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To shade the regions for the absolute value inequalities, you'll need to follow these steps:

1. **Plot the boundary**: The boundary of each inequality is derived by converting the inequality into an equation. For example, for \( y > 2|x - 3| \), first plot the equation \( y = 2|x - 3| \).
2. **Identify the region**: Based on whether the inequality is "greater than" or "less than", shade the appropriate region. 
   - For \( > \) or \( \geq \), shade the region **above** the boundary.
   - For \( < \) or \( \leq \), shade the region **below** the boundary.

Let’s go through each inequality one by one and determine the correct shading:

1. **\( y > 2|x - 3| \)**
   - Plot the V-shaped graph of \( y = 2|x - 3| \), with the vertex at \( (3, 0) \).
   - Since \( y \) is greater than the boundary, shade **above** the V-shape.

2. **\( y \geq -3|x| + 2 \)**
   - Plot \( y = -3|x| + 2 \), an upside-down V-shape with a vertex at \( (0, 2) \).
   - Since \( y \geq \), shade **on and above** the boundary.

3. **\( y \leq -|x| + 3 \)**
   - Plot \( y = -|x| + 3 \), another upside-down V-shape with vertex at \( (0, 3) \).
   - Since \( y \leq \), shade **on and below** the boundary.

4. **\( y < 4|x| - 4 \)**
   - Plot \( y = 4|x| - 4 \), with a V-shape and vertex at \( (0, -4) \).
   - Since \( y < \), shade **below** the V-shape.

5. **\( y \geq 2|x + 3| - 1 \)**
   - Plot \( y = 2|x + 3| - 1 \), with a vertex at \( (-3, -1) \).
   - Since \( y \geq \), shade **on and above** the boundary.

6. **\( y \leq -\frac{1}{2}|x + 2| + 5 \)**
   - Plot \( y = -\frac{1}{2}|x + 2| + 5 \), an upside-down V-shape with vertex at \( (-2, 5) \).
   - Since \( y \leq \), shade **on and below** the boundary.

7. **\( y > \frac{2}{3}|x - 1| - 2 \)**
   - Plot \( y = \frac{2}{3}|x - 1| - 2 \), a V-shape with vertex at \( (1, -2) \).
   - Since \( y > \), shade **above** the boundary.

8. **\( y < \frac{1}{2}|x + 1| - 3 \)**
   - Plot \( y = \frac{1}{2}|x + 1| - 3 \), a V-shape with vertex at \( (-1, -3) \).
   - Since \( y < \), shade **below** the boundary.

Would you like detailed instructions for graphing and shading any specific inequality?

Here are 5 related questions to explore further:

1. How do absolute value inequalities affect the shape of the graph?
2. What happens when you multiply the absolute value by a negative constant?
3. Can you explain why the vertex of the absolute value graph shifts horizontally?
4. How do you test a point to verify if it satisfies the inequality?
5. How can these graphs be used to model real-world situations involving distance or deviations?

**Tip:** For inequalities with \( \geq \) or \( \leq \), don't forget to include the boundary line in the shaded region!

Graph each absolute value inequality and determine where to shade.

Learn how to graph and shade regions for absolute value inequalities, such as y > 2|x - 3| and y ≤ -|x| + 3. This lesson covers key algebraic concepts and provides a step-by-step guide to correctly identifying shaded regions in graphing absolute value functions.

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