GRE Algebra: Equations, Inequalities and Functions

ETS splits GRE Quantitative Reasoning content into four areas: Arithmetic, Algebra, Geometry and Data Analysis. Algebra is the area built on symbol manipulation: solving equations, handling inequalities, factoring quadratics and reading function notation. The official content description lists operations with exponents, factoring and simplifying algebraic expressions, relations and functions, equations and inequalities, solving linear and quadratic equations and inequalities, solving simultaneous equations, setting up equations for word problems, and coordinate geometry under this one heading.

The math stays inside familiar ground. ETS caps GRE content at a second course in high school algebra, with trigonometry and calculus left out entirely. This guide works through linear equations and inequalities, systems of linear equations, factoring and solving quadratics, function notation, and the sign and root errors draining points across all four. Word problem setup gets its own guide at GRE algebra word problems, and the graphing side of algebra lives in coordinate geometry and triangles.

What Counts as GRE Algebra

ETS's own list names the skills grouped under Algebra. This table matches each one to where you are able to practice it. Two of the six skills have their own guides, and the rest are covered in the sections below.

Solving and Graphing Linear Equations and Inequalities

A linear equation has no variable raised to a power above 1, so its graph is a straight line. Linear equations and inequalities use the same core moves: distribute, combine like terms and isolate the variable. Inequalities add one extra rule: the direction of the sign flips when you multiply or divide both sides by a negative number.

Multi-Step Linear Equations

Work every multi-step equation in the same order. The sequence keeps the sign and combining errors covered later in this guide out of your work.

1. Distribute across any parentheses first.
2. Combine like terms on each side.
3. Move variable terms to one side and constants to the other.
4. Divide by the coefficient on the variable.

Take 3(x - 2) + 4 = 2x + 5. Distributing gives 3x - 6 + 4 = 2x + 5, which simplifies to 3x - 2 = 2x + 5. Subtracting 2x from both sides leaves x - 2 = 5, so x = 7.

Linear Inequalities and the Sign Flip Rule

Inequalities solve the same way as equations, with one exception. Multiplying or dividing both sides by a negative number flips the inequality sign. Adding or subtracting a negative number leaves the direction unchanged.

Take -2x + 3 > 9. Subtracting 3 gives -2x > 6. Dividing by -2 flips the sign, so the solution is x < -3. Skipping the flip produces x > -3, the wrong direction entirely.

Graphing Solutions on the Number Line

A strict inequality with < or > draws an open circle at the boundary value. A non-strict inequality with ≤ or ≥ draws a closed circle, since the boundary value itself satisfies the inequality. A compound inequality such as -1 ≤ x < 4 graphs as a segment between two boundary points, one closed and one open in this case.

Systems of Linear Equations

A system pairs two or more linear equations sharing the same variables. A solution is the set of values satisfying every equation at once. ETS lists solving simultaneous equations under the Algebra content area. Two methods solve a two-equation, two-unknown system: substitution and elimination, and both reach the same answer.

Solving by Substitution

Solve one equation for a single variable, then substitute the resulting expression into the other equation. Take y = x + 3 and 2x + y = 12. Substituting the first equation into the second gives 2x + (x + 3) = 12, so 3x + 3 = 12, x = 3, and y = 6. Substitution is fastest when one variable sits alone on one side already.

Solving by Elimination

Elimination adds or subtracts the two equations to cancel one variable. Multiply one or both equations first so the coefficients on one variable match in size and are opposite in sign. Take 3x + 2y = 16 and x - 2y = 0. Adding the equations cancels y directly: 4x = 16, so x = 4, and y = 2.

Recognizing No-Solution and Infinite-Solution Systems

Two linear equations describe two lines, and their graphs settle the outcome before any algebra runs. Compare the slopes and the intercepts to see which of three cases applies.

• Different slopes: the lines cross once, giving one solution.
• Same slope, different intercept: parallel lines, giving no solution.
• Same slope, same intercept: the same line, giving infinite solutions.

Algebraically, a no-solution system reduces to a false statement such as 0 = 5. An infinite-solution system reduces to a true statement such as 0 = 0. Both outcomes appear after the variables cancel and only constants remain.

Factoring and Solving Quadratics

A quadratic equation contains a squared variable term, written in standard form as ax² + bx + c = 0. Move every term to one side first, since factoring, completing the square and the quadratic formula all need standard form. Try factoring first, since it is the shortest route when the roots are whole numbers.

Factoring Methods

Check three factoring patterns in order. The first pattern matching the expression is the shortest route to the roots.

• Greatest common factor: pull the shared factor out of every term.
• Difference of squares: a² - b² factors into (a + b)(a - b).
• Trinomial factoring: find two numbers multiplying to c and adding to b in x² + bx + c.

x² - 9 factors into (x + 3)(x - 3), using the difference of squares pattern. Setting each factor to zero gives x = -3 and x = 3. For x² + 5x + 6, the two numbers multiplying to 6 and adding to 5 are 2 and 3, so it factors into (x + 2)(x + 3) with roots x = -2 and x = -3.

The Quadratic Formula

Not every quadratic factors with whole numbers. The quadratic formula solves any quadratic in standard form: x = (-b ± √(b² - 4ac)) / 2a. For x² + 4x - 1 = 0, a = 1, b = 4 and c = -1. Plugging in gives x = (-4 ± √(16 + 4)) / 2, which simplifies to x = -2 ± √5.

The value under the root, b² - 4ac, is the discriminant. A positive discriminant gives two real solutions, zero gives one repeated solution, and a negative discriminant gives no real solution. Compute the discriminant alone when a question asks only for the number of solutions, since the full formula is unnecessary there.

Function Notation and Evaluating f(x)

Function notation writes a rule as f(x) instead of y. f(x) is a name for the output, not multiplication of f by x. The letter inside the parentheses is the input, and the rule to the right of the equals sign tells you what to do with it.

Evaluating f(x) at a Value

Given f(x) = 3x - 2, evaluate at any input by replacing every x in the rule with the input value, then computing the result. f(5) means substitute 5 for x: f(5) = 3(5) - 2 = 13. The input is also allowed to be an expression, so f(a + 1) = 3(a + 1) - 2 = 3a + 1.

Composite Functions f(g(x))

A composite function feeds the output of one function into another. f(g(x)) means evaluate g(x) first, then plug the result into f. With f(x) = x + 1 and g(x) = 2x, f(g(3)) starts with g(3) = 6, then f(6) = 7. Order matters: g(f(3)) reverses the steps, giving f(3) = 4 and then g(4) = 8.

Domain Restrictions

A function's domain is the set of inputs producing a valid output. Two situations remove values from the domain.

• A denominator equal to zero: exclude any x value setting the denominator to zero.
• A negative number under an even root: exclude any x value making the expression under a square root negative.

For f(x) = 1 / (x - 4), x = 4 is excluded, since it sets the denominator to zero. For f(x) = √(x - 4), the domain is x ≥ 4, since any smaller input puts a negative number under the root. Check the domain before answering a question asking for allowed inputs or a range of valid values.

Common Algebra Traps: Sign Errors and Extraneous Roots

Sign errors and extraneous roots cost points on questions you already know how to solve. Both come from execution rather than from the concept. This section covers where each one appears and how to catch it before you select an answer.

Sign Flips and Distribution Errors

Two moments create sign errors: distributing a negative across parentheses, and dividing an inequality by a negative number. Take -(2x - 5): distributing the negative gives -2x + 5, not -2x - 5. Dropping the sign on the second term produces a clean-looking result, so the error survives a fast review. Rewrite a subtraction as adding a negative when the distribution is hard to track.

Extraneous Roots from Squaring and Cross-Multiplying

Squaring both sides of an equation, or cross-multiplying a rational equation, sometimes introduces a solution outside the domain of the original equation. The extra root satisfies the squared equation and fails the original one. Check every solution back in the original equation before selecting an answer.

Take √(x + 2) = x. Squaring gives x + 2 = x², so x² - x - 2 = 0, which factors into (x - 2)(x + 1) = 0, giving x = 2 or x = -1. Checking x = -1 in the original equation gives √1 = -1, which is false, so x = -1 is extraneous. Only x = 2 solves the original equation.

Practice GRE Algebra Questions

Every correct answer on GRE Quant earns one point, with no separate penalty for a wrong answer or a blank, per ETS scoring rules. Attempt every algebra question, even under time pressure, since an educated guess costs nothing extra. Your Quant score reports on a 130 to 170 scale in 1-point increments, and algebra accuracy moves this score directly, since it sits inside every Quant section alongside arithmetic, geometry and data analysis.

Our free drills split algebra practice into linear equations and inequalities, and quadratics and functions, so weak spots are isolated. Word problem setups are covered separately at GRE algebra word problems, and the graphing and coordinate plane skills live in coordinate geometry and triangles. Practice is free and needs no signup on our GRE practice page, alongside full-length timed tests and the rest of the Quant content areas covered in our GRE Quant Reasoning guide.

Sources

• ETS: GRE Quantitative Reasoning Content Overview
• ETS: What You Need to Know About the GRE General Test (PDF)