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Let's analyze the first inequality as an example to find the five points and the solution set. The process will be similar for the others. 

### 1) \( y > 2 |x - 3| \)

#### Step 1: Understanding the inequality
We need to graph the inequality \( y > 2 |x - 3| \), which is a transformation of the basic absolute value function \( y = |x| \). The expression \( |x - 3| \) shifts the graph 3 units to the right, and multiplying by 2 stretches the graph vertically by a factor of 2.

#### Step 2: Finding 5 points
We pick values of \( x \), substitute them into the equation \( y = 2|x - 3| \), and then plot these points.

- When \( x = 1 \): 
  \( y = 2|1 - 3| = 2(2) = 4 \)
  Point: \( (1, 4) \)

- When \( x = 2 \): 
  \( y = 2|2 - 3| = 2(1) = 2 \)
  Point: \( (2, 2) \)

- When \( x = 3 \): 
  \( y = 2|3 - 3| = 2(0) = 0 \)
  Point: \( (3, 0) \)

- When \( x = 4 \): 
  \( y = 2|4 - 3| = 2(1) = 2 \)
  Point: \( (4, 2) \)

- When \( x = 5 \): 
  \( y = 2|5 - 3| = 2(2) = 4 \)
  Point: \( (5, 4) \)

#### Step 3: Graphing the inequality
Once these points are plotted, we graph the line and then shade the area above the line because the inequality is \( y > 2 |x - 3| \). This shaded region represents the solution set.

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Would you like to go through another inequality from the list or need more details on this one?

### Follow-up Questions:
1. How does the transformation affect the graph of an absolute value function?
2. What happens if we have a "less than" inequality instead of "greater than"?
3. How do vertical stretches or compressions affect the steepness of the graph?
4. What is the difference between \( y = |x| \) and \( y = |x - h| \)?
5. How does shifting horizontally or vertically change the solution set?

### Tip:
When graphing absolute value inequalities, always plot the "equal to" line first and then determine the region to shade based on the inequality symbol.

Graph each inequality using 5 points. Pick a point and test your solution set.

Learn how to graph absolute value inequalities using 5 key points. This guide covers graphing inequalities such as y > 2 |x - 3|, transformations of absolute value functions, and finding solution sets by plotting points.

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