Given: \[ P=\frac{u}{2}+\sqrt{\frac{4}{u+1}} \] 1. For what value(s) of \( u \) will \( P \) be a real number?

1. To ensure \( P = \frac{u}{2} + \sqrt{\frac{4}{u+1}} \) is real, the expression under the square root must be nonnegative. 2. The expression under the square root is \( \frac{4}{u+1} \). Since the numerator is positive (\(4 > 0\)), the fraction is nonnegative if and only if the denominator is positive. 3. Therefore, we require: \[ u+1 > 0 \] Solving the inequality: \[ u > -1 \] 4. Also, note that \( u = -1 \) would make the denominator zero, which is not allowed. Hence, it is excluded. 5. Thus, \( P \) is real for: \[ u > -1 \]