For what values of [tex]\( x \)[/tex] is the expression below defined?

[tex]\[ \sqrt{3x^2} \div \sqrt{4x} \][/tex]

A. [tex]\( x=0 \)[/tex]
B. [tex]\( x\ \textless \ 1 \)[/tex]
C. [tex]\( x\ \textless \ 0 \)[/tex]
D. [tex]\( x\ \textgreater \ 0 \)[/tex]



Answer :

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]