To determine which inequality represents all values of [tex]\( x \)[/tex] for which the quotient
[tex]\[ \frac{\sqrt{6x^2}}{\sqrt{4x}} \][/tex]
is defined, let's go through the problem step-by-step.
1. Simplify the Expression:
Start by combining the square roots in the expression:
[tex]\[
\frac{\sqrt{6x^2}}{\sqrt{4x}} = \sqrt{\frac{6x^2}{4x}}
\][/tex]
2. Simplify Inside the Square Root:
Simplify the expression inside the square root:
[tex]\[
\frac{6x^2}{4x} = \frac{6}{4} \cdot \frac{x^2}{x} = \frac{3}{2} \cdot x = 1.5x
\][/tex]
So, the expression becomes:
[tex]\[
\sqrt{1.5x}
\][/tex]
3. Determine When It Is Defined:
For the square root to be defined, the quantity under the square root must be non-negative:
[tex]\[
1.5x \geq 0
\][/tex]
4. Solve the Inequality:
Solve for [tex]\( x \)[/tex]:
[tex]\[
1.5x \geq 0 \implies x \geq 0
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which the quotient [tex]\(\frac{\sqrt{6x^2}}{\sqrt{4x}}\)[/tex] is defined are all [tex]\( x \geq 0 \)[/tex].
So, the correct inequality is:
[tex]\[ x \geq 0 \][/tex]
Thus, the answer is option:
D. [tex]\( x \geq 0 \)[/tex]
