To solve this problem, we need to use the information given about the angles formed at point $U$. The sum of the angles around point $U$ should be equal to $180^\circ$ because they form a straight line.

Given:

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

The sum of these angles is:

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

Now, let's solve this equation step-by-step:

1. Combine like terms:

$9x + 7x + 5x + 1 - 9 - 11 = 180$

$21x - 19 = 180$

2. Add 19 to both sides:

$21x = 199$

3. Divide by 21:

$x = \frac{199}{21}$

Now, let's calculate the value of $x$.The value of $x$ is approximately $9.48$.

Would you like further details or have any questions? Here are some related questions for further practice:

1. How do you determine the sum of angles around a point?
2. What steps are involved in combining like terms in an equation?
3. How do you check if your solution for $x$ is correct in the context of the problem?
4. What are the properties of linear pairs of angles?
5. How can this problem be solved graphically?

Tip: Always double-check your calculations, especially when dealing with angles and equations, to avoid simple arithmetic errors!