To determine the values of [tex]\( x \)[/tex] for which the product [tex]\( \sqrt{x-5} \cdot \sqrt{x+2} \)[/tex] is defined, we must ensure that both square roots are defined, which means that the expressions inside both square roots must be non-negative.
Step-by-step solution:
1. Expression under the first square root:
[tex]\[
\sqrt{x - 5}
\][/tex]
For this square root to be defined, the expression inside the square root must be greater than or equal to zero:
[tex]\[
x - 5 \geq 0
\][/tex]
Solving this inequality:
[tex]\[
x \geq 5
\][/tex]
2. Expression under the second square root:
[tex]\[
\sqrt{x + 2}
\][/tex]
For this square root to be defined, the expression inside the square root must be greater than or equal to zero:
[tex]\[
x + 2 \geq 0
\][/tex]
Solving this inequality:
[tex]\[
x \geq -2
\][/tex]
3. Combining both conditions:
For the product [tex]\( \sqrt{x-5} \cdot \sqrt{x+2} \)[/tex] to be defined, both individual conditions must be satisfied simultaneously. Therefore, [tex]\( x \)[/tex] must satisfy both:
[tex]\[
x \geq 5 \quad \text{and} \quad x \geq -2
\][/tex]
Since [tex]\( x \geq 5 \)[/tex] is more restrictive than [tex]\( x \geq -2 \)[/tex], we take the more restrictive condition. Thus:
[tex]\[
x \geq 5
\][/tex]
Therefore, the inequality representing all values of [tex]\( x \)[/tex] for which [tex]\( \sqrt{x-5} \cdot \sqrt{x+2} \)[/tex] is defined is:
[tex]\[
\boxed{x \geq 5}
\][/tex]
This corresponds to option D.
Answer: D. [tex]\( x \geq 5 \)[/tex]
