To solve for $x$ in this diagram, let’s analyze the information given.

1. Parallel Lines and Transversal:

   • Since lines $m$ and $n$ are parallel ($m \parallel n$), and there is a transversal line intersecting them, the angles formed have specific relationships.

2. Angle Relationships:

   • The two angles labeled $(x - 3)^\circ$ and $(4x + 3)^\circ$ are alternate interior angles since they are on opposite sides of the transversal and lie between the parallel lines.
   • For alternate interior angles formed by a transversal intersecting two parallel lines, the angles are equal.

3. Setting Up the Equation:

   • Since these angles are equal, we can set up the following equation: $x - 3 = 4x + 3$

4. Solving the Equation:

   • Move all terms involving $x$ to one side of the equation and constants to the other: $x - 4x = 3 + 3$ $-3x = 6$
   • Now, divide by $-3$ to solve for $x$: $x = -2$

Conclusion:

The value of $x$ is $-2$.

Would you like more details or have any questions on the solution?

Here are five additional questions that extend this topic:

1. How do we identify alternate interior angles in different configurations?
2. What are corresponding angles, and how are they different from alternate interior angles?
3. If the angles were supplementary instead of equal, how would the equation change?
4. How can we use similar angle relationships to solve for unknowns in non-parallel lines?
5. What other properties arise when two lines are parallel and intersected by a transversal?

Tip: Remember that alternate interior angles are only equal when the lines they lie between are parallel.