The problem asks us to find the value of $x$, given the angles:

$m\angle TUV = (9x + 1)^\circ, \quad m\angle TUW = (7x - 9)^\circ, \quad m\angle WUV = (5x - 11)^\circ.$

The key to solving this is recognizing that these three angles share a common vertex $U$ and form a straight line. Since the sum of the angles on a straight line is $180^\circ$, we can write the following equation:

$m\angle TUV + m\angle TUW + m\angle WUV = 180^\circ.$

Substitute the given expressions for the angles:

$(9x + 1) + (7x - 9) + (5x - 11) = 180.$

Let's solve this equation step by step.

1. Combine like terms on the left side: $9x + 7x + 5x + 1 - 9 - 11 = 180.$

2. Simplify further: $21x - 19 = 180.$

3. Add 19 to both sides: $21x = 199.$

4. Finally, divide both sides by 21 to find $x$: $x = \frac{199}{21}.$

This gives us: $x \approx 9.476.$

Would you like to know more details or have any questions?

Here are 5 related questions:

1. How do you find the value of an unknown angle when given a straight line?
2. Can you explain the concept of linear pairs of angles?
3. What other types of angle relationships are useful in geometry?
4. How would this problem change if the angles formed a full circle instead of a straight line?
5. What are alternate methods to check the correctness of the value of $x$?

Tip: When working with angles that form a straight line, always check if their sum is $180^\circ$, as this can simplify calculations.