We are asked to solve the equation $\frac{2}{5x} - \frac{5}{2x} = \frac{1}{10}$ for $x$.

1. Problem Description

We are asked to solve the equation

$\frac{2}{5x} - \frac{5}{2x} = \frac{1}{10}$ for $x$.

2. Solution Steps

First, we find a common denominator for the terms on the left-hand side of the equation. The common denominator for $5x$ and $2x$ is $10x$.

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

$\frac{2(2)}{5x(2)} - \frac{5(5)}{2x(5)} = \frac{1}{10}$

$\frac{4}{10x} - \frac{25}{10x} = \frac{1}{10}$

$\frac{4-25}{10x} = \frac{1}{10}$

$\frac{-21}{10x} = \frac{1}{10}$

Now we can cross-multiply:

$-21(10) = 1(10x)$

$-210 = 10x$

Divide both sides by 10:

$x = \frac{-210}{10}$

$x = -21$

3. Final Answer

$x = -21$