We can solve the equation:

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

### Step 1: Check for restrictions

First, let's check for any restrictions in the values of \(x\). Since we have denominators, we need to ensure that the denominators are not zero.

- \(x - 4 \neq 0 \implies x \neq 4\)
- \(x + 2 \neq 0 \implies x \neq -2\)

Thus, \(x \neq 4\) and \(x \neq -2\).

### Step 2: Simplify the equation

Now, simplify both sides. If \(x + 2 \neq 0\), we can cancel out \(x + 2\) from both sides:

\[
\frac{x+2}{x-4} = \frac{x+2}{x+2} \implies  \frac{1}{x-4} = 1
\]

### Step 3: Solve for \(x\)

Now solve the equation:

\[
\frac{1}{x-4} = 1
\]

Multiply both sides by \(x - 4\):

\[
1 = x - 4
\]

Add 4 to both sides:

\[
x = 5
\]

### Final Solution

The solution is \(x = 5\), but remember, \(x \neq 4\) and \(x \neq -2\), so the solution is valid.

Would you like further details or have any questions?

Here are 5 related questions to extend your understanding:

1. How would you solve if there were no common factors in the numerators?
2. What happens if \(x = -2\)? Why is it excluded?
3. Can we approach the same problem graphically? What would the graph look like?
4. How do we handle similar rational equations with higher degrees?
5. What other methods can we use to solve rational equations?

**Tip**: Always check for excluded values when solving rational equations to avoid division by zero errors!

Learn how to solve the rational equation (x+2)/(x-4) = (x+2)/(x+2) with detailed step-by-step instructions. This problem explains the importance of simplifying rational expressions, checking for restrictions, and solving algebraic equations. Suitable for students in Grades 9-11.

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