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The problem asks to determine which graph corresponds to the given piecewise function:

\[
f(x) = \begin{cases} 
-0.5x + 5 & \text{if } x < 1 \\
-0.5(x + 5) & \text{if } x \geq 1 
\end{cases}
\]

Let's analyze each piece of the function and its behavior.

### First Case: \( f(x) = -0.5x + 5 \) (when \( x < 1 \))
- This is a linear function with a slope of \( -0.5 \) and a y-intercept of \( 5 \).
- For \( x = 1 \), the function gives:
  \[
  f(1) = -0.5(1) + 5 = 4.5
  \]
  But this value is not included because the inequality is strict \( x < 1 \), so at \( x = 1 \), there should be an open circle.

### Second Case: \( f(x) = -0.5(x + 5) \) (when \( x \geq 1 \))
- This is also a linear function, but with a different slope-intercept form: \( f(x) = -0.5x - 2.5 \).
- For \( x = 1 \):
  \[
  f(1) = -0.5(1 + 5) = -0.5(6) = -3
  \]
  At \( x = 1 \), the point \( (1, -3) \) is included (so a closed circle should appear here).

### Graph Features:
- The graph should have two distinct parts:
  1. A line segment with a slope of \( -0.5 \) starting at \( y = 5 \) and continuing until \( x = 1 \), where there is an open circle at \( (1, 4.5) \).
  2. A second line with a slope of \( -0.5 \) starting at \( (1, -3) \) with a closed circle and continuing to the right.

Looking at the provided graphs:
- **Graph c** matches these characteristics:
  - The line before \( x = 1 \) is decreasing with a slope of \( -0.5 \), starting from \( y = 5 \).
  - There's an open circle at \( (1, 4.5) \).
  - After \( x = 1 \), the second segment starts at \( (1, -3) \) with a closed circle, also having a slope of \( -0.5 \).

### Conclusion:
The correct graph is **graph c**.

Would you like more details on any part of this explanation?

### Related Questions:
1. What is the significance of an open vs. closed circle on a graph?
2. How do you calculate the slope of a line segment in a piecewise function?
3. How can you verify the intersection points of piecewise functions?
4. What are the key differences between the two linear equations in this problem?
5. How do the domain restrictions affect the behavior of a piecewise function?

#### Tip:
When analyzing piecewise functions, always check the points where the function "splits" to understand how the graph behaves around these points!

Determine which is the graph of the given piecewise function: f(x) = -0.5x + 5 if x < 1, and f(x) = -0.5(x + 5) if x ≥ 1.

Learn how to graph the piecewise function defined by two linear equations: f(x) = -0.5x + 5 for x < 1 and f(x) = -0.5(x + 5) for x ≥ 1. This problem covers key algebraic concepts such as graphing, slopes, and piecewise functions. Suitable for students in Grades 9-11.

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