Final answer:
The greatest possible integer value of x for which the square root of √x - 5 is an imaginary number is 4, since x must be an integer less than 5 for the expression to be negative and thus yield an imaginary number.
Step-by-step explanation:
The question requires us to determine the maximum integer value of x for which the expression √x - 5 will result in an imaginary number. We know that the square root of a negative number yields an imaginary result. Thus, we need to find the greatest integer value of x such that when we subtract 5, the result is negative.
If √x - 5 is imaginary, then x - 5 must be less than zero. To find the largest integer value of x where this is true, we can solve the inequality x - 5 < 0, which simplifies to x < 5. Therefore, since we are looking for the largest integer value of x, the greatest possible integer value just below 5 is 4.