The expression $\sqrt{x - 5}$ will be an imaginary number when $x - 5$ is negative, because the square root of a negative number is imaginary.

To find the greatest possible integer value of $x$ for which $\sqrt{x - 5}$ is imaginary, we need to solve the inequality:

$x - 5 < 0$

Solving for $x$:

$x < 5$

The greatest integer less than 5 is $4$. Therefore, the greatest possible integer value of $x$ that makes $\sqrt{x - 5}$ imaginary is $4$.

Answer: 4

Would you like more details or have any questions? Here are some related questions to consider:

1. What happens when $x = 5$ for $\sqrt{x - 5}$?
2. What is the smallest possible integer value of $x$ for which $\sqrt{x - 5}$ is real?
3. Can $\sqrt{x - 5}$ ever be undefined?
4. How do we define imaginary numbers in general?
5. What are some applications of imaginary numbers in real-world problems?

Tip: Imaginary numbers arise when taking square roots of negative numbers, and they form the basis of complex numbers.