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How many integers lie in the domain of f(x) = sqrt(log₀.5((5 - 2x)/x))?

1. 1
2. 3
3. 2
4. 0

Correct answer: 1

Solution

Need log₀.5((5-2x)/x) >= 0. With base < 1 this requires 0 < (5-2x)/x <= 1. First, (5-2x)/x > 0 gives 0 < x < 5/2. Second, (5-2x)/x <= 1 => (5-3x)/x <= 0 => x <= 0 or x >= 5/3. Intersection: 5/3 <= x < 5/2, i.e. [1.667, 2.5). The only integer there is x = 2, so exactly 1 integer.

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