https://file.aiquickdraw.com/mathbot/file/17288791029442x6ed2ij.jpg

To solve the inequalities using the given graph, let's analyze the behavior of the function \( f(x) \) at different intervals based on its sign (whether it's above or below the x-axis).

### (a) \( f(x) < 0 \) 
This represents the intervals where the graph of \( f(x) \) is below the x-axis, meaning the values of \( f(x) \) are negative.

From the graph, \( f(x) \) is below the x-axis in the following intervals:
- \( x \in (0, 1) \)
- \( x \in (3, \infty) \)

Thus, the solution for \( f(x) < 0 \) is:
\[
x \in (0, 1) \cup (3, \infty)
\]

### (b) \( f(x) \geq 0 \)
This represents the intervals where the graph of \( f(x) \) is on or above the x-axis, meaning the values of \( f(x) \) are non-negative.

From the graph, \( f(x) \) is on or above the x-axis in the following intervals:
- \( x \in (-\infty, 0] \)
- \( x \in [1, 3] \)

Thus, the solution for \( f(x) \geq 0 \) is:
\[
x \in (-\infty, 0] \cup [1, 3]
\]

### Would you like further details or have any questions?

Here are five relative questions for further exploration:
1. How would you solve inequalities if the graph was more complex?
2. What is the difference between strict and non-strict inequalities on a graph?
3. Can you determine whether the function is increasing or decreasing in the different intervals?
4. How would the solutions change if you were given a piecewise function instead of a continuous graph?
5. What additional information can be derived from analyzing the turning points of the graph?

**Tip:** When analyzing a graph for inequalities, always focus on where the graph crosses the x-axis—this helps in identifying where the function changes sign!

Use the graph of the function f to solve the inequality.

Learn how to solve the inequalities f(x) < 0 and f(x) ≥ 0 by analyzing the graph of the function. This problem explores intervals where the function is negative or non-negative, covering key concepts in graph analysis and inequality solving. Suitable for students in Grades 9-11.

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