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The image provided shows a geometry problem involving a diagram with angles. Here's a breakdown of what I see:

- The angle ∠PEC is given as \(36^\circ\).
- An expression \(5x + 17^\circ\) is also associated with an arc, likely representing the measure of an angle or arc on the circle.
- Another angle or arc measure is labeled as \(128^\circ\).
- The diagram includes four possible equations to solve for \(x\):
  - \(22x - 36 = 128\)
  - \(5x - 19 = 128\)
  - \(22x + 36 = 128\)
  - \(5x + 53 = 128\)

The problem asks which equation can be used to solve for \(x\).

### Approach:
Given that the angles and arcs are part of a circle, the equation likely relates to the properties of inscribed angles or the sum of angles along a straight line. Without further details, it seems that the relationship between the given angles, the arc, and the expression \(5x + 17\) would help form an equation. The goal is to find the correct equation from the options that matches the geometry of the problem.

### Solution:
Since the sum of angles forming a straight line should equal \(180^\circ\), and the angle \(36^\circ\) is given, the arc measure \(128^\circ\) might be involved in this reasoning.

The most likely correct equation is:
\[ 5x + 17 = 128 \]

Rewriting this gives:
\[
5x = 128 - 17
\]
\[
5x = 111
\]
Thus:
\[
x = \frac{111}{5} = 22.2
\]

Would you like more details on the geometric reasoning or further explanations?

### Related Questions:
1. How are angles and arcs related in a circle?
2. What is the relationship between inscribed angles and the arc they subtend?
3. How do we use the sum of angles on a straight line in circle geometry?
4. What are the different types of angles formed in a circle?
5. How can we use algebra to solve geometric problems involving expressions?

### Tip:
When solving geometric problems with expressions, always focus on identifying key relationships between angles, arcs, and lines to form accurate equations.

Which equation can be used to solve for x based on the geometry diagram involving angle measures and an arc?

Learn how to solve for x using angle measures and arc relationships in circle geometry. This problem demonstrates algebraic methods and angle properties, suitable for students in Grades 9-11.

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