GRE algebra strategies

GRE algebra tests linear equations, quadratic equations, factoring, functions, and the quantitative comparison format where you decide which quantity is greater instead of solving for a single value. These strategies cover setting up equations before solving, the trap of assuming a variable is positive, and a worked example you can follow step by step.

The core principle

Set up the equation before you solve it. Most algebra errors happen in the translation from words to symbols, not in the solving. If a problem says a number is 5 less than twice another number, the equation is $x = 2y - 5$ before you touch a single operation. Define your variables, write the equation, then solve. In quantitative comparison questions, the same principle applies: translate both columns into expressions before you compare them.

Quantitative comparison: test the edge cases

Many GRE algebra questions use the quantitative comparison format: Column A versus Column B, and you choose whether A is greater, B is greater, they are equal, or the answer cannot be determined. The fastest approach is to plug in numbers. Test a positive integer, zero, a negative number, and a fraction. If the relationship changes across cases, the answer is D (cannot be determined). If the relationship holds for every case, commit to it.

The most common trap

Assuming a variable is positive

The question says $x^2 = 25$. You conclude $x = 5$, so Column A equals 5. But $x$ could also be $-5$. In a quantitative comparison, this changes the answer from a definite comparison to D (cannot be determined). The test writers include the answer that assumes $x$ is positive because they know this mistake. Always consider negative values when a variable is squared or when the problem does not state that the variable is positive.

Step-by-step strategy

When to use: any algebra question, including quantitative comparison and multiple-choice formats.

1Define variables for every unknown, and write down what the question asks for separately.
2Translate the words into equations, one sentence at a time. For quantitative comparison, translate both columns first.
3Solve using substitution, elimination, or factoring. For quantitative comparison, test a positive, zero, a negative, and a fraction.
4Verify the answer matches what the question asked for. If any test case changes the relationship, choose D.

When not to use: when the problem is pure arithmetic or when you can backsolve from the answer choices faster than setting up.

Worked example

Quantitative comparison: $x$ is a positive integer such that $x^2 < 30$ . | Column A | Column B | |---|---| | $x$ | 5 | Step 1: Translate. Column A is $x$ , Column B is 5. Step 2: Find the constraint. $x$ is a positive integer and $x^2 < 30$ , so $x$ can be 1, 2, 3, or 5 (since $5^2 = 25 < 30$ ). Wait: $5^2 = 25$ , which is less than 30, so $x = 5$ is valid. And $6^2 = 36$ , which is not less than 30, so the maximum is 5. Step 3: Test. When $x = 5$ , the columns are equal. When $x = 1, 2, 3$ , Column A is less than Column B. The relationship changes depending on the value of $x$ . Step 4: The answer is D (cannot be determined), because sometimes Column A is smaller and sometimes they are equal.

Algebra practice questions

Free practice questions with full explanations.

GRE practice

Take a full-length GRE exam for free.

Arithmetic strategies

Integer properties, divisibility, and edge cases.

Geometry strategies

Drawing the figure and the scale trap.

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