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How many integers are there satisfying 1 < [3 - x/2] < 4?

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How many integers are there satisfying 1 < [3 - x/2] < 4?

by Max@Math Revolution » Tue Jan 28, 2020 11:52 pm
[GMAT math practice question]

How many integers are there satisfying 1 < [3 - x/2] < 4?
([x] means the greatest integer less than or equal to x)

A. 2
B. 3
C. 4
D. 5
E. 6
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Re: How many integers are there satisfying 1 < [3 - x/2] < 4?

by Max@Math Revolution » Fri Jan 31, 2020 1:44 am
=>

Since 1 < [3 - x/2] < 4, we have [3 - x/2] = 2 or [3 - x/2] = 3.

If [3 - x/2] = 2, then we have 2 ≤ 3 - x/2 < 3 or -1 ≤ - x/2 < 0 (subtracting 3). This is equivalent to 0 < x ≤ 2 (multiplying by -2, which changes the direction of the inequality signs).
If [3 - x/2] = 3, then we have 3 ≤ 3 - x/2 < 4 or 0 ≤ - x/2 < 1 (subtracting 3). This is equivalent to -2 < x ≤ 0 (multiplying by -2, which changes the direction of the inequality signs).

Thus, we have -2 < x ≤ 2 and the integer values of x are -1, 0, 1, and 2.
We have 4 integer solutions.

Therefore, C is the answer.
Answer: C
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