We're looking for values of x such that: \(0 < \frac{x^2}{4} + \frac{x}{2} -4 < 4\)

To eliminate the fractions, we can multiply all sides of the inequality by 4 (the LCM of 2 and 4)

When we do this, we get: \(0 < x^2 + 2x - 16 < 16\)

Add 16 to all sides to get: \(16 < x^2 + 2x < 32\)

Factor the middle to get: \(16 < x(x + 2) < 32\)

At this point, it's probably fastest to just check the answer choices....

Answer choice A doesn't work
Answer choice B doesn't work
Answer choice C doesn't work
Answer choice D WORKS
Answer choice E doesn't work
Answer choice F doesn't work


Answer: D

Cheers,
Brent