GRE Quantitative: GRE Quantitative: Algebra Practice Questions
Test yourself on GRE Quantitative: Algebra with 10 original GRE practice questions. Pick an answer to see instant feedback and a full explanation.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of GRE.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. If 3x − 7 = 2x + 5, what is the value of x?
Explanation. Subtract 2x from both sides: x − 7 = 5. Add 7: x = 12. Check: 3(12)−7 = 29 and 2(12)+5 = 29. Correct.
2. If x² − 5x + 6 = 0, what is the sum of all possible values of x?
Explanation. Factor: (x−2)(x−3) = 0, so x = 2 or x = 3. Sum = 5. (Quick check: for ax²+bx+c, sum of roots = −b/a = 5.)
3. If 2^(x+3) = 32, what is x?
Explanation. 32 = 2^5, so x + 3 = 5, giving x = 2.
5. If a + b = 7 and a − b = 3, what is the value of a² − b²?
Explanation. a² − b² = (a+b)(a−b) = 7 × 3 = 21. No need to solve for a and b individually.
6. The expression (x² − 9)/(x − 3) simplifies to which of the following for x ≠ 3?
Explanation. x² − 9 = (x−3)(x+3). Dividing by (x−3) leaves x + 3 (valid for x ≠ 3).
7. If 1/x + 1/y = 5 and xy = 2, what is x + y?
Explanation. 1/x + 1/y = (x+y)/(xy). So (x+y)/2 = 5, giving x + y = 10.
8. For how many integer values of x is the inequality |2x − 5| < 7 satisfied?
Explanation. |2x−5|<7 means −7 < 2x−5 < 7, so −2 < 2x < 12, thus −1 < x < 6. Integers are 0,1,2,3,4,5 — that's 6 values.
9. If x and y are positive integers with x + y = 12 and x² − y² = 48, what is x?
Explanation. x² − y² = (x+y)(x−y) = 12(x−y) = 48, so x−y = 4. With x+y = 12, add: 2x = 16, x = 8.
10. If the system 2x + 3y = 12 and 4x + 6y = k has infinitely many solutions, what is the value of k?
Explanation. The second equation is exactly 2 times the first (4x+6y = 2(2x+3y)), so it must equal 2(12) = 24 for the lines to coincide and give infinitely many solutions.
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FAQ
What algebra topics appear most on GRE Quant?
Common topics include linear and quadratic equations, systems of equations, inequalities, absolute value, exponents and roots, factoring (especially the difference of squares and quadratic factoring), and functions. The GRE rewards recognizing structures like (a+b)(a−b) rather than brute-force solving.
How can I solve algebra problems faster on the GRE?
Look for shortcuts before grinding through algebra: use factoring identities (a²−b² = (a+b)(a−b)), substitute target expressions directly when possible, and on Quantitative Comparison try plugging in smart values (0, 1, negatives, fractions). Also, the on-screen calculator is for arithmetic, not for setting up the algebra.
Should I use plugging-in numbers or pure algebra?
Both are valid. Use algebra when the relationships are clear and direct. Use plugging-in (picking concrete values for variables) when answer choices contain variables, when the problem is abstract, or to verify a tricky inequality. Test edge cases like negatives and fractions to avoid false matches.