The problem states that the two angles are supplementary. Supplementary angles add up to 180°. The two given angles are $(5x + 6)^\circ$ and $(3x - 8)^\circ$.

We can set up the following equation to represent the relationship:

$(5x + 6) + (3x - 8) = 180$

Now, let's solve for $x$:

1. Combine like terms: $5x + 3x + 6 - 8 = 180$ $8x - 2 = 180$

2. Add 2 to both sides: $8x = 182$

3. Divide both sides by 8: $x = \frac{182}{8} = 22.75$

Thus, $x = 22.75$.

Would you like further details or explanations? Here are five related questions you might consider:

1. What are complementary angles, and how do they differ from supplementary angles?
2. Can two obtuse angles be supplementary?
3. How would you verify that the angles are supplementary with the calculated value of $x$?
4. What is the value of each angle after substituting $x = 22.75$?
5. How would you solve for $x$ if the angles were complementary instead of supplementary?

Tip: Always verify your solution by substituting the value of $x$ back into the original expressions for the angles!