Let's solve the equation:

\[
\frac{-5}{5x} + 2 = -2 + \frac{2}{10x}
\]

### Step 1: Simplify the fractions
- The fraction \(\frac{-5}{5x}\) simplifies to \(\frac{-1}{x}\).
- The fraction \(\frac{2}{10x}\) simplifies to \(\frac{1}{5x}\).

The equation becomes:

\[
\frac{-1}{x} + 2 = -2 + \frac{1}{5x}
\]

### Step 2: Eliminate the fractions
Multiply the entire equation by \(5x\) to eliminate the denominators:

\[
5x \left(\frac{-1}{x} + 2\right) = 5x \left(-2 + \frac{1}{5x}\right)
\]

This simplifies to:

\[
-5 + 10x = -10x + 1
\]

### Step 3: Combine like terms
Move all terms involving \(x\) to one side and constants to the other side:

\[
10x + 10x = 1 + 5
\]

This simplifies to:

\[
20x = 6
\]

### Step 4: Solve for \(x\)
Now, divide both sides by 20:

\[
x = \frac{6}{20} = \frac{3}{10}
\]

### Final Answer:
\[
x = \frac{3}{10}
\]

Would you like more details on any of these steps or another clarification?

### Related Questions:
1. How do you multiply both sides of an equation with a variable denominator?
2. What other methods can be used to solve fractions in equations?
3. Can you solve an equation if the variable is in multiple fractions?
4. How do you check if a solution to an equation is correct?
5. How can cross-multiplication help with fractional equations?

**Tip:** Always simplify fractions early in the equation to make solving easier.

Learn how to solve the rational equation -5/5x + 2 = -2 + 2/10x through fraction simplification and linear equation solving. This detailed step-by-step solution covers key algebraic concepts, perfect for students in Grades 8-10.

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