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To find the measure of each segment in the given diagram, we'll need to set up and solve the equations based on the segment relationships.

### Step 1: Understand the Diagram
- The segment \( \overline{CD} \) is labeled with the expression \( 3x - 6 \).
- The segment \( \overline{DE} \) is labeled with the expression \( 2(x + 1) \).

### Step 2: Find the Measures of the Segments
Assuming that \( \overline{CD} \) and \( \overline{DE} \) are parts of a straight line segment \( \overline{CE} \), we can find the individual lengths.

1. **Measure of \( \overline{CD} \):**
   \[
   \overline{CD} = 3x - 6
   \]

2. **Measure of \( \overline{DE} \):**
   \[
   \overline{DE} = 2(x + 1)
   \]

### Step 3: Set up an Equation (if there is additional information)
If we know that \( \overline{CE} \) (which would be the sum of \( \overline{CD} \) and \( \overline{DE} \)) equals some specific value, or if \( D \) is a midpoint, then we could set up an equation.

For example, if \( D \) is the midpoint, then:
\[
\overline{CD} = \overline{DE}
\]
\[
3x - 6 = 2(x + 1)
\]
Solve for \( x \):

\[
3x - 6 = 2x + 2
\]
\[
3x - 2x = 2 + 6
\]
\[
x = 8
\]

### Step 4: Plug \( x \) back into the expressions
1. Measure of \( \overline{CD} \):
   \[
   \overline{CD} = 3(8) - 6 = 24 - 6 = 18
   \]
2. Measure of \( \overline{DE} \):
   \[
   \overline{DE} = 2(8 + 1) = 2(9) = 18
   \]

Thus, both segments \( \overline{CD} \) and \( \overline{DE} \) are 18 units in length.

Would you like to confirm the assumptions or explore other potential conditions in this problem? Also, here are 5 related questions for practice:

1. What if \( D \) is not the midpoint? How would that change the equation?
2. How do you find the total length of \( \overline{CE} \)?
3. What would happen if the length of \( \overline{CE} \) were given?
4. How would you solve for \( x \) if \( \overline{CE} \) were twice as long as \( \overline{DE} \)?
5. Can you write a general equation for \( \overline{CE} \) given different lengths of \( \overline{CD} \) and \( \overline{DE} \)?

**Tip:** Always check if the diagram or problem description mentions specific conditions, like midpoints or equal lengths, as they can guide you in setting up the correct equation.

Discover how to find the measure of each segment in a geometric diagram using linear equations and segment length relationships. This problem involves solving equations for segment lengths, suitable for Grades 9-12 students studying geometry and algebra.

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