https://file.aiquickdraw.com/mathbot/file/1726950341352dgghllwh.jpg

The equation in the image is:

\[
-\frac{2}{2x-1} + \frac{1}{x+3} = -\frac{3}{2x+6}
\]

### Steps to solve the equation:

1. **Factor where possible**:
   - The denominator \( 2x + 6 \) can be factored as \( 2(x + 3) \), so we rewrite the equation as:
   \[
   -\frac{2}{2x-1} + \frac{1}{x+3} = -\frac{3}{2(x+3)}
   \]

2. **Find a common denominator**:
   - The denominators involved are \( 2x - 1 \) and \( 2(x + 3) \). So, we need to eliminate the fractions by multiplying through by the least common denominator (LCD). The LCD is \( (2x - 1)(2(x + 3)) \).

   Multiply the entire equation by \( (2x - 1)(2(x + 3)) \):

   \[
   (2x - 1)(2(x + 3))\left( -\frac{2}{2x - 1} \right) + (2x - 1)(2(x + 3))\left( \frac{1}{x+3} \right) = (2x - 1)(2(x + 3))\left( -\frac{3}{2(x+3)} \right)
   \]

3. **Simplify** each term:
   - First term: \( -\frac{2}{2x - 1} \) becomes \( -2(2(x + 3)) = -4(x + 3) \).
   - Second term: \( \frac{1}{x + 3} \) becomes \( (2x - 1) \cdot 2 = 2(2x - 1) \).
   - Third term: \( -\frac{3}{2(x+3)} \) becomes \( -3(2x - 1) \).

   After simplifying, we have:
   \[
   -4(x + 3) + 2(2x - 1) = -3(2x - 1)
   \]

4. **Expand the terms**:
   - Expand each term:
     - \( -4(x + 3) = -4x - 12 \)
     - \( 2(2x - 1) = 4x - 2 \)
     - \( -3(2x - 1) = -6x + 3 \)

   Now the equation is:
   \[
   -4x - 12 + 4x - 2 = -6x + 3
   \]

5. **Combine like terms**:
   - On the left-hand side: \( -4x + 4x = 0 \), so the equation simplifies to:
   \[
   -12 - 2 = -6x + 3
   \]

   Which becomes:
   \[
   -14 = -6x + 3
   \]

6. **Solve for \( x \)**:
   - Subtract 3 from both sides:
   \[
   -14 - 3 = -6x
   \]
   \[
   -17 = -6x
   \]
   - Divide by \( -6 \):
   \[
   x = \frac{17}{6}
   \]

So, the solution is \( x = \frac{17}{6} \).

---

Would you like further details or have any questions about this solution?

Here are 5 related questions you might find useful:
1. How do you handle equations with multiple fractions like this one?
2. What is the best way to find a common denominator in rational equations?
3. How can you check if your solution is valid in the original equation?
4. How does factoring help in simplifying rational equations?
5. How would this process change if there were more terms on both sides?

**Tip:** Always factor wherever possible to simplify terms early in equations with rational expressions.

-\frac{2}{2x-1} + \frac{1}{x+3} = -\frac{3}{2x+6}

This problem focuses on solving a rational equation by factoring and finding a common denominator. Learn how to simplify the equation -2/(2x-1) + 1/(x+3) = -3/(2x+6) step by step. Suitable for students in Grades 9-11.

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