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  3. If there is a least integer that satisfies the inequality 9/X > 2, what is that least integer?

If there is a least integer that satisfies the inequality 9/X > 2, what is that least integer?

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问题 If there is a least integer that satisfies the inequality 9/X > 2, what is that least integer?
选项 A、0
B、1
C、4
D、5
E、There is not a least integer that satisfies the inequality.
答案B
解析 It is clear that no negative integer satisfies the inequality (because 9/negative > 2 is false) and zero does not satisfy the inequality (because 9/0 is undefined). Thus, the integers, if any, that satisfy 9/x > 2 must be among 1,2,3,4, ....The least of these integers is 1, and it is easy to see that x = 1 satisfies the inequality 9/x  ≥ 2. Therefore, the least integer that satisfies the inequality is 1.
Alternatively, the inequality can be solved algebraically. It will be convenient to consider three cases according to whether x < 0, x = 0, and x>0.
Case 1: Assume x < 0.Then multiplying both sides of the inequality by x, which is negative, gives 9 ≤ 2x, or x≥ 4.5. Because we are assuming x < 0, there are no solutions to x ≥ 4.5. Therefore, no solutions exist in Case 1.
Case 2: Assume x = 0. Then 9/x is not defined, and thus x = 0 cannot be a solution.
Case 3: Assume x > 0. Then multiplying both sides of the inequality by x, which is positive, gives 9 ≥ 2x, or x ≤ 4.5. Because we are assuming x > 0, the solutions that exist in Case 2 are all real numbers x such that 0 < x ≤ 4.5.
The set of all solutions to the inequality 9/x > 2x will be all solutions found in Cases 1,2, and 3.
Therefore, the solutions to the inequality consist of all real numbers x such that 0 < x ≤ 4.5. The least of these solutions that is an integer is 1.
The correct answer is B.
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