What must a counterexample show?
FNEGE describes the Data Sufficiency subtest as evaluating whether additional information is sufficient. To practice this idea, our method is to find two situations allowed by the prompt and by the one piece of information being tested. If they yield different answers to the question, that information is insufficient.
Each case must respect all remaining constraints: integer or real, positive or unrestricted, figure dimensions, and any known total. But when testing statement (1) alone, ignore statement (2). To prove that (1) alone is sufficient, your reasoning must also cover cases where (1) is true and (2) is not.
This asymmetry is useful: two different results prove insufficiency, but two trials with the same result do not prove sufficiency. Sufficiency requires an argument that covers every admissible case. Often, you can provide one using an equation, property, or relationship directly tied to what is being asked.
How can two signs help you build a counterexample?
The answer choices in this workshop are written out in full to focus on the reasoning. For the complete subtest format, continue with the Data Sufficiency guide.
x est un nombre réel. Quelle est la valeur de x ? Information (1) : (x − 4)² = 25. Information (2) : x > 5.
Choose an answer.
The solution, step by step
Together, the statements are sufficient; neither alone is sufficient.
The method in action
- With (1) alone, x − 4 can be 5 or −5, so x = 9 or x = −1. The answers differ, so (1) is insufficient.
- With (2) alone, x = 6 and x = 7 are both possible. It is therefore insufficient too. Do not carry (x − 4)² = 25 into this step.
- With both statements, x is in {−1, 9} and must be greater than 5. Only 9 remains, so the answer is determined.
Do you need to know x to determine x²?
Changing the quantity being asked for can change the sufficiency verdict. Read the question before trying to determine every unknown.
x est un nombre réel. Quelle est la valeur de x² ? Information (1) : la valeur absolue de x est 5. Information (2) : x < 0.
Choose an answer.
The solution, step by step
(1) alone is sufficient.
The method in action
- With (1), x is 5 or −5. In either case, x² = 25. This covers all possibilities, so (1) is sufficient for the question asked.
- With (2), x = −1 gives x² = 1, while x = −2 gives x² = 4. The two permitted cases produce different answers, so (2) is insufficient.
- Uncertainty about the sign of x does not prevent you from determining its square. Finding x itself would answer a different question.
The pitfall
A forbidden case does not disprove a conclusion
If x is specified as strictly positive, −3 is not an admissible counterexample. If x is an integer, 1.5 is excluded. Check the domain of the unknown before drawing a conclusion. A striking example that violates a constraint does not prove insufficiency.
Also, do not confuse different calculation paths with different answers. What must vary is the requested result. Two different configurations may produce the same perimeter, sum, or yes/no answer.
How can you choose test cases without guessing randomly?
Start with possibilities left open by the information: sign, ordering of two numbers, zero if allowed, equality, or inequality. Use simple numbers. The goal is to vary the answer while keeping the statement being tested true, not to create complicated calculations.
In your review, keep one line per statement: case A, case B, and the results. If you conclude that a statement is sufficient, replace the list of trials with a general proof. This distinction makes your verdict verifiable even when you do not need to calculate the final value.
Key takeaways
- Two admissible cases with different answers prove insufficiency.
- Test each statement alone without carrying over the other.
- Sufficiency concerns the quantity requested, not every unknown.