2. 外语
  3. If a < b < 0, which of the following numbers must be positive? Indicate all such numbers

If a < b < 0, which of the following numbers must be positive? Indicate all such numbers

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问题 If a < b < 0, which of the following numbers must be positive?
Indicate all such numbers
选项 A、a - b
B、a2-b2
C、ab
D、a2b
E、a2b + ab2
答案B，C
解析 In this question, you are given that a < b < 0 and are asked to determine which of the answer choices must be positive. Note that the condition a < b < 0 means that a and b are negative and that a     Choice A: a - b. In the question, it is given that a     Choice B: a2 - b2. Since a and b are negative, you can square both sides of the inequality a < b to get the inequality a2 > b2. Then you can subtract b2 from both sides of the inequality a2 > b2 to conclude that a2 - b2 > 0. So a2 - b2 must be positive. Alternatively, note that a2 - b2 can be factored as(a - b)(a + b). The factor a - b is, Choice A, which must be negative, and the factor a + b is the sum of two negative numbers, which also must be negative. Thus, a2 - b2 is the product of two negative numbers, so it must be positive.
    Choice C: ab. Because a and b are negative, you can conclude that their product ab must be positive.
    Choice D: a2b. Because a2b can be written as(a)(a)(b), which is the product of three negative numbers, you can conclude that a2b must be negative.
    Choice E: a2b + ab2. By the reasoning in the explanation of Choice D, Choice E is the sum of two negative numbers. Therefore, you can conclude that a22b + ab2 must be negative.
    Choices B and C must be positive, and Choices A, D, and E must be negative. The correct answer consists of Choices B and C.
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