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!