https://file.aiquickdraw.com/mathbot/file/17258417145572vcu7gvx.jpg

The image shows the following equation:

\[
\frac{3}{x + 3} - \frac{2}{x + 5} = 0
\]

To solve for \(x\), we need to find a common denominator and solve the resulting equation. Let's go step-by-step:

### Step 1: Find a Common Denominator

The denominators are \(x + 3\) and \(x + 5\). The common denominator is \((x + 3)(x + 5)\).

### Step 2: Rewrite the Equation with the Common Denominator

\[
\frac{3(x + 5) - 2(x + 3)}{(x + 3)(x + 5)} = 0
\]

### Step 3: Expand and Simplify the Numerator

\[
3(x + 5) - 2(x + 3) = 3x + 15 - 2x - 6 = x + 9
\]

So, the equation becomes:

\[
\frac{x + 9}{(x + 3)(x + 5)} = 0
\]

### Step 4: Set the Numerator Equal to Zero

Since a fraction is zero when its numerator is zero, we set:

\[
x + 9 = 0
\]

### Step 5: Solve for \(x\)

\[
x = -9
\]

### Step 6: Check for Extraneous Solutions

Ensure that the solution does not make any denominator zero:
- For \(x = -9\), \(x + 3 = -6\) and \(x + 5 = -4\), both of which are not zero.

Thus, the solution is valid.

### Final Answer:
\[
x = -9
\]

Would you like more details or have any questions?

### Related Questions:

1. How would the solution change if the equation were \(\frac{3}{x + 2} - \frac{2}{x + 5} = 0\)?
2. Can you solve a similar equation involving three fractions instead of two?
3. What are the methods to solve rational equations with different types of denominators?
4. How do you solve rational equations with quadratic denominators?
5. What are potential pitfalls when solving rational equations involving polynomials?

### Tip:
Always check your solutions in rational equations to ensure they do not make any denominator zero!

Solve the equation \(\frac{3}{x + 3} - \frac{2}{x + 5} = 0\).

Learn how to solve the rational equation \(\frac{3}{x + 3} - \frac{2}{x + 5} = 0\) with a detailed step-by-step solution. This problem covers key algebraic concepts, including finding a common denominator and solving linear equations. Suitable for students in Grades 9-11.

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