Which inequality represents all values of [tex][tex]$x$[/tex][/tex] for which the product below is defined?

[tex] \sqrt{5x} \cdot \sqrt{x+3} [/tex]

A. [tex] x \geq -3 [/tex]
B. [tex] x \geq 0 [/tex]
C. [tex] x \ \textgreater \ 0 [/tex]
D. [tex] x \leq -3 [/tex]



Answer :

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]