The given equation is:

\[
\frac{20}{x} - \frac{20}{x-2} = \frac{4}{x}
\]

To solve for \( x \), follow these steps:

1. **Identify a common denominator**: The common denominator for the fractions on the left side of the equation is \( x(x-2) \).

2. **Rewrite the equation using the common denominator**:

\[
\frac{20(x-2) - 20x}{x(x-2)} = \frac{4}{x}
\]

3. **Simplify the numerator on the left side**:

\[
\frac{20x - 40 - 20x}{x(x-2)} = \frac{4}{x}
\]

This simplifies to:

\[
\frac{-40}{x(x-2)} = \frac{4}{x}
\]

4. **Cross-multiply to eliminate the fractions**:

\[
-40x = 4x(x-2)
\]

5. **Expand and simplify**:

\[
-40x = 4x^2 - 8x
\]

6. **Move all terms to one side of the equation**:

\[
4x^2 - 8x + 40x = 0
\]

\[
4x^2 + 32x = 0
\]

7. **Factor the equation**:

\[
4x(x + 8) = 0
\]

8. **Solve for \( x \)**:

\[
4x = 0 \quad \text{or} \quad x + 8 = 0
\]

\[
x = 0 \quad \text{or} \quad x = -8
\]

Since \( x = 0 \) would make the original equation undefined, the valid solution is \( x = -8 \).

### Final Answer:
The solution is \( x = -8 \).

Would you like more details, or do you have any questions about this problem?

### Related Questions:
1. How do you solve equations with multiple fractions?
2. What are common denominators, and how are they used in solving equations?
3. Can you explain how cross-multiplication works in equations?
4. What does it mean for a solution to be undefined in a fraction-based equation?
5. How can we check the validity of a solution in a fractional equation?

### Tip:
Always check the potential solutions in the original equation to ensure they do not make any denominator zero, which would make the solution invalid.

Learn how to solve the equation 20/x - 20/(x-2) = 4/x step-by-step with detailed explanations. This problem involves solving equations with fractions and cross-multiplication. Suitable for high school students studying algebra.

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