How many answers should you select?
ETS distinguishes questions that ask for one answer from those that allow multiple answers. For the latter, the prompt may specify the number of choices or ask you to select all that apply. Sometimes only one answer is correct. The official instructions require a complete selection with no extra choices to earn credit.
Before calculating, also identify the kind of question. “Which values are possible?” asks for values that satisfy the conditions. “Which statements must be true?” asks for statements that hold in every allowed case. A confirming example may answer the first question, but it does not prove the second.
Exercise 1: a strict inequality
First describe the solution set, then compare the answer choices with it. An excluded endpoint remains excluded even if it is very close to a solution.
The integer n satisfies |2n − 3| < 7. Which of the following could be n? Select all that apply.
Select all correct answers.
The solution, step by step
B, C, and D: −1, 0, and 4.
The method in action
- The inequality is equivalent to −7 < 2n − 3 < 7, then −4 < 2n < 10, and finally −2 < n < 5. Of the choices, −1, 0, and 4 fall in this interval.
- The endpoints −2 and 5 give an absolute value of 7, so they are excluded by the strict inequality. For 6, the absolute value is 9. The other integers in the interval that are not listed do not need to be selected.
Exercise 2: translate a double condition
The word “strictly,” repeated in the prompt, excludes both endpoints. You can calculate the interval once instead of recomputing the full average for each choice.
The arithmetic mean of x, 4, and 9 is strictly greater than 5 and strictly less than 7. Which of the following values could be x? Select all that apply.
Select all correct answers.
The solution, step by step
B, C, and D: 3, 5, and 7.
The method in action
- Write 5 < (x + 13) / 3 < 7. Multiplying by 3 preserves the inequalities: 15 < x + 13 < 21. Therefore, 2 < x < 8.
- The values 3, 5, and 7 are possible. With 2, the average would be exactly 5; with 8, exactly 7; and with 9, it would exceed 7. The prompt does not ask x to equal the average; x is one of the three data values.
Exercise 3: test each statement
When the unknown ranges over an interval, a statement that is true for one value is not necessarily true for all the others. Look for a counterexample to choices that narrow the resulting interval further.
The real number x satisfies x² < x. Which statements must be true? Select all that apply.
Select all correct answers.
The solution, step by step
A, B, and D.
The method in action
- x² < x is equivalent to x(x − 1) < 0. The factors have opposite signs when 0 < x < 1. Therefore, A and B must be true.
- Multiplying x² < x by x, which is strictly positive, gives x³ < x², so D is true. For C, x = 3/4 satisfies the prompt but not the statement; this counterexample is enough to rule it out.
- For 0 < x < 1, the reciprocal 1/x is greater than 1, so E is false. Dividing too early without knowing the sign of x can obscure this reasoning.
Exercise 4: accept a single selection
Checkboxes describe a possible format. They do not indicate how many correct answers a particular exercise has.
The positive number x satisfies (x − 2)(x + 5) = 0. Which of the following could be x? Select all that apply.
Select all correct answers.
The solution, step by step
D only: 2.
The method in action
- The product is zero when x = 2 or x = −5. The positive condition rules out −5. None of the other choices makes either factor zero.
- Selecting both algebraic roots would ignore part of the prompt. Your final check must cover every condition, not just the equation.
The pitfall
Stopping as soon as one answer works
Confirming one value and immediately opening the explanation repeats the habit of a single-answer question. In this format, you are not done until you have assessed every choice. Make one final pass down the list to check that you have not simply missed an answer.
Conversely, selecting an extra value “just in case” does not protect a correct answer. If a choice is uncertain, return to the domain or construct a counterexample. Uncertainty is not a mathematical reason to select it.
How can you correct an incomplete or overinclusive selection?
Separate calculation errors from selection errors. If your interval was correct but your list was incomplete, practicing the operations will not solve the problem; you need to review each answer choice. If the interval was wrong, revisit the transformation that changed an endpoint or sign.
In your practice notes, write a short reason for rejecting each choice, then compare with the worked solution. Next, try a different question with the same point of difficulty. These exercises provide feedback on the skills practiced, not a conversion to a GRE score.
Key takeaways
- Identify the domain and the number of answers requested.
- Establish the solution set before selecting answers when possible.
- Look for counterexamples to universal statements.
- Check every choice, including those left blank.