We are given that $UV = 3x$, $VW = x + 2$, and $UW = 10$. From the diagram, we can infer that $UW$ is the total length of the segment, which is the sum of $UV$ and $VW$. Thus, we can set up the equation:

$UV + VW = UW$

Substitute the given expressions:

$3x + (x + 2) = 10$

Now, solve for $x$:

$3x + x + 2 = 10$ $4x + 2 = 10$ $4x = 10 - 2$ $4x = 8$ $x = \frac{8}{4} = 2$

Now that we have $x = 2$, substitute it back into the expression for $UV$:

$UV = 3x = 3(2) = 6$

Thus, $UV = 6$.

Let me know if you would like more details or have any questions. Here are some related questions for further exploration:

1. What is the value of $VW$?
2. How can you verify the total length of $UW$ using the values of $UV$ and $VW$?
3. If the total length $UW$ was 15 instead of 10, how would the value of $UV$ change?
4. How would the problem change if $VW = 2x + 3$?
5. How can you use this type of linear equation problem in real-life scenarios?

Tip: Always verify your solution by substituting the value of the variable back into the original equation.