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For the following exercise, solve for x. State all x-values that are excluded from the solution set.

The given equation is:

\[
\frac{x}{x^2 - 9} + \frac{4}{x + 3} = \frac{3}{x^2 - 9}
\]
with the condition \( x \neq 3, -3 \) to avoid division by zero.

### Step-by-Step Solution:

1. **Factor the denominator \( x^2 - 9 \):**

\[
x^2 - 9 = (x - 3)(x + 3)
\]

So the equation becomes:

\[
\frac{x}{(x - 3)(x + 3)} + \frac{4}{x + 3} = \frac{3}{(x - 3)(x + 3)}
\]

2. **Find a common denominator:**

The common denominator is \( (x - 3)(x + 3) \), so we rewrite each term with this denominator:

\[
\frac{x}{(x - 3)(x + 3)} + \frac{4(x - 3)}{(x + 3)(x - 3)} = \frac{3}{(x - 3)(x + 3)}
\]

Now we have:

\[
\frac{x + 4(x - 3)}{(x - 3)(x + 3)} = \frac{3}{(x - 3)(x + 3)}
\]

3. **Simplify the numerator:**

\[
x + 4(x - 3) = x + 4x - 12 = 5x - 12
\]

So the equation becomes:

\[
\frac{5x - 12}{(x - 3)(x + 3)} = \frac{3}{(x - 3)(x + 3)}
\]

4. **Cancel the denominators (since they are the same on both sides):**

\[
5x - 12 = 3
\]

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

\[
5x = 3 + 12 = 15
\]

\[
x = \frac{15}{5} = 3
\]

6. **Check for excluded values:**

Since \( x = 3 \) is an excluded value (given at the start), there is **no solution** to this equation.

### Final Answer:

There is **no solution** to this equation because the only possible solution, \( x = 3 \), is excluded.

---

Do you need further clarification on any part of the process?

Here are five related questions for practice:

1. How do you solve equations with rational expressions?
2. What is the importance of identifying excluded values in a rational equation?
3. Can you factor other types of quadratic expressions like \(x^2 - 16\)?
4. What happens when the solution to an equation is an excluded value?
5. How would you solve a similar equation where the denominators differ?

**Tip:** Always check for excluded values before simplifying, as they can eliminate possible solutions.

Solve for x. State all x-values that are excluded from the solution set.

Learn how to solve the rational equation x / (x^2 - 9) + 4 / (x + 3) = 3 / (x^2 - 9). The problem focuses on factoring the difference of squares, finding a common denominator, and checking for excluded values. Suitable for students in Grades 9-12.

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