The problem asks to find $x$, given the following angle relationships:

• $m\angle IJA = 3x - 2$
• $m\angle IJK = 128^\circ$
• $m\angle AJK = 11x - 10$

By looking at the diagram, we can see that angles $\angle IJA$, $\angle IJK$, and $\angle AJK$ all form a straight line at point $J$, which means their sum must equal $180^\circ$. This leads to the equation:

$m\angle IJA + m\angle AJK + m\angle IJK = 180^\circ$

Now, substituting the given expressions:

$(3x - 2) + (11x - 10) + 128 = 180$

Let's solve this equation step by step.

Simplifying the equation:

$3x - 2 + 11x - 10 + 128 = 180$ Combine like terms: $14x + 116 = 180$ Subtract 116 from both sides: $14x = 64$ Divide by 14: $x = \frac{64}{14} = 4.57 \text{ (rounded to two decimal places)}$

Thus, $x \approx 4.57$.

Would you like a detailed explanation of the steps or further assistance with this?

Here are 5 questions to deepen understanding:

1. What is the sum of angles on a straight line?
2. How does angle addition work in linear arrangements?
3. How do we solve linear equations when angles are expressed in terms of variables?
4. Why is it important to combine like terms before solving for $x$?
5. What are other common types of angle problems involving straight lines?

Tip: Always double-check your algebraic work when dealing with angles and geometric figures to avoid small errors in calculations.