https://file.aiquickdraw.com/mathbot/file/1726545468714ciequgvi.jpg

The equation in the image is:

\[
\frac{1}{x-3} + \frac{1}{x+3} = \frac{10}{x^2 - 9}
\]

### Solution steps:

1. Notice that \( x^2 - 9 \) is a difference of squares:
   \[
   x^2 - 9 = (x - 3)(x + 3)
   \]
   So the equation becomes:
   \[
   \frac{1}{x-3} + \frac{1}{x+3} = \frac{10}{(x-3)(x+3)}
   \]

2. The left-hand side has two fractions with different denominators. To add these fractions, find the common denominator, which is \( (x - 3)(x + 3) \):
   \[
   \frac{1}{x-3} + \frac{1}{x+3} = \frac{(x+3) + (x-3)}{(x-3)(x+3)}
   \]

3. Simplify the numerator on the left-hand side:
   \[
   (x + 3) + (x - 3) = 2x
   \]
   So, the equation becomes:
   \[
   \frac{2x}{(x-3)(x+3)} = \frac{10}{(x-3)(x+3)}
   \]

4. Since the denominators are the same, you can equate the numerators:
   \[
   2x = 10
   \]

5. Solve for \( x \):
   \[
   x = \frac{10}{2} = 5
   \]

### Solution:
The value of \( x \) is \( x = 5 \).

### Important Note:
We must check if this value of \( x \) leads to any division by zero in the original equation. The denominators in the original equation involve \( x - 3 \) and \( x + 3 \), so we need to ensure \( x \neq 3 \) and \( x \neq -3 \). Since \( x = 5 \), there are no issues with division by zero.

Would you like more details on this solution or further explanations?

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Here are 5 related questions you might want to explore:
1. What happens if \( x = 3 \) or \( x = -3 \) in the original equation?
2. How can you solve similar rational equations where the denominators are not easily factored?
3. Can this equation be solved using an alternative method such as cross-multiplication?
4. How does factoring help in solving equations with differences of squares?
5. What are the general strategies for solving equations that involve fractions?

**Tip:** Always check for any restrictions on the variable (like division by zero) when solving equations with fractions.

Solve the equation 1/(x-3) + 1/(x+3) = 10/(x^2-9).

This webpage explores the process of solving rational equations by employing the difference of squares and finding a common denominator. A step-by-step guide helps to solve the equation 1/(x-3) + 1/(x+3) = 10/(x^2-9), suitable for high school level students.

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