题目信息
Is x2 − y2 a positive number?
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B:Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D:EACH statement ALONE is sufficient.
E:Statements (1) and (2) TOGETHER are NOT sufficient.
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已选答案：
正确答案：
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Algebra Factoring
Since x2 − y2 = (x + y)(x − y), it follows that x2 − y2 will be a positive number if both x + y and x − y are positive numbers or if both are negative numbers.
It is given that x − y is a positive number. If, for example, x = 2 and y = −3, then x − y is a positive number, but x + y is not a positive number, so x2 − y2 is not a positive number. On the other hand, if x = 3 and y = 1, then x − y is a positive number and x + y is a positive number, so x2 − y2 is a positive number; NOT sufficient. It is given that x + y is a positive number. If, for example, x = −2 and y = 4, then x + y is a positive number, but x − y is not a positive number, so x2 − y2 is not a positive number. On the other hand, if x = 3 and y = 1, then x + y is a positive number and x − y is a positive number, so x2 − y2 is a positive number; NOT sufficient.
Taking (1) and (2) together, both x + y and x − y are positive numbers, so x2 − y2 is a positive number.
The correct answer is C;both statements together are sufficient.
Since x2 − y2 = (x + y)(x − y), it follows that x2 − y2 will be a positive number if both x + y and x − y are positive numbers or if both are negative numbers.
It is given that x − y is a positive number. If, for example, x = 2 and y = −3, then x − y is a positive number, but x + y is not a positive number, so x2 − y2 is not a positive number. On the other hand, if x = 3 and y = 1, then x − y is a positive number and x + y is a positive number, so x2 − y2 is a positive number; NOT sufficient. It is given that x + y is a positive number. If, for example, x = −2 and y = 4, then x + y is a positive number, but x − y is not a positive number, so x2 − y2 is not a positive number. On the other hand, if x = 3 and y = 1, then x + y is a positive number and x − y is a positive number, so x2 − y2 is a positive number; NOT sufficient.
Taking (1) and (2) together, both x + y and x − y are positive numbers, so x2 − y2 is a positive number.
The correct answer is C;both statements together are sufficient.
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