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To solve this problem, we need to analyze the equation \( y = |2x - 5| \) and answer the following questions:

1. **Graph the equation** by selecting the correct graph from the options.
2. **Find the x-intercept**.
3. **Determine the intervals where the graph is increasing and decreasing**.

### Step-by-Step Solution:

#### Step 1: Analyze the Equation
The equation given is \( y = |2x - 5| \). This is an absolute value function, which creates a V-shaped graph.

#### Step 2: Find the Vertex and X-Intercept
1. **Vertex**: The expression inside the absolute value, \(2x - 5\), equals zero when \( x = \frac{5}{2} = 2.5 \). So, the vertex is at \( \left(2.5, 0\right) \).
2. **X-Intercept**: Since the vertex is where the function touches the x-axis, the x-intercept is at \( \left(2.5, 0\right) \).

#### Step 3: Determine Increasing and Decreasing Intervals
1. For \( x < 2.5 \): The slope of \(2x - 5\) is negative (since it’s decreasing on this interval), so the function will be decreasing.
2. For \( x > 2.5 \): The slope of \(2x - 5\) is positive, making the function increase.

Thus:
- **Increasing on** \( (2.5, \infty) \)
- **Decreasing on** \( (-\infty, 2.5) \)

### Step 4: Match with the Graph
Now, based on these characteristics, we should look for a V-shaped graph with a vertex at \( (2.5, 0) \), decreasing on the left and increasing on the right. Looking at the options:

- **Option C** has the correct vertex and interval characteristics.

### Final Answers
- **Correct Graph**: Option C
- **X-Intercept**: \( (2.5, 0) \)
- **Increasing Interval**: \( (2.5, \infty) \)
- **Decreasing Interval**: \( (-\infty, 2.5) \)

Would you like further details on each step, or do you have any specific questions?

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Here are five additional questions related to this topic:
1. How do you determine the vertex of an absolute value function?
2. What changes in the graph if the absolute value term is multiplied by a negative sign?
3. How would the graph change if we had \( y = |2x + 3| \) instead?
4. Can you describe how to find intercepts for any absolute value function?
5. What are the differences in intervals if the slope within the absolute value changes?

**Tip:** For absolute value functions, the vertex is located where the expression inside the absolute value equals zero.

Graph the equation y = |2x - 5|. (a) Find the x-intercept. (b) Determine where the graph is increasing and where it is decreasing.

Learn how to graph the absolute value equation y = |2x - 5|. Find the x-intercept and determine the intervals where the graph is increasing and decreasing. This problem covers concepts of absolute value functions, graphing techniques, and interval notation. Suitable for students in Grades 9-10.

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