Find x if `(sqrt(3x +1) + sqrt(3x - 6))/(sqrt(3x + 1) - sqrt(3x - 6)) = 7 `

Options

A. 2
B. 5
C. 3
D. 7

Answer

To solve the equation

\[ \frac{\sqrt{3x + 1} + \sqrt{3x - 6}}{\sqrt{3x + 1} - \sqrt{3x - 6}} = 7, \]

we will follow these steps:

### Step 1: Apply the Compando and Dividendo Rule

According to the compando and dividendo rule, if

\[ \frac{a}{b} = \frac{c}{d}, \]

then

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

In our case, we can set:

- \( a = \sqrt{3x + 1} \)
- \( b = \sqrt{3x - 6} \)
- \( c = 7 \)
- \( d = 1 \)

So we can rewrite the equation as:

\[ \frac{\sqrt{3x + 1} + \sqrt{3x - 6} + \sqrt{3x + 1} - \sqrt{3x - 6}}{\sqrt{3x + 1} + \sqrt{3x - 6} - \sqrt{3x + 1} + \sqrt{3x - 6}} = \frac{7 + 1}{7 - 1}. \]

### Step 2: Simplify the Left Side

The left side simplifies to:

\[ \frac{2\sqrt{3x + 1}}{2\sqrt{3x - 6}} = \frac{8}{6}. \]

Cancelling the 2's gives:

\[ \frac{\sqrt{3x + 1}}{\sqrt{3x - 6}} = \frac{4}{3}. \]

### Step 3: Cross Multiply

Cross multiplying gives:

\[ 3(\sqrt{3x + 1}) = 4(\sqrt{3x - 6}). \]

### Step 4: Square Both Sides

Squaring both sides results in:

\[ 9(3x + 1) = 16(3x - 6). \]

### Step 5: Expand Both Sides

Expanding both sides gives:

\[ 27x + 9 = 48x - 96. \]

### Step 6: Rearrange the Equation

Rearranging the equation results in:

\[ 27x + 9 + 96 = 48x, \]

which simplifies to:

\[ 105 = 48x - 27x, \]

or

\[ 21x = 105. \]

### Step 7: Solve for x

Dividing both sides by 21 gives:

\[ x = \frac{105}{21} = 5. \]

Thus, the solution is:

\[ \boxed{5}. \]

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