https://file.aiquickdraw.com/mathbot/file/1725482830220mon8oc73.jpg

The problem shown in the image is to solve the rational equation:

\[
\frac{4}{x + 2} = \frac{9}{x - 4}
\]

### Step-by-Step Solution:

**Step 1: Determine the domain of the variable in the rational equation.**

The variable \( x \) cannot take values that would cause any denominator to be zero because division by zero is undefined. 

- For the denominator \( x + 2 \), \( x \) cannot be \( -2 \).
- For the denominator \( x - 4 \), \( x \) cannot be \( 4 \).

So, the domain of \( x \) is \( x \neq -2 \) and \( x \neq 4 \).

**Step 2: Determine the least common denominator (LCD) of all the denominators.**

The denominators in this problem are \( x + 2 \) and \( x - 4 \). The LCD is simply the product of these two expressions:

\[
\text{LCD} = (x + 2)(x - 4)
\]

Next steps would typically involve multiplying both sides of the equation by the LCD to eliminate the denominators, simplifying, and solving the resulting equation.

Would you like to continue with the solution, or do you have any specific questions about the steps provided?

### Follow-up Questions:
1. How do we eliminate the denominators by multiplying by the LCD?
2. What would the resulting equation look like after eliminating the denominators?
3. How can we solve the equation after eliminating the denominators?
4. What are the possible solutions, and how do we check if they are within the domain?
5. How would the solution change if the equation had more complex denominators?

### Tip:
When solving rational equations, always check your solutions against the domain to ensure they do not make any denominator zero.

Solve the rational equation: 4 / (x + 2) = 9 / (x - 4)

Learn how to solve the rational equation 4/(x + 2) = 9/(x - 4) by determining the domain, finding the least common denominator (LCD), and solving the equation step by step. This problem highlights core algebraic concepts and is suitable for students in Grades 8-10.

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