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To solve for \( x \) in the given geometry problem, we can use the properties of similar triangles. 

### Analyzing the Problem:
Given:
- \( AB = 36 \)
- \( AC = 24 \)
- \( DE = 5x + 1 \)
- \( EC = 2x + 6 \)
- \( AB \parallel DE \)

Since \( AB \parallel DE \), the triangles \( \triangle ABC \) and \( \triangle DEC \) are similar by the Basic Proportionality Theorem (or Thales' theorem). This means the corresponding sides of these triangles are proportional:

\[
\frac{AB}{DE} = \frac{AC}{EC}
\]

### Setting up the Equation:
Substituting the given values:

\[
\frac{36}{5x + 1} = \frac{24}{2x + 6}
\]

### Solving for \( x \):
1. Cross-multiply to get rid of the fractions:

\[
36(2x + 6) = 24(5x + 1)
\]

2. Expand both sides:

\[
72x + 216 = 120x + 24
\]

3. Move all terms involving \( x \) to one side and constant terms to the other:

\[
216 - 24 = 120x - 72x
\]

\[
192 = 48x
\]

4. Solve for \( x \):

\[
x = \frac{192}{48} = 4
\]

### Answer:
The value of \( x \) is \( 4 \).

Would you like more details or explanations? Here are some related questions for further practice:

1. What is the length of \( DE \) when \( x = 4 \)?
2. How would the solution change if \( AB \) and \( DE \) were not parallel?
3. Can you prove that triangles \( ABC \) and \( DEC \) are similar using angles?
4. What are the possible values of \( x \) if \( AB \) was not given?
5. How would you solve the problem if only \( AC \) and \( EC \) were given?

**Tip:** Always check for similar triangles when you see parallel lines and transversals in geometric problems; it can simplify your calculations greatly!

Learn how to solve a geometry problem involving similar triangles. Given AB = 36, AC = 24, DE = 5x + 1, EC = 2x + 6, and AB parallel to DE, discover the value of x using the properties of similar triangles and proportionality theorems.

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