To determine the values of [tex]\( x \)[/tex] for which the expression
[tex]\[
\frac{\sqrt{3x^2}}{\sqrt{4x}}
\][/tex]
is defined, we need to check the conditions under which all parts of the expression are valid.
1. Expression Under the Square Roots:
- Numerator: [tex]\(\sqrt{3x^2}\)[/tex], we need [tex]\(3x^2 \geq 0\)[/tex].
- This inequality is always true for all real values of [tex]\( x \)[/tex] because the square of any real number [tex]\( x \)[/tex] is always non-negative, making [tex]\( 3x^2 \geq 0 \)[/tex].
- Denominator: [tex]\(\sqrt{4x}\)[/tex], we need [tex]\(4x \geq 0\)[/tex].
- For this inequality to be true, [tex]\( x \)[/tex] must be non-negative since [tex]\( 4x \geq 0 \implies x \geq 0 \)[/tex].
2. Additional Condition to Avoid Division by Zero:
- The denominator [tex]\(\sqrt{4x}\)[/tex] must be non-zero, implying [tex]\( 4x \neq 0 \)[/tex].
- For [tex]\( 4x \neq 0 \)[/tex], [tex]\( x \neq 0 \)[/tex].
Combining these conditions, we have:
- [tex]\( x \geq 0 \)[/tex] from the requirement that the argument of [tex]\(\sqrt{4x}\)[/tex] is non-negative.
- [tex]\( x \neq 0 \)[/tex] to avoid division by zero.
Therefore, these two combined conditions mean [tex]\( x \)[/tex] must be strictly greater than 0.
Conclusion:
The expression [tex]\(\frac{\sqrt{3x^2}}{\sqrt{4x}}\)[/tex] is defined for [tex]\( x > 0 \)[/tex].
Hence, the correct answer is:
D. [tex]\(x > 0\)[/tex]
