GRE Multiple Choice (Select One): Algebra Questions

Algebra is the largest content domain on the GRE Quantitative Reasoning section, and Multiple Choice Select One is the most common format you will encounter. These questions present five answer choices displayed as circles, and you must select exactly one correct answer. The problems range from direct equation solving and exponent simplification to multi-step word problems, function composition, custom-defined operations, and coordinate geometry. Below you will learn nine question patterns the GRE uses, work through two interactive examples step by step, and then practice with six guided questions drawn from realistic problem sets.

What Are MC Select One Algebra Questions?

Algebra MC Select One questions test your ability to work with variables, equations, inequalities, functions, and coordinate geometry. You are given a problem involving algebraic relationships, presented with five answer choices (A through E) displayed as circles, and you must select exactly one correct answer. The correct answer is the result of correctly applying algebraic rules and completing all steps.

These questions range from direct simplification — for example, simplifying an exponent expression — to multi-step word problems requiring you to set up and solve equations. They also include function evaluation, inequality solving, coordinate geometry, and problems involving custom-defined operations. The algebra is rarely beyond what you learned in a first-year college math course, but the difficulty comes from time pressure, carefully designed distractors, and the need to execute multiple steps without error.

Format note: Multiple Choice Select One is the most common question format on the GRE Quantitative section. Unlike Select Many questions (which use square checkboxes), Select One questions display circular radio buttons, and exactly one of the five choices is correct. You do not lose points for wrong answers, so you should always select an answer even if you are unsure.

9 Patterns You'll See

Nearly every MC Select One algebra question falls into one of nine stem patterns. Recognizing the pattern immediately tells you which strategy to deploy and what kind of trap to watch for.

Solve for a Variable

Direct equation solving: linear, quadratic, or systems. The stem typically reads 'If [equation], what is the value of x?' Apply standard algebraic manipulation to isolate the variable.

Equivalent Expression

Simplify an expression using exponent rules, factoring, or distribution. The stem reads 'Which of the following is equivalent to...' Write out each step carefully rather than simplifying in your head.

Function Composition

Apply a function multiple times using its graph or formula. For

f(f(x))

, evaluate the inner function first and use that output as the next input. Watch out for piecewise functions where different rules apply.

Graph Intersection or Match

Determine which function's graph intersects another, or match an inequality to its graphical representation. Requires understanding of absolute value, slopes, and how transformations affect graphs.

Series, Averages, and Factoring

Use given conditions to derive averages or sums of expressions. The key technique is factoring out a common term to connect the expression you need to information that is given.

Coordinate Geometry

Find a point using geometric properties such as slope, midpoint, perpendicular bisector, or parallelogram vector addition. These require multiple sequential computations.

Counting Solutions

Count how many integers or values satisfy an equation or inequality. Solve the inequality completely first, then count carefully, including or excluding boundary values based on whether the inequality is strict or non-strict.

Custom-Defined Operations

A novel operation is defined using a symbol (such as a triangle). Apply the definition step by step, evaluating inner parenthetical expressions first. The order of operations matters.

Work-Rate and Word Problems

Translate a real-world scenario (machines, rates, mixtures) into an algebraic equation. The most common setup uses the combined-rate formula:

\frac{1}{a} + \frac{1}{b} = \frac{1}{t}

How to Solve MC Algebra Step by Step

These five strategies cover the techniques you need across all nine patterns. Apply them in order for a systematic approach that avoids the most common errors.

Know these rules and apply them step by step. Power of a power: $(x^a)^b = x^{ab}$ — multiply exponents. Same base multiplication: $x^a \cdot x^b = x^{a+b}$ — add exponents. Same base division: $x^a / x^b = x^{a-b}$ — subtract exponents. Write out each step rather than trying to simplify in your head. The GRE places distractors at every intermediate value.

When a problem asks "which of the following must be true" or "which expression equals...," pick a concrete value for the variable and test each choice. For an odd integer n, try n = 3 or n = 5. For 0 < x < 1, try x = 0.5 or x = 1/4. Avoid $n = 0$ and $n = 1$ unless the problem requires them, as they can make different expressions look equal.

When the question asks "for which value of x..." and gives numerical choices, substitute each choice into the equation. Start with the middle choice. If the result is too large, try smaller; if too small, try larger. This is often faster than solving the equation algebraically, especially for rational equations or equations with absolute values.

When you see expressions like $x + x^2 + x^3 + x^4$ , look for common factors. Factoring out $x$ gives $x(1 + x + x^2 + x^3)$ , which often connects to information given in the problem. This technique is essential for series and average problems and frequently reveals a shortcut that avoids heavy arithmetic.

For parallelogram and vector problems, remember that opposite sides are equal and parallel. If you know three vertices, find the fourth using vector addition. Slope = rise/run, and perpendicular slopes multiply to -1. For midpoint problems, use $\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right)$ . Always verify your answer by checking that it satisfies all stated geometric properties.

Pro tip: If a question gives numerical answer choices arranged in order, backsolving is almost always the fastest approach. Start with the middle value. One substitution typically tells you which direction to go, and you will find the answer in two to three attempts.

Worked Example: Exponent Simplification

Work through each step below. You must answer each mini-challenge correctly to unlock the next step. If you get stuck, a second wrong attempt will reveal the answer so you can keep going.

Interactive Walkthrough0/4 steps

Simplifying

\frac{(x^2)^3}{x}

This problem tests your mastery of exponent rules. The GRE frequently tests whether you can distinguish between adding and multiplying exponents, and whether you remember to complete all steps of the simplification.

x \neq 0

, which of the following is equivalent to

\frac{(x^2)^3}{x}

Step 1: Identify the rule for the numerator

What exponent rule applies to

(x^2)^3

? Power of a power means we:

Step 2: Simplify the numerator

Step 3: Apply the division rule

Step 4: Select the answer

Worked Example: Series and Averages

This example teaches the factoring technique that connects a requested expression to given information. Work through each step to see why factoring is the key insight.

Interactive Walkthrough0/4 steps

Finding the Average of Powers

You are told that

1 + x + x^2 + x^3 = 60

. You need to find the average (arithmetic mean) of

x, x^2, x^3

, and

x^4

. This problem looks like it requires solving for

x

, but a factoring shortcut makes it much simpler.

1 + x + x^2 + x^3 = 60

, then the average (arithmetic mean) of

x, x^2, x^3

, and

x^4

is equal to which of the following?

Step 1: Write the average formula

The average of

x, x^2, x^3

, and

x^4

is their sum divided by how many terms?

Step 2: Factor the numerator

Step 3: Substitute the given value

Step 4: Divide to get the average

Practice Questions

Now apply what you have learned. Each question has a step-by-step solution walkthrough — after you submit your answer, click through the solution one step at a time to compare against your own work.

Question 1 — Sum of Cubes (Hard)

a + b = 5

and

a^2 + b^2 = 15

, what is the value of

a^3 + b^3

Question 2 — Piecewise Function Composition (Hard)

The function

f

is defined by

f(x) = 2x + 1

for

x < 3

and

f(x) = x^2 - 5

for

x \geq 3

. What is the value of

f(f(f(1)))

Question 3 — Custom Operation (Medium)

The operation (triangle) is defined by a (triangle) b = a - b for all numbers a and b. For which value of x is x (triangle) (x (triangle) x) = 2?

Question 4 — Integer Count in Inequalities (Hard)

How many positive integers

n

satisfy the inequality

\frac{n^2 - 9}{n + 1} \leq 5

Question 5 — Work-Rate Problem (Hard)

Working alone, Machine A can complete a job in

x

hours and Machine B can complete the same job in

(x + 6)

hours. Working together, they complete the job in 4 hours. What is the value of

x

Question 6 — Quadratic with Prime Constraint (Hard)

x^2 - 6x + k = 0

has two positive integer solutions and the product of those solutions is a prime number, what is the value of

k

Common Traps

Trap 1 — Power of a power vs. multiplication.

(x^2)^3 = x^6

, not

x^5

. The power-of-a-power rule requires multiplying exponents, not adding them. The GRE almost always includes the "added exponents" result as a distractor. Write out the rule explicitly each time:

(x^a)^b = x^{a \cdot b}

Trap 2 — Forgetting the final step. After computing a numerator like

x^6

\frac{(x^2)^3}{x}

, you must still divide by

x

to get

x^5

. Similarly, after finding an intermediate variable in a word problem, you may need to substitute back to find the quantity the question actually asks for. The GRE places distractors at every intermediate value.

Trap 3 — Composite function order and piecewise boundaries. For

f(f(-1))

, first evaluate

f(-1)

, then use that result as input to

f

again. Do not evaluate

f(-1) \times f(-1)

or apply

f

to -1 twice independently. For piecewise functions, check the boundary condition at every step — the piece that applies may change as the input changes.

Pattern Recognition at a Glance

Use this table to quickly identify the pattern and select your strategy when you see a new algebra question on the GRE.

Stem Clue	Pattern	Best Strategy
"What is the value of x?"	Solve for a variable	Direct algebra or backsolving
"Which is equivalent to..."	Equivalent expression	Apply exponent/factoring rules step by step
"What is $f(f(x))$ ?"	Function composition	Evaluate inside-out, check piecewise boundaries
"Which graph intersects..."	Graph intersection	Test key points or use elimination
"What is the average of..."	Series and averages	Factor to connect to given information
"What are the coordinates?"	Coordinate geometry	Use slope, midpoint, or vector properties
"How many integers satisfy..."	Counting solutions	Solve inequality completely, then count
"If a (symbol) b = ..."	Custom operation	Apply definition, innermost parentheses first
"Working alone / together..."	Work-rate problem	Set up $\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$ and solve

The key to fast performance on GRE algebra is not advanced math knowledge but pattern recognition. Once you identify the pattern, you know exactly which technique to apply and which traps to avoid. Practice recognizing these patterns until the identification step becomes automatic, so you can devote your mental energy to careful execution.

Study Checklist

MC Algebra Mastery Checklist0/8 complete

Frequently Asked Questions

How many algebra questions appear on the GRE Quantitative section?

Algebra questions make up roughly one quarter of the Quantitative Reasoning section. Across two scored sections you can expect 8 to 12 algebra questions, spanning multiple choice, numeric entry, and quantitative comparison formats. Multiple Choice Select One is the most common format for algebra on the GRE, so mastering this format gives you the highest return on your preparation time.

What algebra topics are tested most frequently on the GRE?

The most frequently tested algebra topics are linear equations, exponent rules, inequalities, and function evaluation. You will also see coordinate geometry (slopes and midpoints), systems of equations, and occasionally custom-defined operations. Quadratic equations appear but are less common than linear algebra. The difficulty comes not from advanced topics but from carefully designed distractors that exploit common errors.

Should I solve GRE algebra questions algebraically or by plugging in numbers?

It depends on the question type. For "which must be true" or "which is equivalent to" questions, plugging in numbers is often faster and safer. For direct solve-for-x questions with numerical answer choices, backsolving (testing choices) can be quicker than algebraic manipulation. For problems with variable answer choices — such as the series-and-averages example above — algebraic factoring is usually necessary. The best strategy is to have all three techniques ready and choose based on the stem pattern.

What are the most common mistakes on GRE algebra questions?

The three most common mistakes are: (1) confusing exponent rules, especially using addition instead of multiplication for power-of-a-power; (2) sign errors when distributing negatives or flipping inequality direction; and (3) stopping one step early, such as finding an intermediate variable but not converting to the form the question requests. The GRE places distractors at every intermediate value, so completing all steps is essential.

How should I prepare for GRE algebra questions if I have not studied math recently?

Start by reviewing the core exponent rules, the mechanics of solving linear and quadratic equations, and basic coordinate geometry (slope, midpoint, distance). Then practice with timed questions, focusing on recognizing which technique each problem requires. The key skill is not advanced math but pattern recognition and careful execution under time pressure. Work through the interactive examples and practice questions on this page as a starting point, then move to full-length practice tests.