GRE Algebra: Equations, Inequalities, and Functions (With Common Traps)

As of 2026, the GRE Quantitative Reasoning measure has two scored sections: Section 1 has 12 questions in 21 minutes; Section 2 has 15 questions in 26 minutes. The test is section‑level adaptive, so strong accuracy in the first Quant section leads to a harder second section with more scoring potential. You get an on‑screen basic calculator during Quant, and you can review, change, and mark questions within a section. Official scores typically post about 8–10 days after test day. With fewer total questions than the old format, each algebra problem carries more weight—making clean setup and trap‑proof execution essential.

On GRE Quant, algebra shows up as equations, inequalities, and functions across multiple‑choice (single or multiple answers), Numeric Entry, and Quantitative Comparison. The fastest path to points is consistent translation: 1) Define variables exactly (including units and any domain constraints such as x > 0). 2) Restate the question in algebra before touching the calculator. 3) Choose a solution path (solve symbolically, pick numbers that obey constraints, or compare expressions directly for QC). 4) Verify against the question stem and domain before submitting.

Core patterns: percent/ratio equations, linear systems, quadratics, exponent and radical equations, and rational equations. High‑frequency traps include canceling across addition (illegal), dividing by an expression that might be zero, and creating extraneous solutions by squaring or taking reciprocals. Quick example: If (x + 2)/(x − 3) = 4, note the domain x ≠ 3 before cross‑multiplying. Then x + 2 = 4x − 12, so 14 = 3x and x = 14/3, which is allowed because it doesn’t violate the restriction.

Quadratics can often be handled faster by factoring or by recognizing structure. For example, if x^2 − 9x + 14 = 0, factor to (x − 7)(x − 2) = 0 for x = 7 or x = 2. Radical equations invite extraneous roots: if √(2x + 3) = x − 1, first require x − 1 ≥ 0 (so x ≥ 1), then square to get 2x + 3 = x^2 − 2x + 1. Solve x^2 − 4x − 2 = 0 to get x = 2 ± √6; only x = 2 + √6 satisfies x ≥ 1 and the original equation. Always check radical and reciprocal equations back in the original.

Before cross‑multiplying, state the forbidden values that make denominators zero; no value that violates them can be an answer even if it solves the transformed equation. To solve expressions like (a)/(x − k) = (b)/(x − m), record x ≠ k, m, clear denominators carefully, and check the final values against the original restrictions. The GRE loves trap answers that equal a forbidden value.

For linear inequalities, isolate the variable—but remember that multiplying or dividing by a negative flips the inequality sign. For |ax + b| ≤ c, translate to −c ≤ ax + b ≤ c and solve as a compound inequality; for |ax + b| ≥ c, expect a union of two ranges. For rational inequalities like (x − 4)/(x + 1) < 0, use a sign chart: mark the zeros and vertical asymptotes (x = 4 and x = −1), test intervals, and exclude points where the expression is undefined. Strict inequalities produce open endpoints; non‑strict give closed endpoints where defined.

Absolute value: Solve |2x − 5| < 7. Translate to −7 < 2x − 5 < 7, add 5 to get −2 < 2x < 12, divide by 2 to get −1 < x < 6. Rational sign logic: Solve 1/(x − 3) < 0. The numerator is positive, so the fraction is negative exactly when the denominator is negative, which is x − 3 < 0, i.e., x < 3. Exclude x = 3 since the expression is undefined.

Expect evaluation and composition, simple transformations, piecewise definitions, and domain/range reasoning. Keep these in view: 1) f(g(x)) is not g(f(x)); order matters. 2) If a definition includes a denominator, square root, or logarithm, write the domain before substituting. 3) For linear functions f(x) = mx + b, differences reveal m quickly: if f(2) − f(0) = 10, then 2m = 10 so m = 5. If also f(3) = 19, then 5·3 + b = 19 giving b = 4 and f(5) = 29. 4) For piecewise functions, check which rule applies at the exact input and whether endpoints are open or closed.

Composition example: Let f(x) = x^2 − 1 and g(x) = 2x + 3. Then f(g(−2)) = f(−1) = 0, but g(f(−2)) = g(3) = 9. On the GRE, many wrong choices assume the two are equal. Piecewise example: h(x) = x^2 for x ≤ 1 and h(x) = 2x + 1 for x > 1. Then h(1) = 1 from the first rule (closed at 1), while h(1^+) = 3 from the second—endpoints matter when answering limit‑style or equality questions.

QC rewards strategic comparison rather than full solutions. Use legal test numbers that satisfy any constraints, and include edge cases such as negatives, fractions between 0 and 1, and 0 itself when allowed. Example: Compare Quantity A: √(x^2) and Quantity B: x for real x. If x = 3, they’re equal; if x = −3, Quantity A is 3 while Quantity B is −3, so A > B. Because results differ across valid values, the relationship cannot be determined from the information given. A classic trap answer here is “equal.”

For multiple‑answer items, you must select all correct choices and only those. Read constraints and look for ranges or patterns that let you check choices quickly. For Numeric Entry, the answer box labels (units, percent vs decimal) and rounding instructions matter; if rounding is not specified, provide the exact value. Many trap answers are rounded or in the wrong unit.

The calculator is basic and available during Quant, but many questions are designed so reasoning or estimation is faster and safer. Use it for long arithmetic or square roots, not for every step. With 21 minutes for 12 questions and 26 minutes for 15 questions, your average is about a minute and three‑quarters per problem. Practical approach: 1) First pass answers the quick wins and marks time sinks. 2) Second pass tackles set‑up‑heavy algebra now that you understand the stem. 3) Final minute checks domains, signs, and whether the requested quantity (not its reciprocal or square) is what you’re about to submit. Early accuracy matters for the adaptive second section.

Start with Exambank’s diagnostic to see how you currently handle equations, inequalities, and functions; the platform tags errors by concept and by trap type (for example, flipped‑sign inequality or extraneous‑solution miss). In the Learn phase, review focused lessons such as function notation and absolute‑value inequalities. Then use Solve Together to walk step‑by‑step through GRE‑style algebra problems and compare your setup to expert translations. Finish with Test Yourself mini‑quizzes that mirror the 12‑question and 15‑question section lengths, so pacing feels natural. As you practice, Exambank’s adaptive engine serves more of your “tricky bits,” and the analytics show accuracy by subtopic, time per question, and which traps you’re still falling for—making your next session more targeted than the last.

1. Dividing by an expression that could be zero or canceling across a sum. 2) Forgetting to reverse the inequality when multiplying or dividing by a negative. 3) Treating √(x^2) as x instead of |x|. 4) Ignoring domain restrictions before and after solving (especially with denominators, square roots, and logs). 5) Not checking for extraneous solutions after squaring or clearing fractions. 6) Mixing up f(g(x)) with g(f(x)). 7) Selecting a single correct option on multiple‑answer items. 8) Giving a rounded or unit‑mismatched value on Numeric Entry. 9) Assuming a diagram is to scale (only coordinate graphs and data displays are). 10) Answering the intermediate value instead of the quantity the question actually asks for.

Write the domain, translate, choose the fastest valid method, compute cleanly, and verify against both the original stem and your domain. If time is tight, at least do the domain and verification steps—those two alone prevent a surprising number of wrong clicks.

Algebra on the GRE is less about heavy computation and more about crisp translation and error‑free execution. Hone a repeatable setup, watch the classic traps, and your accuracy will climb quickly. If you’re ready to turn algebra into a scoring advantage with adaptive practice that targets your exact weak spots, sign up to Exambank today.