Solve for $x$ in the equation $\frac{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{5}}}{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}}} = \frac{\sqrt{10} - 1}{x}$.

1. Problem Description

Solve for $x$ in the equation $\frac{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{5}}}{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}}} = \frac{\sqrt{10} - 1}{x}$.

2. Solution Steps

First, simplify the left-hand side of the equation. We can simplify the numerator and the denominator separately and then divide them.

Numerator:

$\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{5}} = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{2}\sqrt{5}} = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{10}}$

Denominator:

$\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{5}} = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{2}\sqrt{5}} = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{10}}$

Now, divide the numerator by the denominator:

$\frac{\frac{\sqrt{5} - \sqrt{2}}{\sqrt{10}}}{\frac{\sqrt{5} + \sqrt{2}}{\sqrt{10}}} = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{5} + \sqrt{2}} = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}}$

Now we need to rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{5} - \sqrt{2}$:

$\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \cdot \frac{\ جذر5 - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{(\sqrt{5} - \sqrt{2})^2}{(\sqrt{5})^2 - (\sqrt{2})^2} = \frac{5 - 2\sqrt{10} + 2}{5 - 2} = \frac{7 - 2\sqrt{10}}{3}$

We are given that

$\frac{7 - 2\sqrt{10}}{3} = \frac{\sqrt{10} - 1}{x}$.

This is incorrect, based on the original problem, we have the simplified equation:

$\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{\sqrt{10} - 1}{x}$

Rationalizing the left-hand side:

$\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{(\sqrt{5} - \sqrt{2})(\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})} = \frac{5 - 2\sqrt{10} + 2}{5 - 2} = \frac{7 - 2\sqrt{10}}{3}$

We are given:

$\frac{7 - 2\sqrt{10}}{3} = \frac{\sqrt{10} - 1}{x}$

$x = \frac{3(\sqrt{10} - 1)}{7 - 2\sqrt{10}}$

Now we rationalize the denominator:

$x = \frac{3(\sqrt{10} - 1)(7 + 2\sqrt{10})}{(7 - 2\sqrt{10})(7 + 2\sqrt{10})} = \frac{3(7\sqrt{10} + 20 - 7 - 2\sqrt{10})}{49 - 40} = \frac{3(5\sqrt{10} + 13)}{9} = \frac{5\sqrt{10} + 13}{3}$

This looks wrong. Let's double-check the problem statement.

Let's go back to

$\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}} = \frac{\sqrt{10}-1}{x}$

$x = \frac{(\sqrt{10}-1)(\sqrt{5}+\sqrt{2})}{\sqrt{5}-\sqrt{2}} = \frac{\sqrt{50}+\sqrt{20}-\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}} = \frac{5\sqrt{2}+2\sqrt{5}-\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}} = \frac{4\sqrt{2}+\sqrt{5}}{\sqrt{5}-\sqrt{2}} = \frac{(4\sqrt{2}+\sqrt{5})(\sqrt{5}+\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})} = \frac{4\sqrt{10}+8+5+\sqrt{10}}{5-2} = \frac{5\sqrt{10}+13}{3}$

3. Final Answer

$x = \frac{5\sqrt{10} + 13}{3}$