Use the properties of proportionality to solve : `(sqrt(12x+1)+sqrt(2x-3))/(sqrt(12x+1)-sqrt(2x-3))=3/2 `

Answer

To solve the equation

\[ \frac{\sqrt{12x+1} + \sqrt{2x-3}}{\sqrt{12x+1} - \sqrt{2x-3}} = \frac{3}{2}, \]

we can use the properties of proportionality, specifically the Componendo and Evendo method.

### Step 1: Apply Componendo and Evendo
Using the Componendo and Evendo method, we can rewrite the equation as:

\[ \frac{\sqrt{12x+1} + \sqrt{2x-3}}{\sqrt{12x+1} - \sqrt{2x-3}} = \frac{3 + 2}{3 - 2}. \]

This simplifies to:

\[ \frac{\sqrt{12x+1} + \sqrt{2x-3}}{\sqrt{12x+1} - \sqrt{2x-3}} = \frac{5}{1}. \]

### Step 2: Cross-multiply
Cross-multiplying gives us:

\[ \sqrt{12x+1} + \sqrt{2x-3} = 5(\sqrt{12x+1} - \sqrt{2x-3}). \]

### Step 3: Expand the equation
Expanding the right side, we have:

\[ \sqrt{12x+1} + \sqrt{2x-3} = 5\sqrt{12x+1} - 5\sqrt{2x-3}. \]

### Step 4: Rearrange the equation
Rearranging the equation gives:

\[ \sqrt{2x-3} + 5\sqrt{2x-3} = 5\sqrt{12x+1} - \sqrt{12x+1}. \]

This simplifies to:

\[ 6\sqrt{2x-3} = 4\sqrt{12x+1}. \]

### Step 5: Divide both sides by 2
Dividing both sides by 2 gives:

\[ 3\sqrt{2x-3} = 2\sqrt{12x+1}. \]

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

\[ (3\sqrt{2x-3})^2 = (2\sqrt{12x+1})^2. \]

This simplifies to:

\[ 9(2x-3) = 4(12x+1). \]

### Step 7: Expand both sides
Expanding both sides yields:

\[ 18x - 27 = 48x + 4. \]

### Step 8: Rearrange to isolate x
Rearranging gives:

\[ 18x - 48x = 4 + 27, \]

which simplifies to:

\[ -30x = 31. \]

### Step 9: Solve for x
Dividing both sides by -30 gives:

\[ x = -\frac{31}{30}. \]

### Final Answer
Thus, the solution to the equation is:

\[ x = -\frac{31}{30}. \]