2. 外语
  3. DIn this question, you are asked to compare 2x with 3x+1, given that x is a negative integer. One way to approach this problem i

DIn this question, you are asked to compare 2x with 3x+1, given that x is a negative integer. One way to approach this problem i

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答案D
解析 In this question, you are asked to compare 2x with 3x+1, given that x is a negative integer. One way to approach this problem is to plug a value of x in both expressions and compare the results.
    You are given that x is a negative integer, so the greatest integer you can plug in for x is -1.
    For x = -l, it follows that 2x = 2-1 = 1/2 and 3x+1 =3-1+1 =30= 1.
    In this case, 2x is less than 3x+1. However, to conclude that Quantity B is greater, it is not sufficient for 2x to be less than 3x+1 for one particular value of x; the relationship would need to be true for all negative integer values of x. To analyze this relationship further, plug in another value of x, for example, -2.
    For x = -2, it follows that 2x = 2-2 == 1/4 and.
    Again, 2x is less than 3x+1 but note that these values are closer together than the previous values of 2x and 3x+1. It appears that the relationship between the quantities may differ for smaller values of x, so now try plugging in —3 for x.
    For x = -3, it follows that 2x = 2-3 ==1/8 and 3x+1 = 3-3+1 = 3-2 == 1/9.
    In this case, 2x is greater than 3x+1.
    Since 2x is less than 3x+1 for x = -1 and 2x is greater than 3x+1 for x = — 3, the relationship between these two quantities cannot be determined from the information given. The correct answer is Choice D.
    Since both quantities are algebraic expressions, another way to approach the comparison is to set up a placeholder relationship, denoted by, between the two quantities and then to simplify to see what conclusions you can draw. As you simplify and draw conclusions, keep in mind that x is a negative integer.

For any value of x(including negative integer values of x), the value of 3x is positive, so dividing by 3x does not affect any inequality that might be represented by the placeholder. Since each step in this simplification is reversible, the simplification reduces the problem to comparingwith 3, given that x is a negative integer. Note that, where n = —x; so the problem can be reduced further to comparingwith 3, given that n is a positive integer.
    Because 3/2 is greater than 1, the value ofbecomes greater as n becomes larger. For small values of n,is less than 3, but for large values of n,is greater than 3. Therefore, the relationship between Quantity A and Quantity B cannot be determined from the given information, and the correct answer is Choice D.
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