To determine the values of [tex]\( x \)[/tex] for which the product [tex]\( \sqrt{5x} \cdot \sqrt{x+4} \)[/tex] is defined, we need to ensure that both square roots are defined. This means both expressions under the square roots must be non-negative.
### Step 1: Determine the condition for [tex]\( \sqrt{5x} \)[/tex] to be defined.
The expression [tex]\( \sqrt{5x} \)[/tex] is defined if and only if:
[tex]\[ 5x \geq 0 \][/tex]
Solving this inequality:
[tex]\[ x \geq 0 \][/tex]
### Step 2: Determine the condition for [tex]\( \sqrt{x+4} \)[/tex] to be defined.
The expression [tex]\( \sqrt{x+4} \)[/tex] is defined if and only if:
[tex]\[ x + 4 \geq 0 \][/tex]
Solving this inequality:
[tex]\[ x \geq -4 \][/tex]
### Step 3: Combine both conditions.
For the product [tex]\( \sqrt{5x} \cdot \sqrt{x+4} \)[/tex] to be defined, both inequalities must be satisfied simultaneously:
[tex]\[ x \geq 0 \quad \text{and} \quad x \geq -4 \][/tex]
The more restrictive condition here is [tex]\( x \geq 0 \)[/tex], because it encompasses the requirement [tex]\( x \geq -4 \)[/tex] (i.e., if [tex]\( x \geq 0 \)[/tex], it is naturally also [tex]\( \geq -4 \)[/tex]).
### Conclusion:
Combining these results, the inequality that represents all values of [tex]\( x \)[/tex] for which the product [tex]\( \sqrt{5x} \cdot \sqrt{x+4} \)[/tex] is defined is:
[tex]\[ x \geq 0 \][/tex]
The correct answer is:
[tex]\[
\boxed{x \geq 0}
\][/tex]
