Using componendo and dividendo, find the value of x if `(sqrt(3x+4)+sqrt(3x-5))/(sqrt(3x+4)-sqrt(3x-5))=9`.

Answer

To solve the equation

\[ \frac{\sqrt{3x+4} + \sqrt{3x-5}}{\sqrt{3x+4} - \sqrt{3x-5}} = 9 \]

using the method of componendo and dividendo, we follow these steps:

### Step 1: Identify the components
Let:
- \( a = \sqrt{3x + 4} \)
- \( b = \sqrt{3x - 5} \)

Thus, the equation can be rewritten as:

\[ \frac{a + b}{a - b} = 9 \]

### Step 2: Apply the componendo and dividendo formula
According to the componendo and dividendo formula, if \( \frac{a}{b} = \frac{c}{d} \), then:

\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]

In our case, we can take \( c = 9 \) and \( d = 1 \). Therefore, we have:

\[ \frac{a + b}{a - b} = \frac{9 + 1}{9 - 1} = \frac{10}{8} = \frac{5}{4} \]

### Step 3: Set up the equation
Now we can equate:

\[ \frac{a + b}{a - b} = \frac{5}{4} \]

Cross-multiplying gives:

\[ 4(a + b) = 5(a - b) \]

### Step 4: Expand and simplify
Expanding both sides:

\[ 4a + 4b = 5a - 5b \]

Rearranging gives:

\[ 4b + 5b = 5a - 4a \]

This simplifies to:

\[ 9b = a \]

### Step 5: Substitute back for \( a \) and \( b \)
Now substituting back for \( a \) and \( b \):

\[ 9\sqrt{3x - 5} = \sqrt{3x + 4} \]

### Step 6: Square both sides
Squaring both sides results in:

\[ 81(3x - 5) = 3x + 4 \]

### Step 7: Expand and collect like terms
Expanding gives:

\[ 243x - 405 = 3x + 4 \]

Rearranging terms leads to:

\[ 243x - 3x = 405 + 4 \]

This simplifies to:

\[ 240x = 409 \]

### Step 8: Solve for \( x \)
Dividing both sides by 240 gives:

\[ x = \frac{409}{240} \]

### Step 9: Simplify the fraction if possible
To simplify \( \frac{409}{240} \), we check for common factors. Since 409 is a prime number and does not share any factors with 240, the fraction is already in its simplest form.

Thus, the value of \( x \) is:

\[ x = \frac{409}{240} \]