Let's solve the equation step by step:

Given equation: $\frac{37}{18} + \frac{2x}{3} = -1 + \frac{2}{3} \left(-\frac{4x}{3} + \frac{9}{4}\right)$

Step 1: Distribute the $\frac{2}{3}$ on the right side.

$\frac{37}{18} + \frac{2x}{3} = -1 + \frac{2}{3} \cdot \left(-\frac{4x}{3}\right) + \frac{2}{3} \cdot \frac{9}{4}$

Calculate each term: $\frac{2}{3} \cdot \left(-\frac{4x}{3}\right) = -\frac{8x}{9}$ $\frac{2}{3} \cdot \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$

So the equation becomes: $\frac{37}{18} + \frac{2x}{3} = -1 - \frac{8x}{9} + \frac{3}{2}$

Step 2: Simplify the constants on the right side.

Combine $-1$ and $\frac{3}{2}$: $-1 + \frac{3}{2} = -\frac{2}{2} + \frac{3}{2} = \frac{1}{2}$

So now we have: $\frac{37}{18} + \frac{2x}{3} = \frac{1}{2} - \frac{8x}{9}$

Step 3: Find a common denominator for all fractions to simplify the equation.

The common denominator for 18, 3, 2, and 9 is 18. Rewriting all terms with a denominator of 18 gives: $\frac{37}{18} + \frac{12x}{18} = \frac{9}{18} - \frac{16x}{18}$

Combine like terms: $\frac{37 + 12x}{18} = \frac{9 - 16x}{18}$

Since the denominators are equal, we can equate the numerators: $37 + 12x = 9 - 16x$

Step 4: Solve for $x$.

Rearrange the terms to isolate $x$: $12x + 16x = 9 - 37$ $28x = -28$ $x = -1$

So, the value of $x$ is $\boxed{-1}$.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you solve a similar equation with different fractions?
2. What are the steps to simplify fractions with different denominators?
3. How do you handle equations with multiple terms containing $x$?
4. What happens if there are more variables in the equation?
5. How can you check if your solution to the equation is correct?

Tip: Always perform the same operation on both sides of an equation to maintain balance and obtain the correct solution.