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Let's work through the problem step-by-step.

### Problem 1: Segment Addition Postulate

We are given three places: West Morris Central (WMC), Six Flags, and Long Beach Island (LBI). The distances they traveled are described as follows:

- From WMC to LBI: \( 5x + 2 \) units.
- On the way back, they traveled \( 3x + \frac{3}{2} \) units to Six Flags.
- Then, from Six Flags to WMC, they traveled \( \frac{3}{2}x + 2 \) units.

We need to find the distance from WMC to LBI in units if the total distance from WMC to LBI is 109 miles.

### Step 1: Setting up the equation

According to the Segment Addition Postulate, the sum of the distances from WMC to Six Flags and Six Flags to LBI should equal the total distance from WMC to LBI.

\[
\left( 3x + \frac{3}{2} \right) + \left( \frac{3}{2}x + 2 \right) = 5x + 2
\]

### Step 2: Solve for \(x\)

We can now solve this equation step by step:

1. Combine like terms on the left-hand side:
   \[
   3x + \frac{3}{2} + \frac{3}{2}x + 2 = 5x + 2
   \]
   Simplifying:
   \[
   3x + \frac{3}{2}x + \frac{3}{2} + 2 = 5x + 2
   \]
   \[
   \left( 3x + \frac{3}{2}x \right) + \left( \frac{3}{2} + 2 \right) = 5x + 2
   \]
   Now, combine the terms:
   \[
   \left( 3x + 1.5x \right) + \left( 1.5 + 2 \right) = 5x + 2
   \]
   Simplifying further:
   \[
   4.5x + 3.5 = 5x + 2
   \]

2. Subtract \(4.5x\) from both sides:
   \[
   3.5 = 0.5x + 2
   \]

3. Subtract 2 from both sides:
   \[
   1.5 = 0.5x
   \]

4. Solve for \(x\):
   \[
   x = \frac{1.5}{0.5} = 3
   \]

### Step 3: Find the distances

Now that we know \(x = 3\), we can plug it into the original expressions to find the distances in units.

- From WMC to LBI:
  \[
  5x + 2 = 5(3) + 2 = 15 + 2 = 17 \, \text{units}
  \]

Thus, the distance from WMC to LBI is 17 units. Given that this distance corresponds to 109 miles, each unit represents:

\[
\text{1 unit} = \frac{109}{17} \approx 6.41 \, \text{miles}
\]

---

### Problem 2: Finding Segment Lengths

We are given a situation where:
- \(S\) is between \(T\) and \(V\),
- \(R\) is between \(S\) and \(T\),
- \(T\) is between \(R\) and \(Q\),
- \(QV = 18\), \(QT = 6\), and \(TR = RS = SV\).

The task is to find the lengths of:
1. \(RS\),
2. \(QS\),
3. \(TS\).

### Step 1: Use the given information

- \(TR = RS = SV\), meaning these segments are all equal.
- Let \(RS = x\).
  
The total distance \(QV = 18\), and we know that \(QT = 6\), so \(TV = 18 - 6 = 12\).

Thus:
\[
TR + RS + SV = TV
\]
Substitute \(RS = TR = SV = x\):
\[
x + x + x = 12
\]
\[
3x = 12
\]
\[
x = 4
\]

### Step 2: Find the required lengths

1. \(RS = 4\).
2. \(QS = QT + TS = 6 + 8 = 14\).
3. \(TS = TR + RS = 4 + 4 = 8\).

---

Would you like more details on any part of this solution? 

Here are five follow-up questions related to the problems:

1. How would the equation change if the return trip distances were different?
2. What if the total distance from WMC to LBI was a different value?
3. How can we verify the calculated value of \(x\) using a different method?
4. What is the significance of the Segment Addition Postulate in geometry?
5. Can we generalize the process of finding segment lengths for more complex line segments?

**Tip:** Always double-check the units and conditions when solving word problems!

Suppose West Morris Central, Six Flags, and Long Beach Island are collinear. This summer, a group of students traveled (5x + 2) units from WMC to LBI. On the way home, they traveled (3x + 3/2) units and stopped at Six Flags, then (3/2x + 2) units to get back to WMC. What is the distance (in units) from WMC to LBI? If that distance is 109 miles, about how many miles is each of the units described here?

Solve an algebraic problem involving segment addition and unit conversion. Learn how to calculate distances and convert units from a real-life scenario, using the Segment Addition Postulate. Suitable for Grades 9-10.

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