GRE Algebra: Topics, Key Rules and 5 Practice Questions
Key Takeaways
- Algebra is one of four GRE Quant content areas: exponents, factoring, linear and quadratic equations, inequalities, functions and coordinate geometry.
- ETS says Quant content goes no higher than a second course in algebra and includes no trigonometry or calculus.
- Three identities save the most time: (a + b) squared, (a - b) squared and the difference of squares.
- The most common traps are dropping the negative root, dividing by a variable that could be zero and forgetting to flip an inequality.
- Quant gives 47 minutes for 27 questions, so recognising the setup matters more than fast arithmetic.
Table of Contents
- What algebra is tested on the GRE?
- How is the GRE Quant section set up?
- Which algebra rules should you know by heart?
- How do you solve GRE equations and inequalities?
- How does algebra appear in each GRE question type?
- What are the common GRE algebra traps?
- GRE algebra practice questions with solutions
- Question 1 (multiple choice, select one)
- Question 2 (Quantitative Comparison)
- Question 3 (numeric entry)
- Question 4 (multiple choice, select one or more)
- Question 5 (multiple choice, select one)
- How should you practise GRE algebra?
Algebra is one of the four content areas of GRE Quantitative Reasoning, alongside arithmetic, geometry and data analysis. It covers exponents, factoring, linear and quadratic equations, inequalities, functions and coordinate geometry, all at the level of a second course in high school algebra. Most GRE algebra questions are not hard algebra: they test whether you set up the equation correctly and avoid the traps in the answer choices.
This guide lists exactly what ETS says the algebra content covers, the rules worth knowing by heart, how algebra shows up in each of the four Quant question types, and five practice questions with full solutions.
The five practice questions are written by Galvanize in the official GRE formats. They are not ETS questions. For the whole Quant section, read our GRE Quant guide.
What algebra is tested on the GRE?
ETS lists the algebra topics on its Quantitative Reasoning content page:
- operations with exponents
- factoring and simplifying algebraic expressions
- relations, functions, equations and inequalities
- solving linear and quadratic equations and inequalities
- solving simultaneous equations and inequalities
- setting up equations to solve word problems
- coordinate geometry, including graphs of functions, equations and inequalities, intercepts and slopes of lines
ETS adds that the Quant content is generally no higher than a second course in algebra, and that it does not include trigonometry, calculus or other higher-level mathematics. So if a question looks like it needs calculus, there is a simpler route you have not spotted yet.
How is the GRE Quant section set up?
The ETS test structure page sets out two Quant sections:
| Quantitative Reasoning | Questions | Time |
|---|---|---|
| Section 1 | 12 | 21 minutes |
| Section 2 | 15 | 26 minutes |
| Total | 27 | 47 minutes |
That is a little over 1.7 minutes a question. You get an on-screen calculator for Quant. ETS does not publish how many questions come from each content area, and algebra often sits inside a word problem or a geometry question rather than standing alone.
Which algebra rules should you know by heart?
| Rule | What it says |
|---|---|
| Multiplying powers | xa × xb = xa+b |
| Dividing powers | xa ÷ xb = xa−b |
| Power of a power | (xa)b = xab |
| Zero and negative exponents | x0 = 1 and x−a = 1 / xa (for x ≠ 0) |
| Square of a sum | (a + b)2 = a2 + 2ab + b2 |
| Square of a difference | (a − b)2 = a2 − 2ab + b2 |
| Difference of squares | a2 − b2 = (a + b)(a − b) |
| Quadratic formula | for ax2 + bx + c = 0, x = (−b ± √(b2 − 4ac)) / 2a |
| Slope of a line | (y2 − y1) / (x2 − x1) |
| Slope-intercept form | y = mx + b, where m is the slope and b is the y-intercept |
The three identities in the middle of the table save the most time. GRE questions are often built so that factoring a2 − b2 or spotting (a + b)2 turns a long calculation into one line.
How do you solve GRE equations and inequalities?
Linear equations. Collect the variable on one side and the numbers on the other. Expand brackets first, and check your answer by substituting it back.
Quadratic equations. Move everything to one side so it equals zero, then factor. x2 − 5x + 6 = 0 factors to (x − 2)(x − 3) = 0, so x = 2 or x = 3. A quadratic usually has two solutions, and GRE questions often turn on that second solution.
Simultaneous equations. With two equations and two unknowns, add or subtract the equations to remove one variable. Before solving fully, check what the question asks: if it wants x + y, you may get it in one step without finding x and y.
Inequalities. Solve them like equations, with one rule: when you multiply or divide both sides by a negative number, flip the inequality sign. −2x > 6 becomes x < −3.
Absolute value. |x − 2| < 3 means x is less than 3 units from 2, so −1 < x < 5. |x − 2| > 3 means x is more than 3 units away, so x < −1 or x > 5.
Functions. f(x) = 2x + 1 is a rule: put in x, get out 2x + 1. f(3) = 7. The GRE also uses made-up symbols, such as “a ⋄ b = a2 − b for all numbers a and b”. Treat these as functions and substitute carefully.
How does algebra appear in each GRE question type?
ETS lists four Quant question types. Algebra can appear in any of them:
- Quantitative Comparison: you compare Quantity A with Quantity B and choose whether A is greater, B is greater, they are equal, or the relationship cannot be determined. With variables, test several values, including negatives, zero and fractions. If different values give different results, the answer is “cannot be determined”.
- Multiple choice, select one: five choices, one correct. You can often substitute the answer choices back into the equation, starting with the middle value.
- Multiple choice, select one or more: you must pick every correct choice and no wrong ones. There is no partial credit, so test each choice on its own.
- Numeric entry: you type the answer. There are no choices to check against, so substitute your answer back before moving on.
What are the common GRE algebra traps?
- Dropping a solution. x2 = 9 means x = 3 or x = −3. Forgetting the negative root is the most common error in Quantitative Comparison.
- Dividing by a variable. If x could be zero, dividing both sides by x loses that solution. Factor instead.
- Forgetting to flip the sign. Multiplying an inequality by a negative number reverses it.
- Answering a different question. You solve for x, but the question asked for 2x or x + y. Re-read the last line before choosing.
- Assuming integers. Unless the question says x is an integer, it can be a fraction or a negative number.
GRE algebra practice questions with solutions
These five practice questions are written by Galvanize in the official GRE formats. Try each one before opening the solution.
Question 1 (multiple choice, select one)
If 2(x − 3) = 3x − 11, what is the value of x?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
Answer: (C) 5. Expand the left side: 2x − 6 = 3x − 11. Subtract 2x from both sides: −6 = x − 11. Add 11: x = 5. Check: 2(5 − 3) = 4 and 3(5) − 11 = 4.
Question 2 (Quantitative Comparison)
x2 − 5x + 6 = 0
Quantity A: x Quantity B: 2
(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.
Answer: (D). The equation factors to (x − 2)(x − 3) = 0, so x = 2 or x = 3. If x = 2 the quantities are equal; if x = 3 Quantity A is greater. Two different results means the relationship cannot be determined. Choosing (C) after finding only x = 2 is the trap.
Question 3 (numeric entry)
If x + y = 14 and x − y = 4, what is the value of xy?
Answer: 45. Add the two equations: 2x = 18, so x = 9. Then y = 14 − 9 = 5. So xy = 9 × 5 = 45.
Question 4 (multiple choice, select one or more)
Which of the following values of x satisfy |x − 2| < 3? Indicate all such values.
(A) −2 (B) −1 (C) 0 (D) 3 (E) 5 (F) 6
Answer: (C) and (D). |x − 2| < 3 means −3 < x − 2 < 3, so −1 < x < 5. Only 0 and 3 lie strictly between −1 and 5. −1 and 5 are excluded because the inequality is strict: at those values |x − 2| equals 3, which is not less than 3.
Question 5 (multiple choice, select one)
If 2x+1 = 32, what is the value of 4x?
(A) 16 (B) 64 (C) 128 (D) 256 (E) 512
Answer: (D) 256. 32 = 25, so x + 1 = 5 and x = 4. Then 44 = 256. A faster route: 4x = (22)x = 22x = 28 = 256. Choice (B) is the trap for anyone who stops at 43.
How should you practise GRE algebra?
- Learn the identities until you see them. Write the three square identities from memory each day for a week.
- Practise setting up equations. Many GRE algebra questions are word problems. Translate each sentence into an equation before you calculate anything.
- Test values in Quantitative Comparison. Try a positive number, a negative number, zero and a fraction before you commit to an answer.
- Practise under timing. At just over 1.7 minutes a question, speed comes from recognising the setup, not from fast arithmetic.
To see how your algebra holds up across the whole section, take the free GRE mock test. For more practice in every content area, try our GRE sample questions with answers and the GRE probability questions with solutions. To turn a practice score into a target, use the GRE score calculator.
If you are taking the GRE for a master’s abroad, our counsellors can help you plan your applications around your target score. Talk to an admissions counsellor.
Frequently Asked Questions
What algebra is on the GRE?
ETS lists operations with exponents, factoring and simplifying expressions, functions, equations and inequalities, linear and quadratic equations, simultaneous equations, setting up equations for word problems, and coordinate geometry such as slopes and intercepts.
How hard is GRE algebra?
The content is no higher than a second course in high school algebra, according to ETS. The difficulty comes from setting up the problem correctly and avoiding traps, not from advanced techniques.
Is there calculus or trigonometry on the GRE?
No. ETS states that Quantitative Reasoning does not include trigonometry, calculus or other higher-level mathematics.
Do I need the quadratic formula for the GRE?
It helps to know it, but most GRE quadratics factor neatly. Try factoring first and use the formula only when the numbers do not factor.
How many algebra questions are on the GRE?
ETS does not publish a count per content area. Algebra also appears inside word problems and geometry questions, so it affects more questions than a count would suggest.
Can I use a calculator for GRE algebra?
Yes, the GRE gives you an on-screen calculator for the Quant sections. It helps with arithmetic, but algebra questions are usually faster by hand.
What is the best strategy for algebra in Quantitative Comparison?
Test several values for each variable, including a negative number, zero and a fraction. If different values give different results, the answer is that the relationship cannot be determined.
Is coordinate geometry part of GRE algebra?
Yes. ETS lists coordinate geometry, including graphs of functions, intercepts and slopes of lines, under algebra.
Should I plug in the answer choices?
Often, yes. On multiple-choice questions with numeric answers, substituting the choices back into the equation, starting with the middle value, can be faster than solving.
How long should I spend on a GRE algebra question?
Quant gives 47 minutes for 27 questions, a little over 1.7 minutes each on average. Quick equation questions buy time for longer word problems and data sets.
