We are given that point $V$ is on line segment $\overline{UW}$, and the following segment lengths:

• $VW = x + 7$
• $UW = 3x - 9$
• $UV = 8$

Since $V$ is between $U$ and $W$, we use the segment addition postulate:

$UV + VW = UW$

Substituting the given expressions:

$8 + (x + 7) = 3x - 9$

Step 1: Solve for $x$

$8 + x + 7 = 3x - 9$

$15 + x = 3x - 9$

Subtract $x$ from both sides:

$15 = 2x - 9$

Add 9 to both sides:

$24 = 2x$

Divide by 2:

$x = 12$

Step 2: Find $UW$

$UW = 3x - 9$

Substituting $x = 12$:

$UW = 3(12) - 9 = 36 - 9 = 27$

Thus, the numerical length of $\overline{UW}$ is 27.

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Would you like a detailed explanation of any step?

Here are five related questions to expand your understanding:

1. How would the solution change if $V$ were not between $U$ and $W$?
2. What is the length of $VW$ using the found value of $x$?
3. How can you verify the solution by checking if the segment addition postulate holds?
4. What if the given values were expressed differently, such as $VW = 2x + 5$? How would that change the solution?
5. How does this problem illustrate the properties of collinear points?

Tip:

When solving for a variable in an equation involving segment lengths, always check that the final values satisfy the given conditions.