We are given a geometric problem involving a line segment $\overline{TV}$, with the point $U$ located on the line between $T$ and $V$. The lengths of different parts of the segment are given in terms of $x$.

Given:

• $UV = 2$
• $TU = 5x$
• $TV = 3x + 6$

Since $U$ is on the line segment $\overline{TV}$, the total length $TV$ is the sum of the lengths of $TU$ and $UV$: $TU + UV = TV$ Substituting the given values: $5x + 2 = 3x + 6$ Now, solve for $x$.

1. Subtract $3x$ from both sides: $5x - 3x + 2 = 6$ This simplifies to: $2x + 2 = 6$

2. Subtract 2 from both sides: $2x = 4$

3. Divide both sides by 2: $x = 2$

Now, substitute $x = 2$ into the expression for $TV$: $TV = 3x + 6 = 3(2) + 6 = 6 + 6 = 12$

Thus, the length of $\overline{TV}$ is $12$.

Would you like further details or have any questions?

Here are some related questions to think about:

1. What is the length of $TU$ when $x = 2$?
2. Can you find the midpoint of $\overline{TV}$?
3. If $U$ is the midpoint of $\overline{TV}$, how does that change the values of $TU$ and $UV$?
4. What would happen if $TV = 4x + 5$ instead of $3x + 6$?
5. How would you solve the problem if the lengths of $TU$ and $UV$ were given in decimals?

Tip: When solving equations involving geometric segments, always check if the sum of the parts equals the whole.