https://file.aiquickdraw.com/mathbot/file/1729097444231gt73tplg.jpg

Let's analyze the graph and solve the four sub-questions.

The graph shows a function \( f(x) \), and we need to answer the following questions based on its behavior.

### Part A: Solution to the inequality \( f(x) > 0 \)
We need to determine where the function is above the x-axis. From the graph, the function appears to be greater than 0 in two intervals:
- From \( x = -6 \) to \( x = -2 \) (the curve is above the x-axis in this range),
- From \( x = 2 \) to \( x \to \infty \).

So the solution is:
\[
f(x) > 0 \quad \text{for} \quad (-6, -2) \cup (2, \infty)
\]

### Part B: One \( x \)-value for which \( f(x) = 0 \)
From the graph, the function crosses the x-axis at \( x = 1 \), so one \( x \)-value for which \( f(x) = 0 \) is:
\[
x = 1
\]

### Part C: Solution to the inequality \( f(x) \geq 0 \)
This includes where the function is greater than or equal to 0, so we add the points where the function touches the x-axis:
- From \( x = -6 \) to \( x = -2 \),
- \( x = 1 \) (since \( f(x) = 0 \) here),
- From \( x = 2 \) to \( \infty \).

Thus, the solution is:
\[
f(x) \geq 0 \quad \text{for} \quad [-6, -2] \cup [1, 1] \cup [2, \infty)
\]

### Part D: Solution to the inequality \( f(x) \leq 0 \)
This includes where the function is less than or equal to 0. The graph is below the x-axis in the intervals:
- From \( x = -2 \) to \( x = 1 \).

So the solution is:
\[
f(x) \leq 0 \quad \text{for} \quad [-2, 1]
\]

Let me know if you need more details or have any questions! Here are 5 related questions you might consider:

1. How do you determine the intervals where \( f(x) \) is positive or negative?
2. How do you handle inequalities when the function touches the x-axis at specific points?
3. What are the differences between solving \( f(x) > 0 \) and \( f(x) \geq 0 \)?
4. Can a function be positive and negative on the same interval?
5. How can you use the graph of the derivative of a function to solve inequalities like these?

**Tip**: Always pay attention to the open and closed intervals when solving inequalities involving \( f(x) \) because it indicates whether the value at the boundary is included or not!

A) What is the solution to the inequality f(x) > 0? B) What is one x value for which f(x) = 0? C) What is the solution to the inequality f(x) ≥ 0? D) What is the solution to the inequality f(x) ≤ 0?

This problem involves solving inequalities by analyzing a function graph. Learn how to find the intervals where the function is greater than, less than, or equal to 0. Suitable for Grades 10-12, this covers key concepts in graphing and interval notation.

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