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The given equation is \( y = |2x - 3| \).

### (a) Find the x-intercept:
To find the x-intercept, set \( y = 0 \) and solve for \( x \):
\[
0 = |2x - 3|
\]
The absolute value equation \( |2x - 3| = 0 \) is true when:
\[
2x - 3 = 0 \implies x = \frac{3}{2}
\]

So, the x-intercept is \( \left(\frac{3}{2}, 0\right) \).

### (b) Determine where the graph is increasing and where it is decreasing:
The graph of an absolute value function like \( y = |2x - 3| \) forms a V-shape. It has a vertex where the expression inside the absolute value equals zero:
\[
2x - 3 = 0 \implies x = \frac{3}{2}
\]

- **Increasing**: For \( x > \frac{3}{2} \), the function increases because the slope of \( 2x - 3 \) is positive.
- **Decreasing**: For \( x < \frac{3}{2} \), the function decreases as the slope is negative on this side of the vertex.

### Graph Identification:
Given the V-shape nature of the graph and the vertex at \( x = \frac{3}{2} \), the correct graph should show a V with its vertex at \( \left(\frac{3}{2}, 0\right) \). From the provided options (A, B, C, D), we need to select the graph that matches this description.

Based on the x-intercept and the shape of the graph around \( x = \frac{3}{2} \), the correct answer is **Option D**.

Would you like further details or have any questions about this solution?

### Related Questions:
1. What is the vertex form of an absolute value function?
2. How do absolute value functions change when you alter the coefficients?
3. How do you find the y-intercept of \( y = |2x - 3| \)?
4. What is the general form of the graph of an absolute value function?
5. How does the graph of \( y = |2x - 3| \) compare to \( y = 2x - 3 \)?

### Tip:
For absolute value functions of the form \( y = |ax + b| \), the graph always has a V-shape with the vertex at the point where \( ax + b = 0 \).

Graph the equation y = |2x - 3| by hand.
(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 - 3|. This step-by-step solution covers finding the x-intercept and determining where the graph is increasing or decreasing. Suitable for high school algebra students.

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