The Data Sufficiency Decision
A student is solving a data sufficiency problem: 'What is the value of x?' Statement (1): x² = 16. Statement (2): x is positive. The student must determine whether the statements provide sufficient information to answer the question. The student knows that in data sufficiency, each statement should be considered independently, and the answer choices are: A) Statement (1) alone is sufficient, but statement (2) alone is not; B) Statement (2) alone is sufficient, but statement (1) alone is not; C) Both statements together are sufficient, but neither alone is; D) Each statement alone is sufficient; E) Statements together are not sufficient.
1. Evaluate each statement independently. Does statement (1) alone give a unique value for x? Does statement (2) alone give a unique value for x? Based on that, what is the correct answer choice?
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Statement (1): x² = 16 → x = 4 or x = -4. Not unique, so not sufficient alone. Statement (2): x is positive → x could be any positive number. Not unique, so not sufficient alone. Together: x² = 16 and x > 0 → x = 4. Unique, so together sufficient. Therefore, the correct answer is C.
2. Suppose the question is changed to 'What is the value of x?' and the statements are: (1) x³ = 27, (2) x is an integer. Re-evaluate. Does either statement alone suffice? What is the answer choice?
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Statement (1): x³ = 27 → x = 3 (since cube root is unique for real numbers). Sufficient alone. Statement (2): x is an integer → many possibilities. Not sufficient alone. Since statement (1) alone works, the answer is A.