To determine the numerical length of line segment $\overline{TV}$, we can use the information given:

• $TU = 4$
• $UV = 5x$
• $TV = 3x + 6$

Since $U$ is a point on the line segment $\overline{TV}$, we know that:

$TV = TU + UV$

Substituting the given values:

$3x + 6 = 4 + 5x$

Now, solve this equation for $x$:

$3x + 6 = 4 + 5x$

Subtract $3x$ from both sides:

$6 = 4 + 2x$

Subtract 4 from both sides:

$2 = 2x$

Divide both sides by 2:

$x = 1$

Now that we know $x = 1$, substitute it back into the expression for $TV$:

$TV = 3x + 6 = 3(1) + 6 = 3 + 6 = 9$

Therefore, the numerical length of $\overline{TV}$ is $\boxed{9}$.

Would you like to see more details or have any questions about this? Here are five related questions:

1. How would the problem change if $U$ were the midpoint of $\overline{TV}$?
2. What is the length of $UV$ given that $x = 1$?
3. How would you set up the problem if $TU = 3x + 2$ and $UV = x + 4$?
4. Can you verify the solution by substituting back into the original expressions?
5. What happens to the length of $\overline{TV}$ if $x$ were to double?

Tip: Always double-check your solution by substituting the found value back into the original equation to ensure it satisfies all given conditions.