Let's solve the equation:

\[
\frac{x + 5}{x - 5} = \frac{x - 4}{x + 4}
\]

### Step 1: Cross-multiply the terms.
Multiply both sides of the equation by \((x - 5)(x + 4)\):

\[
(x + 5)(x + 4) = (x - 5)(x - 4)
\]

### Step 2: Expand both sides.

On the left-hand side:
\[
(x + 5)(x + 4) = x^2 + 4x + 5x + 20 = x^2 + 9x + 20
\]

On the right-hand side:
\[
(x - 5)(x - 4) = x^2 - 4x - 5x + 20 = x^2 - 9x + 20
\]

### Step 3: Set the equation equal.

\[
x^2 + 9x + 20 = x^2 - 9x + 20
\]

### Step 4: Simplify the equation.

Subtract \(x^2 + 20\) from both sides:

\[
9x = -9x
\]

### Step 5: Combine like terms.

\[
18x = 0
\]

### Step 6: Solve for \(x\).

\[
x = 0
\]

### Step 7: Verify the solution.

Substitute \(x = 0\) into the original equation:

\[
\frac{0 + 5}{0 - 5} = \frac{0 - 4}{0 + 4}
\]

This simplifies to:

\[
\frac{5}{-5} = \frac{-4}{4}
\]

\[
-1 = -1
\]

Thus, \(x = 0\) is the correct solution.

Would you like further clarification or more details? 

Here are 5 related questions to explore:
1. How would the solution change if we modified the constants in the equation?
2. Can we have other types of solutions when dealing with rational equations?
3. What are the restrictions for \(x\) in this equation?
4. How can we graph both sides of the equation to confirm the solution?
5. How would solving this type of equation change if it were a system of equations?

**Tip:** Always check for extraneous solutions when solving rational equations, especially when the denominator could be zero.

Learn how to solve the rational equation (X+5)/(X-5) = (X-4)/(X+4) with a detailed step-by-step solution. This problem covers key algebraic concepts, including cross-multiplication and solving linear equations. Suitable for students in Grades 9-12.

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