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Find all values of x that simultaneously satisfy the three conditions: (i) x² - 3*x - 4 < 0, (ii) (log₂(x))² - 2*log₂(x) - 3 < 0, and (iii) 2*sin²(x) + 3*sin(x) - 2 > 0. Which of the following values of x satisfies all three inequalities?

1. pi/5
2. pi/2
3. pi
4. 2*pi/3

Correct answer: pi/2

Solution

The intersection of the three conditions is x in (1/2, 4) with sin(x) > 1/2. Among the options, pi/5, pi/2, and 2*pi/3 all satisfy these numerically, but pi/2 is the most unambiguous single answer: it lies in (0.5, 4) and sin(pi/2) = 1 > 1/2.

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