https://file.aiquickdraw.com/mathbot/file/1724339125982uzaltw4s.jpg

The equation in the image appears to be:

\[
\frac{x}{x^2 - 9} + \frac{4}{x + 3} = \frac{3}{x^2 - 9}
\]

This equation involves rational expressions, and we can solve it by finding a common denominator. Let's go through the steps to solve this equation.

### Step 1: Factor the denominators.
Notice that the quadratic \( x^2 - 9 \) is a difference of squares and can be factored as:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
So the equation becomes:
\[
\frac{x}{(x - 3)(x + 3)} + \frac{4}{x + 3} = \frac{3}{(x - 3)(x + 3)}
\]

### Step 2: Find a common denominator.
The common denominator is \( (x - 3)(x + 3) \). The left side of the equation needs to be rewritten to have this common denominator.

- The first fraction \( \frac{x}{(x - 3)(x + 3)} \) already has the common denominator.
- The second fraction \( \frac{4}{x + 3} \) needs to be multiplied by \( (x - 3) \) in both the numerator and denominator.

So, the equation becomes:
\[
\frac{x}{(x - 3)(x + 3)} + \frac{4(x - 3)}{(x - 3)(x + 3)} = \frac{3}{(x - 3)(x + 3)}
\]

### Step 3: Combine the fractions.
Now that all the fractions have the same denominator, we can combine the numerators:
\[
\frac{x + 4(x - 3)}{(x - 3)(x + 3)} = \frac{3}{(x - 3)(x + 3)}
\]

Simplify the numerator:
\[
x + 4(x - 3) = x + 4x - 12 = 5x - 12
\]

The equation now is:
\[
\frac{5x - 12}{(x - 3)(x + 3)} = \frac{3}{(x - 3)(x + 3)}
\]

### Step 4: Solve the equation.
Since the denominators are the same, we can equate the numerators:
\[
5x - 12 = 3
\]

### Step 5: Solve for \( x \).
Add 12 to both sides:
\[
5x = 15
\]
Now divide by 5:
\[
x = 3
\]

### Step 6: Check for extraneous solutions.
Since the original equation involves rational expressions, we need to check if \( x = 3 \) leads to any undefined expressions (division by zero). The denominators involve the factors \( x - 3 \) and \( x + 3 \). When \( x = 3 \), the factor \( x - 3 \) becomes zero, which would make the denominator undefined.

Therefore, \( x = 3 \) is **not a valid solution** because it would make the original equation undefined.

### Conclusion:
There is **no valid solution** to the equation because the only possible solution \( x = 3 \) leads to a division by zero, which is not allowed.

Would you like further clarification or have additional questions on this?

Here are some related questions to think about:
1. How do you handle rational equations with multiple solutions?
2. Can you always cancel denominators when solving rational equations?
3. What other techniques can be used to solve rational equations?
4. How do you determine if a solution is extraneous in rational equations?
5. What role does factoring play in solving rational equations?

**Tip:** Always check for undefined values in rational equations by considering when the denominator could equal zero. This helps avoid extraneous solutions.

Learn how to solve the rational equation (x / (x^2 - 9)) + (4 / (x + 3)) = (3 / (x^2 - 9)) step-by-step. Understand the process of factoring, finding common denominators, and identifying extraneous solutions. Suitable for advanced high school students and those preparing for math analysis exams.

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