If `(1/sqrt(9) - 1/sqrt(11))/(1/sqrt(9) + 1/sqrt(11)) xx (10 + sqrt(99))/x =1/2`, then x equal

Options

A. 2
B. 3
C. 10
D. `1//10`

Answer

To solve the equation

\[ \frac{\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{11}}}{\frac{1}{\sqrt{9}} + \frac{1}{\sqrt{11}}} \cdot \frac{10 + \sqrt{99}}{x} = \frac{1}{2}, \]

we will follow these steps:

### Step 1: Simplify the fractions
First, we simplify the fractions in the equation.

\[ \frac{1}{\sqrt{9}} = \frac{1}{3}, \quad \text{and} \quad \frac{1}{\sqrt{11}} = \frac{1}{\sqrt{11}}. \]

So, we can rewrite the equation as:

\[ \frac{\frac{1}{3} - \frac{1}{\sqrt{11}}}{\frac{1}{3} + \frac{1}{\sqrt{11}}} \cdot \frac{10 + \sqrt{99}}{x} = \frac{1}{2}. \]

### Step 2: Multiply by the conjugate
To simplify the fraction, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{\left(\frac{1}{3} - \frac{1}{\sqrt{11}}\right) \left(\frac{1}{3} - \frac{1}{\sqrt{11}}\right)}{\left(\frac{1}{3} + \frac{1}{\sqrt{11}}\right) \left(\frac{1}{3} - \frac{1}{\sqrt{11}}\right)}. \]

This results in:

\[ \frac{\left(\frac{1}{3} - \frac{1}{\sqrt{11}}\right)^2}{\left(\frac{1}{3}\right)^2 - \left(\frac{1}{\sqrt{11}}\right)^2}. \]

### Step 3: Calculate the squares
Calculating the squares gives:

\[ \left(\frac{1}{3} - \frac{1}{\sqrt{11}}\right)^2 = \frac{1}{9} - 2 \cdot \frac{1}{3 \sqrt{11}} + \frac{1}{11}, \]

and

\[ \left(\frac{1}{3}\right)^2 - \left(\frac{1}{\sqrt{11}}\right)^2 = \frac{1}{9} - \frac{1}{11}. \]

### Step 4: Find a common denominator
The common denominator for \(9\) and \(11\) is \(99\). Thus,

\[ \frac{1}{9} = \frac{11}{99}, \quad \text{and} \quad \frac{1}{11} = \frac{9}{99}. \]

So,

\[ \frac{1}{9} - \frac{1}{11} = \frac{11 - 9}{99} = \frac{2}{99}. \]

### Step 5: Substitute back into the equation
Now we substitute back into the equation:

\[ \frac{\frac{1}{9} - 2 \cdot \frac{1}{3 \sqrt{11}} + \frac{1}{11}}{\frac{2}{99}} \cdot \frac{10 + \sqrt{99}}{x} = \frac{1}{2}. \]

### Step 6: Solve for x
Cross-multiplying gives:

\[ \left(\frac{1}{9} - 2 \cdot \frac{1}{3 \sqrt{11}} + \frac{1}{11}\right) \cdot (10 + \sqrt{99}) = \frac{1}{2} \cdot \frac{2}{99} \cdot x. \]

### Step 7: Simplify further
After simplifying, we find:

\[ x = 2. \]

### Final Answer
Thus, the value of \(x\) is

\[ \boxed{2}. \]