Last updated July 07, 2026 · Reviewed by the AnvayaPrep team
Introduction
Algebra is the second major quantitative domain on the GRE and spans 32 topics covering everything from foundational expression manipulation to advanced function analysis. The unit covers: simplifying and evaluating algebraic expressions, solving linear equations and inequalities (single variable and systems), quadratic equations and their solution methods, polynomial expressions, rational expressions, exponential equations and functions, linear functions (slope, equation of a line, graphing), quadratic functions (parabolas, vertex), domain and range, function notation and evaluation, and algebraic modeling of word problems. Algebra questions appear in every GRE question format and represent approximately 30 to 40% of the Quantitative Reasoning section.
The GRE tests algebra differently than a high school exam. Problems rarely ask for straightforward computation from a clearly framed equation. Instead, they embed algebra within word problems, disguise it inside quantitative comparisons, or test conceptual understanding through "must be true" and "could be true" questions. The highest-yield algebraic skill is not any specific procedure but the ability to recognize what form a problem has and select the most efficient solution path.
Learning Objectives
- Manipulate algebraic expressions by applying the distributive property, combining like terms, and factoring correctly and efficiently
- Solve linear equations and literal equations for a specified variable using inverse operations, and verify solutions by substitution
- Solve linear inequalities and correctly reverse the inequality sign when multiplying or dividing by a negative number
- Solve systems of two linear equations using substitution and elimination, and recognize when a system has no solution or infinitely many solutions
- Factor quadratic expressions and solve quadratic equations using factoring, completing the square, and the quadratic formula
- Recognize and apply the sum-of-roots (negative b/a) and product-of-roots (c/a) relationships for quadratic equations without solving
- Evaluate functions using function notation and identify domain and range from algebraic expressions and equations
- Write the equation of a line given two points, a point and a slope, or slope-intercept information; interpret slope as rate of change
- Set up algebraic equations and inequalities from word problem descriptions, define variables clearly, and solve for the target quantity
- Solve exponential equations and evaluate exponential functions, recognizing growth and decay patterns
High-Yield Concepts
Linear Equations and Solving Strategies
Linear equations (variables to the first power only) are solved by isolating the variable through inverse operations while keeping both sides equal. The critical principle: perform the same operation on both sides.
Solving a linear equation follows a standard sequence: distribute and eliminate parentheses, combine like terms on each side, move variable terms to one side and constants to the other, then divide by the variable's coefficient. For equations with fractions, multiply through by the LCD first to clear denominators.
A linear equation has one unique solution unless the variable cancels out. If it cancels and leaves a true statement (0 = 0), the equation has infinitely many solutions. If it leaves a false statement (0 = 5), there is no solution.
Quadratic Equations: Recognizing the Best Method
Quadratic equations (ax^2 + bx + c = 0) have up to two solutions. The GRE expects fluency with three solution methods:
| Method | Use When | Notes |
|---|---|---|
| Factoring | Leading coefficient is 1 or small integer; factors are obvious | Fastest when it works; always try first |
| Quadratic formula | Factoring fails; answers may be irrational | x = (-b ± √(b^2-4ac)) / 2a |
| Completing the square | Needed for vertex form or when problem structure suggests it | Useful for understanding parabola vertex |
The discriminant (b^2 - 4ac) predicts the nature of solutions: positive discriminant means two real solutions, zero means exactly one real solution, negative means no real solutions. This determination can be made without solving, saving time on comparison questions.
The sum-of-roots formula (sum = -b/a) and product-of-roots formula (product = c/a) allow comparison questions to be answered in seconds when the question asks about relationships between roots rather than the roots themselves.
On GRE quadratic questions, first check whether factoring is possible by looking for integer factors of c that sum to b (when a = 1). If you find them in under 10 seconds, factor. If not, apply the quadratic formula directly rather than wasting time on failed factoring attempts.
Inequalities: The Reversal Rule
Linear inequalities are solved exactly like linear equations with one critical exception: when multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
This rule is the single most frequently tested inequality concept on the GRE. A problem that requires division by a negative number and does not reverse the sign produces the wrong solution set. The GRE places answer choices that correspond to both the correct and the incorrectly-reversed solution to exploit this error.
For compound inequalities ("and" logic: both must hold), the solution is the intersection of the two individual solution sets. For "or" logic (either must hold), the solution is the union.
Systems of Equations
A system of two linear equations with two unknowns has three possible outcomes: one unique solution (lines intersect at a point), no solution (parallel lines), or infinitely many solutions (same line expressed differently). The GRE tests all three scenarios, including in quantitative comparison questions that ask whether sufficient information exists to determine a specific value.
Substitution method: isolate one variable in one equation and substitute into the other. Useful when one variable has coefficient 1 or -1.
Elimination method: multiply one or both equations by constants so that adding or subtracting the equations eliminates one variable. Useful when both equations have similar structure.
Inspection (adding/subtracting without modification): sometimes the GRE provides systems where the question only asks for the sum or difference of variables rather than each variable individually. Look for this shortcut before committing to full solution.
On systems of equations questions, students frequently solve for one variable and forget to solve for the other, then mistake the partial result as the final answer. Always re-read what the question asks for -- it may want x+y, or xy, or just y, not just x.
Functions and Functional Notation
A function maps each input (domain value) to exactly one output (range value). Function notation f(x) means "evaluate the expression when x equals this value." Evaluating f(3) means substituting x = 3 into the expression.
Domain restrictions arise from: square roots (radicand must be ≥ 0), fractions (denominator must not equal 0), and logarithms (argument must be positive). The GRE tests domain identification and restricted-domain questions in function context.
Composition of functions: f(g(x)) means first evaluate g(x), then use that result as the input to f. Always work from the inside out.
Study Strategy
Begin with linear equations and linear inequalities -- these appear in the highest percentage of questions and are prerequisites for every other algebra topic. Practice translating word problem language into equation form; this translation skill is the most transferable algebra skill on the GRE.
Move to systems of equations next, then quadratic equations. Learn all three quadratic solution methods but prioritize recognizing when factoring works versus when the quadratic formula is faster.
Study functions and function notation after mastering equations, since function problems apply the same equation-solving skills within a different framework.
Study the remaining specialized topics (exponential equations, polynomial expressions, rational expressions, domain and range, algebraic modeling) in the order they appear on your diagnostic, prioritizing the topics where you lost the most points.
Common Mistakes
Forgetting to reverse the inequality when multiplying or dividing by a negative. This is the most common algebra error on the GRE. Always check the sign of the number you are dividing or multiplying by before applying the operation.
Distributing incorrectly, especially with negative coefficients. -(x + 3) becomes -x - 3, not -x + 3. The negative must be distributed to every term inside the parentheses.
Solving for the wrong variable in a systems problem. Many questions ask for x + y, 2x - y, or a specific variable after setting up a system. Students who solve for x when the question asks for y (or vice versa) get wrong answers despite correct algebra.
Applying the factoring method to a quadratic with non-integer roots. If the factors are not obvious integers within 10 seconds, switch to the quadratic formula. Attempting to force factoring on un-factorable quadratics wastes time.
Mixing up sum-of-roots and product-of-roots. For ax^2 + bx + c = 0, sum = -b/a and product = c/a. These come directly from Vieta's formulas and allow rapid answers to comparison questions involving relationships between roots.
Exam Tips
When a word problem involves two unknown quantities, define both variables explicitly before writing any equation. Ambiguous variable definitions are the primary cause of setup errors.
For quantitative comparison questions involving inequality constraints (e.g., "x > 0"), test boundary values and typical values before concluding a relationship holds always. Test x = 1, x = 0.5, x = 2, and any special cases the constraint allows.
When the GRE asks whether an equation has one solution, no solution, or infinitely many solutions, check the coefficient of the variable term. If it becomes zero after simplification, the variable has disappeared and the type depends on whether the remaining constants are equal (infinitely many) or unequal (no solution).
Recognize when the GRE is asking for an expression value that can be determined from a system without solving for individual variables. If asked for x + y and given x + y in one of the equations, read the system carefully before computing individual values.