What is the value of x in the equation `sqrt((1+x)/(x))-sqrt((x)/(1+x))=(1)/(sqrt(6))`?

Options

A. `-2`
B. 3
C. 2
D. None of these

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square roots.

\[ \left( \sqrt{\frac{1+x}{x}} - \sqrt{\frac{x}{1+x}} \right)^2 = \left( \frac{1}{\sqrt{6}} \right)^2. \]

This gives us:

\[ \frac{1+x}{x} - 2\sqrt{\frac{(1+x)x}{x(1+x)}} + \frac{x}{1+x} = \frac{1}{6}. \]

### Step 2: Simplify the left side
The middle term simplifies to 2, since:

\[ \sqrt{\frac{(1+x)x}{x(1+x)}} = 1. \]

So we have:

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

### Step 3: Combine fractions
Next, we need to combine the fractions on the left side. The common denominator for \(\frac{1+x}{x}\) and \(\frac{x}{1+x}\) is \(x(1+x)\):

\[ \frac{(1+x)^2 + x^2 - 2x(1+x)}{x(1+x)} = \frac{1}{6}. \]

### Step 4: Expand and simplify
Expanding the numerator gives:

\[ (1 + 2x + x^2 + x^2 - 2x - 2x^2) = 1 - x^2. \]

So we have:

\[ \frac{1 - x^2}{x(1+x)} = \frac{1}{6}. \]

### Step 5: Cross multiply
Cross multiplying gives:

\[ 6(1 - x^2) = x(1+x). \]

### Step 6: Expand and rearrange
Expanding both sides results in:

\[ 6 - 6x^2 = x + x^2. \]

Rearranging gives:

\[ 7x^2 + x - 6 = 0. \]

### Step 7: Factor the quadratic
Now we factor the quadratic equation:

\[ (7x - 6)(x + 1) = 0. \]

### Step 8: Solve for x
Setting each factor to zero gives:

1. \(7x - 6 = 0 \Rightarrow x = \frac{6}{7}\)
2. \(x + 1 = 0 \Rightarrow x = -1\)

### Step 9: Check for valid solutions
Since we are dealing with square roots, we need to ensure that the solutions are valid in the context of the original equation.

Substituting \(x = -1\) into the original equation results in an undefined expression, so we discard this solution.

Thus, the only valid solution is:

\[ \boxed{\frac{6}{7}}. \]