To solve the equation $\frac{9}{x+1} - \frac{5}{2} = \frac{6}{3x+3}$, we can follow these steps:

Step 1: Simplify the equation

Notice that $3x + 3$ can be factored as $3(x + 1)$, so we rewrite the equation as:

$\frac{9}{x+1} - \frac{5}{2} = \frac{6}{3(x+1)}$

We can simplify $\frac{6}{3(x+1)}$ to $\frac{2}{x+1}$:

$\frac{9}{x+1} - \frac{5}{2} = \frac{2}{x+1}$

Step 2: Combine like terms

Now, subtract $\frac{2}{x+1}$ from $\frac{9}{x+1}$:

$\frac{9 - 2}{x+1} - \frac{5}{2} = 0$

This simplifies to:

$\frac{7}{x+1} = \frac{5}{2}$

Step 3: Cross-multiply to solve for $x$

Cross-multiply to eliminate the fractions:

$7 \times 2 = 5 \times (x + 1)$

Simplify:

$14 = 5x + 5$

Step 4: Isolate $x$

Subtract 5 from both sides:

$9 = 5x$

Now, divide by 5:

$x = \frac{9}{5}$

Final Answer:

The solution is $x = \frac{9}{5}$ or $x = 1.8$.

Would you like further details or explanations? Here are some related questions:

1. How do we solve similar equations with different fractions?
2. What if the equation had a quadratic term instead?
3. How can we verify our solution?
4. What happens if we have more complicated denominators?
5. How does cross-multiplication work with inequalities?

Tip: Always look for common factors or opportunities to simplify fractions before solving the equation.