, multiply it by 3, then subtract 1." The letter f is just the name of the function. To evaluate it, replace x with the input and compute:

• f(4) = 3(4) − 1 = 12 − 1 = 11
• f(0) = 3(0) − 1 = −1
• f(−2) = 3(−2) − 1 = −6 − 1 = −7

Domain = all x-values you are allowed to plug in. Most functions accept every real number, but two situations create restrictions:

• Can't divide by zero: f(x) = 1/(x−3). If you plug in x = 3, the denominator becomes 0 — undefined. So x = 3 is excluded. Domain: all real numbers except 3.
• Can't take square root of a negative: f(x) = √(x−2). The expression under the root must be ≥ 0, so x − 2 ≥ 0 → x ≥ 2. Domain: x ≥ 2.

Range = all possible output values. Example: f(x) = x². No matter what number you square, the result is always ≥ 0 (e.g., (−5)² = 25). So the range is y ≥ 0 — the function can never output a negative number.

Composite functions f(g(x)) chain two functions together. The notation means: first apply g to x, then apply f to that result. Always work inside out.

Example: f(x) = 2x + 1, g(x) = x². Find f(g(3)).

1. Inner function first: g(3) = 3² = 9
2. Outer function next: f(9) = 2(9) + 1 = 19

G) Symbol / Operator Functions

The GMAT sometimes defines a custom operation using a symbol (★, #, @, ⊕, etc.) and asks you to evaluate it. These look strange but are pure substitution — exactly like f(x), just with two inputs and a custom rule.

Process: (1) read the definition, (2) substitute the given numbers, (3) if nested, evaluate inside out.

Example 1: a ★ b = 3a − 2b. Find 5 ★ 2 → 3(5) − 2(2) = 15 − 4 = 11.

Example 2 (nested): same definition. Find 3 ★ (1 ★ 4).
Inner first: 1 ★ 4 = 3(1) − 2(4) = 3 − 8 = −5.
Outer: 3 ★ (−5) = 3(3) − 2(−5) = 9 + 10 = 19.

• Order matters: a ★ b ≠ b ★ a in most custom operations. Always substitute in the given order.
• Fixed-point questions: "for what value of x does x ★ x = x?" Substitute x for both variables, set equal to x, and solve like a normal equation.

H) Sequences

A sequence is a function whose domain is a set of consecutive positive integers. The terms are listed as a₁, a₂, a₃, … where the subscript is the position. A sequence can be finite (fixed number of terms, e.g. a₁, a₂, …, aₙ) or infinite (continues without end). An arithmetic sequence has the form aₙ = b + cn for constants b and c.

A series is the sum of the terms of a sequence. The partial sum Sₙ is the sum of the first n terms: Sₙ = a₁ + a₂ + … + aₙ. An infinite series is the sum of all terms of an infinite sequence — it converges to a finite value only under certain conditions (see geometric series below).

• Arithmetic: each term = previous + constant d. n^th term = a₁ + (n−1)d. Sum of n terms = (n/2)(a₁ + aₙ) or (n/2)(2a₁ + (n−1)d).
  Example: 3, 7, 11, 15, … (d=4). 10th term = 3 + 9(4) = 39. Sum of 10 terms = (10/2)(3+39) = 5 × 42 = 210.
• Geometric: each term = previous × constant r. n^th term = a₁ × rⁿ⁻¹. Sum of n terms = a₁(1 − rⁿ) / (1 − r) for r ≠ 1.
  Example: 2, 6, 18, 54, … (r=3). 5th term = 2 × 3⁴ = 162. Sum of 5 terms = 2(1−243)/(1−3) = 2(−242)/(−2) = 242.
• Infinite geometric series: if |r| < 1, sum = a₁ / (1 − r). If |r| ≥ 1, the series diverges (no finite sum).
  Example: 1 + 1/2 + 1/4 + … = 1/(1−1/2) = 2.

I) AM-GM Inequality

For any two positive numbers a and b: (a+b)/2 ≥ √(ab). The arithmetic mean is always ≥ the geometric mean. Equality holds only when a = b. Use this on optimization problems asking for the maximum product given a fixed sum, or minimum sum given a fixed product.

Example: x + y = 10, both positive. Maximum product xy occurs when x = y = 5 → max product = 25. At any other split (e.g., x=3, y=7), product = 21 < 25.

J) Harmonic Mean

The harmonic mean of a and b: HM(a,b) = 2ab/(a+b). It always produces the correct average speed for equal-distance round trips (which is why the simple average of two speeds is wrong).

• Inequality chain: AM ≥ GM ≥ HM for positive numbers.
• Relationship: AM × HM = GM² (useful to verify or derive values).

K) Inequalities

An inequality is a statement using one of these symbols:

• ≠   is not equal to
• >   is greater than
• ≥   is greater than or equal to
• <   is less than
• ≤   is less than or equal to

Example: 5x − 3 < 9 and 6x ≥ y are both inequalities.

Solve a linear inequality the same way you solve a linear equation — isolate the unknown — with one critical rule: multiplying or dividing both sides by a negative number reverses the inequality sign.

• 2x − 3 > 7 → 2x > 10 → x > 5 ✓ (positive division: no sign flip)
• −3x > 12 → x < −4 ✓ (negative division: sign flips)
• Adding or subtracting the same value from both sides never changes the sign.

L) Compound Inequalities

• a < x < b: x lies strictly between a and b. Perform the same operation on all three parts simultaneously.
• Example: −2 < 3x + 1 < 10 → subtract 1: −3 < 3x < 9 → divide by 3: −1 < x < 3.

M) Quadratic Inequalities

Factor first, then use a sign chart or test values in each region.

• (x − 2)(x − 3) > 0: product positive when both factors positive (x > 3) or both negative (x < 2). Solution: x < 2 or x > 3.
• (x − 2)(x − 3) < 0: product negative when one factor positive, one negative. Solution: 2 < x < 3.

N) Key Inequality Properties for GMAT

• If x > 0 and y > 0: x + y > 0, xy > 0, x/y > 0.
• If x > 0 and y < 0: xy < 0.
• x² ≥ 0 for all real x. x² > 0 if and only if x ≠ 0.
• If a > b and c > d, then a + c > b + d. You cannot subtract inequalities without caution.
• If a > b > 0, then a² > b². But if a or b could be negative, you cannot square both sides directly.
• Triangle inequality: |a + b| ≤ |a| + |b|. The absolute value of a sum ≤ sum of absolute values.