To determine the values of [tex]\( x \)[/tex] for which the product [tex]\( \sqrt{5x} \cdot \sqrt{x + 3} \)[/tex] is defined, we need to consider the domain of the square root functions involved.
1. First Square Root: [tex]\( \sqrt{5x} \)[/tex]
The expression inside the square root, [tex]\( 5x \)[/tex], must be non-negative for the square root to be defined. This means:
[tex]\[
5x \geq 0
\][/tex]
Solving this inequality, we divide both sides by 5:
[tex]\[
x \geq 0
\][/tex]
2. Second Square Root: [tex]\( \sqrt{x + 3} \)[/tex]
The expression inside this square root, [tex]\( x + 3 \)[/tex], must also be non-negative. This means:
[tex]\[
x + 3 \geq 0
\][/tex]
Solving this inequality, we subtract 3 from both sides:
[tex]\[
x \geq -3
\][/tex]
3. Combining the Conditions
For the product [tex]\( \sqrt{5x} \cdot \sqrt{x + 3} \)[/tex] to be defined, both conditions [tex]\( x \geq 0 \)[/tex] and [tex]\( x \geq -3 \)[/tex] must be satisfied simultaneously.
The condition [tex]\( x \geq 0 \)[/tex] is more restrictive than [tex]\( x \geq -3 \)[/tex]. Therefore, we must use the more restrictive condition to find the domain where both inequalities hold true.
4. Conclusion
The inequality [tex]\( x \geq 0 \)[/tex] encompasses all the values that make both square roots defined.
Thus, the correct inequality representing all values of [tex]\( x \)[/tex] for which the product [tex]\( \sqrt{5x} \cdot \sqrt{x + 3} \)[/tex] is defined is:
[tex]\[
\boxed{x \geq 0}
\][/tex]
