In the provided image, there are two triangles, each with three angle expressions in terms of $x$. To solve for $x$ in each triangle, we'll use the fact that the sum of the interior angles in a triangle is $180^\circ$.

Problem 5:

The angles in the triangle are labeled as $5x + 8$, $9x - 24$, and the third angle is $x$.

1. Set up the equation for the sum of the angles: $(5x + 8) + (9x - 24) + x = 180$

2. Combine like terms: $15x - 16 = 180$

3. Add 16 to both sides: $15x = 196$

4. Divide by 15: $x = \frac{196}{15} \approx 13.07$

Problem 6:

The angles in the triangle are labeled as $2x - 8$, $5x$, and $3x - 28$.

1. Set up the equation for the sum of the angles: $(2x - 8) + (5x) + (3x - 28) = 180$

2. Combine like terms: $10x - 36 = 180$

3. Add 36 to both sides: $10x = 216$

4. Divide by 10: $x = 21.6$

Let me know if you would like a more detailed explanation or have any questions.

Follow-Up Questions:

1. How would the approach change if we had a quadrilateral instead of a triangle?
2. What is the importance of angle sum properties in different polygons?
3. Can we determine individual angle measures once $x$ is found?
4. How can this type of problem be applied in real-life scenarios involving shapes?
5. What are the common mistakes students make when solving for angles in algebraic expressions?

Tip: Always double-check your calculations, especially when combining like terms, to avoid simple errors.