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.