Answer :
To determine which inequality represents all values of [tex]\( x \)[/tex] for which the product [tex]\(\sqrt{5x} \cdot \sqrt{x+3}\)[/tex] is defined, we need to consider the conditions under which each square root expression is defined.
1. First Expression: [tex]\(\sqrt{5x}\)[/tex]
A square root [tex]\(\sqrt{5x}\)[/tex] is defined only when the expression inside the square root is non-negative. So, we need:
[tex]\[ 5x \geq 0 \][/tex]
Dividing both sides by 5:
[tex]\[ x \geq 0 \][/tex]
2. Second Expression: [tex]\(\sqrt{x+3}\)[/tex]
Similarly, the square root [tex]\(\sqrt{x+3}\)[/tex] is defined only when the expression inside is non-negative. Thus, we need:
[tex]\[ x+3 \geq 0 \][/tex]
Subtracting 3 from both sides:
[tex]\[ x \geq -3 \][/tex]
3. Combining the Inequalities
The product [tex]\(\sqrt{5x} \cdot \sqrt{x+3}\)[/tex] is only defined if both individual square roots are defined. Therefore, [tex]\( x \)[/tex] must satisfy both inequalities simultaneously:
[tex]\[ x \geq 0 \quad \text{and} \quad x \geq -3 \][/tex]
The more restrictive condition here is [tex]\( x \geq 0 \)[/tex]. In other words, if [tex]\( x \geq 0 \)[/tex], then it automatically satisfies the [tex]\( x \geq -3 \)[/tex] condition as well.
Therefore, the inequality that represents 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]
So, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
1. First Expression: [tex]\(\sqrt{5x}\)[/tex]
A square root [tex]\(\sqrt{5x}\)[/tex] is defined only when the expression inside the square root is non-negative. So, we need:
[tex]\[ 5x \geq 0 \][/tex]
Dividing both sides by 5:
[tex]\[ x \geq 0 \][/tex]
2. Second Expression: [tex]\(\sqrt{x+3}\)[/tex]
Similarly, the square root [tex]\(\sqrt{x+3}\)[/tex] is defined only when the expression inside is non-negative. Thus, we need:
[tex]\[ x+3 \geq 0 \][/tex]
Subtracting 3 from both sides:
[tex]\[ x \geq -3 \][/tex]
3. Combining the Inequalities
The product [tex]\(\sqrt{5x} \cdot \sqrt{x+3}\)[/tex] is only defined if both individual square roots are defined. Therefore, [tex]\( x \)[/tex] must satisfy both inequalities simultaneously:
[tex]\[ x \geq 0 \quad \text{and} \quad x \geq -3 \][/tex]
The more restrictive condition here is [tex]\( x \geq 0 \)[/tex]. In other words, if [tex]\( x \geq 0 \)[/tex], then it automatically satisfies the [tex]\( x \geq -3 \)[/tex] condition as well.
Therefore, the inequality that represents 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]
So, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
