Answer :
To determine which choice is equivalent to the fraction [tex]\(\frac{4}{4 - \sqrt{6x}}\)[/tex] when [tex]\(x\)[/tex] is an appropriate value, we can follow the hint by rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.
The original fraction is:
[tex]\[ \frac{4}{4 - \sqrt{6x}} \][/tex]
### Step 1: Multiply by the Conjugate
The conjugate of [tex]\(4 - \sqrt{6x}\)[/tex] is [tex]\(4 + \sqrt{6x}\)[/tex]. Multiply the numerator and the denominator by this conjugate:
[tex]\[ \frac{4}{4 - \sqrt{6x}} \cdot \frac{4 + \sqrt{6x}}{4 + \sqrt{6x}} = \frac{4(4 + \sqrt{6x})}{(4 - \sqrt{6x})(4 + \sqrt{6x})} \][/tex]
### Step 2: Simplify the Denominator
The denominator involves a difference of squares:
[tex]\[ (4 - \sqrt{6x})(4 + \sqrt{6x}) = 4^2 - (\sqrt{6x})^2 = 16 - 6x \][/tex]
### Step 3: Simplify the Numerator
Now, simplify the numerator:
[tex]\[ 4(4 + \sqrt{6x}) = 16 + 4\sqrt{6x} \][/tex]
### Step 4: Combine and Simplify
Combining the simplified numerator and denominator, we get:
[tex]\[ \frac{16 + 4\sqrt{6x}}{16 - 6x} \][/tex]
### Step 5: Check Choices
We need to match this expression [tex]\(\frac{16 + 4\sqrt{6x}}{16 - 6x}\)[/tex] against the given choices:
A. [tex]\(\frac{8 + 2\sqrt{6x}}{8 - 6x}\)[/tex]
B. [tex]\(\frac{8 + 2\sqrt{6x}}{16 - 6x}\)[/tex]
C. [tex]\(\frac{2 + \sqrt{6x}}{4 - 6x}\)[/tex]
D. [tex]\(\frac{8 + 2\sqrt{6x}}{8 - 3x}\)[/tex]
Upon comparing, choice B is indeed equivalent to our expression:
[tex]\[ \frac{16 + 4\sqrt{6x}}{16 - 6x} \equiv \frac{8 + 2\sqrt{6x}}{16 - 6x} \quad \text{(by factoring out 2 from the numerator)} \][/tex]
### Conclusion
Thus, the correct answer is:
[tex]\[ \boxed{\text{B}} \quad \frac{8 + 2\sqrt{6x}}{16 - 6x} \][/tex]
The original fraction is:
[tex]\[ \frac{4}{4 - \sqrt{6x}} \][/tex]
### Step 1: Multiply by the Conjugate
The conjugate of [tex]\(4 - \sqrt{6x}\)[/tex] is [tex]\(4 + \sqrt{6x}\)[/tex]. Multiply the numerator and the denominator by this conjugate:
[tex]\[ \frac{4}{4 - \sqrt{6x}} \cdot \frac{4 + \sqrt{6x}}{4 + \sqrt{6x}} = \frac{4(4 + \sqrt{6x})}{(4 - \sqrt{6x})(4 + \sqrt{6x})} \][/tex]
### Step 2: Simplify the Denominator
The denominator involves a difference of squares:
[tex]\[ (4 - \sqrt{6x})(4 + \sqrt{6x}) = 4^2 - (\sqrt{6x})^2 = 16 - 6x \][/tex]
### Step 3: Simplify the Numerator
Now, simplify the numerator:
[tex]\[ 4(4 + \sqrt{6x}) = 16 + 4\sqrt{6x} \][/tex]
### Step 4: Combine and Simplify
Combining the simplified numerator and denominator, we get:
[tex]\[ \frac{16 + 4\sqrt{6x}}{16 - 6x} \][/tex]
### Step 5: Check Choices
We need to match this expression [tex]\(\frac{16 + 4\sqrt{6x}}{16 - 6x}\)[/tex] against the given choices:
A. [tex]\(\frac{8 + 2\sqrt{6x}}{8 - 6x}\)[/tex]
B. [tex]\(\frac{8 + 2\sqrt{6x}}{16 - 6x}\)[/tex]
C. [tex]\(\frac{2 + \sqrt{6x}}{4 - 6x}\)[/tex]
D. [tex]\(\frac{8 + 2\sqrt{6x}}{8 - 3x}\)[/tex]
Upon comparing, choice B is indeed equivalent to our expression:
[tex]\[ \frac{16 + 4\sqrt{6x}}{16 - 6x} \equiv \frac{8 + 2\sqrt{6x}}{16 - 6x} \quad \text{(by factoring out 2 from the numerator)} \][/tex]
### Conclusion
Thus, the correct answer is:
[tex]\[ \boxed{\text{B}} \quad \frac{8 + 2\sqrt{6x}}{16 - 6x} \][/tex]
