https://file.aiquickdraw.com/mathbot/file/17312848872432sxltjir.jpg

Let's analyze each part of the question based on the graph and the conditions provided.

### Given Functions and Lines
1. **\( f(x) \)** is represented by the blue line.
2. **\( g(x) \)** and **\( h(x) \)** are represented by horizontal red lines:
   - \( g(x) \) is at approximately \( y = 4 \).
   - \( h(x) \) is at approximately \( y = -3 \).

### Problem Analysis
1. We are asked to determine intervals of \( x \) that satisfy specific relationships between \( f(x) \), \( g(x) \), and \( h(x) \).

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#### Part (a): \( f(x) \leq h(x) \) or \( f(x) > g(x) \)

- For \( f(x) \leq h(x) \):
  - From the graph, \( f(x) \leq h(x) \) occurs where the blue line \( f(x) \) is at or below \( y = -3 \).
  - This condition holds for \( x \leq -4 \).
  
- For \( f(x) > g(x) \):
  - The condition \( f(x) > g(x) \) holds for \( x < -2 \) based on where \( f(x) \) is above the line at \( y = 4 \).

- Combining both conditions:
  - The solution for part (a) is \( (-\infty, -4] \cup (-2, 2) \).

#### Part (b): \( h(x) < f(x) \leq g(x) \)

- \( h(x) < f(x) \leq g(x) \) means that \( f(x) \) must be above \( h(x) \) (i.e., above \( y = -3 \)) and at or below \( g(x) \) (i.e., below or equal to \( y = 4 \)).
- This interval appears to be between \( x = -4 \) and \( x = 2 \).
- Therefore, the solution for part (b) is \( (-4, 2] \).

#### Part (c): \( f(x) > h(x) \)

- The graph shows that \( f(x) > h(x) \) for \( x \in (-\infty, 2) \).
- This solution is confirmed in the image as \( (-\infty, 2) \).

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### Final Answers

- **(a)** \( (-\infty, -4] \cup (-2, 2) \)
- **(b)** \( (-4, 2] \)
- **(c)** \( (-\infty, 2) \) (already provided)

Would you like further details on these calculations or have additional questions?

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Here are some follow-up questions that could extend this topic:

1. What are other methods to solve inequalities involving multiple functions?
2. How would the intervals change if \( h(x) \) or \( g(x) \) were different values?
3. What is the significance of intersection points in solving inequalities?
4. How does interval notation represent continuous and discrete sets?
5. How would you approach if asked for \( x \) values where \( f(x) < g(x) \)?

**Tip:** Always double-check the intersections of graphs visually and confirm values by substitution if possible to avoid interpretation errors.

Use interval notation to answer the following:

a. Give all values of x where f(x) ≤ h(x) or f(x) > g(x).
b. Give all values of x where h(x) < f(x) ≤ g(x).
c. Give all values of x where f(x) > h(x).

Given functions:
- f(x): Represented by a blue line.
- g(x): Horizontal red line at approximately y = 4.
- h(x): Horizontal red line at approximately y = -3.

Learn how to solve inequalities involving functions using interval notation with a detailed step-by-step solution. This problem involves interpreting a graph of f(x), g(x), and h(x) and finding intervals where f(x) ≤ h(x) or f(x) > g(x), among other conditions. Suitable for students in Grades 9-11.

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