To determine the greatest possible integer value of [tex]\( x \)[/tex] for which [tex]\( \sqrt{x-5} \)[/tex] is an imaginary number, let's follow these steps:
1. Understanding Imaginary Numbers: A square root function produces an imaginary number whenever the value inside the square root is negative. Therefore, we need [tex]\( x-5 \)[/tex] to be less than 0 for [tex]\(\sqrt{x-5}\)[/tex] to be imaginary.
2. Setting Up the Inequality: To find when [tex]\(\sqrt{x-5}\)[/tex] is imaginary, we set up the inequality:
[tex]\[
x - 5 < 0
\][/tex]
3. Solving the Inequality: Solving the inequality for [tex]\( x \)[/tex]:
[tex]\[
x - 5 < 0 \implies x < 5
\][/tex]
This tells us that [tex]\( x \)[/tex] must be less than 5 for the square root to be imaginary.
4. Identifying the Greatest Integer: We are looking for the greatest integer that satisfies the inequality [tex]\( x < 5 \)[/tex]. The greatest integer less than 5 is 4.
Thus, the greatest possible integer value of [tex]\( x \)[/tex] for which [tex]\(\sqrt{x-5}\)[/tex] is an imaginary number is:
[tex]\[
\boxed{4}
\][/tex]
