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find the measure of (overline{de}).
diagram: points c, d, e on a line segment; ( overline{cd} ) labeled ( 2x + 7 ), ( overline{de} ) labeled ( 4(x - 3) )
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Explanation:

Step1: Assume CD = DE (since the segments are marked equally)

From the diagram, \( CD = 2x + 7 \) and \( DE = 4(x - 3) \). Since \( CD = DE \) (the tick marks indicate they are congruent), we set up the equation:
\( 2x + 7 = 4(x - 3) \)

Step2: Solve for x

Expand the right side: \( 2x + 7 = 4x - 12 \)
Subtract \( 2x \) from both sides: \( 7 = 2x - 12 \)
Add 12 to both sides: \( 19 = 2x \)
Divide by 2: \( x = \frac{19}{2} = 9.5 \) Wait, no, wait, let's check again. Wait, maybe I made a mistake. Wait, let's re - solve \( 2x + 7 = 4(x - 3) \)
\( 2x+7 = 4x - 12 \)
Subtract \( 2x \): \( 7=2x - 12 \)
Add 12: \( 2x=19 \)? No, that can't be. Wait, maybe the segments are such that CD and DE have the same length (since the number of tick marks is the same). Wait, maybe I misread. Wait, the diagram shows C---D---E, with CD having length \( 2x + 7 \) and DE having length \( 4(x - 3) \), and the tick marks on CD and DE are the same (one tick on CD, one tick on DE), so CD = DE.

Wait, let's solve \( 2x + 7 = 4(x - 3) \) correctly:

\( 2x+7 = 4x - 12 \)

Subtract \( 2x \) from both sides: \( 7 = 2x - 12 \)

Add 12 to both sides: \( 2x=7 + 12=19 \)? No, 7 + 12 is 19? Wait, 7+12 = 19? Yes. Then \( x=\frac{19}{2}=9.5 \). But that seems odd. Wait, maybe the problem is that CD and DE are equal in length (since the segments are marked with the same number of ticks). Wait, maybe I made a mistake in the equation. Wait, let's check again.

Wait, maybe the segments are CD and DE, and since the tick marks are the same, CD = DE. So:

\( 2x + 7=4(x - 3) \)

\( 2x+7 = 4x-12 \)

\( 7 + 12=4x - 2x \)

\( 19 = 2x \)

\( x = 9.5 \). Then DE is \( 4(x - 3)=4(9.5 - 3)=4\times6.5 = 26 \). Wait, but let's check CD: \( 2x + 7=2\times9.5+7 = 19 + 7=26 \). Oh, right, so CD and DE are both 26. So the measure of DE is 26.

Wait, maybe I messed up the first calculation. Let's do it again.

Step1: Set CD = DE (congruent segments)

\( 2x + 7=4(x - 3) \)

Step2: Expand the right - hand side

\( 2x + 7 = 4x-12 \)

Step3: Move x terms to one side and constants to the other

Subtract \( 2x \) from both sides: \( 7=2x - 12 \)

Add 12 to both sides: \( 2x=7 + 12=19 \)? Wait, no, 7+12 is 19? Yes. Then \( x = 9.5 \). Then DE is \( 4(x - 3)=4\times(9.5 - 3)=4\times6.5 = 26 \). And CD is \( 2\times9.5+7 = 19 + 7 = 26 \), so they are equal. So DE is 26.

Answer:

26