The problem is to solve the equation $\frac{2}{5x} - \frac{5}{2x} = \frac{1}{10}$ for $x$.

1. Problem Description

The problem is to solve the equation $\frac{2}{5x} - \frac{5}{2x} = \frac{1}{10}$ for $x$.

2. Solution Steps

We are given the equation $\frac{2}{5x} - \frac{5}{2x} = \frac{1}{10}$.

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

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

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

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

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

Next, we can cross-multiply to solve for $x$:

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

$-210 = 10x$

Now, divide both sides by 10 to isolate $x$:

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

$x = -21$

3. Final Answer

$x = -21$