In the image, we are given two parallel lines $m \parallel n$, and a transversal intersects them, forming two angles:

1. $(9x - 8)^\circ$
2. $(4x + 22)^\circ$

Since the lines are parallel, these two angles are alternate interior angles, which means they are equal. Therefore, we can set the two expressions for the angles equal to each other:

$9x - 8 = 4x + 22$

Solving for $x$:

1. Subtract $4x$ from both sides: $9x - 4x - 8 = 22$ $5x - 8 = 22$

2. Add 8 to both sides: $5x = 30$

3. Divide both sides by 5: $x = 6$

Thus, the value of $x$ is $\boxed{6}$.

Would you like a detailed explanation on why alternate interior angles are equal? Here are some follow-up questions for practice:

1. What are the properties of alternate interior angles?
2. How do you determine whether two lines are parallel using a transversal?
3. What other angle relationships exist when a transversal intersects two parallel lines?
4. If $x = 6$, what is the measure of each angle in the diagram?
5. How would the solution change if the lines were not parallel?

Tip: When solving equations with angles, always check if the lines involved are parallel, as this determines angle relationships.