The image shows a graph $y = f(x)$ with related questions asking to solve inequalities and write solution sets in interval notation. Let me analyze the graph and assist you in solving each part.

Analyzing the Graph:

The graph appears to show intervals where $f(x)$ is positive, negative, or zero. Key points include where the curve crosses the x-axis, as this helps determine intervals for each inequality.

(a) $f(x) < 0$:

This represents where the graph is below the x-axis.

• From the graph, $f(x) < 0$ on:
  • $(-\infty, -3)$ and $(0, 2)$.

Solution Set: $(-\infty, -3) \cup (0, 2)$.

(b) $f(x) \leq 0$:

This includes where $f(x)$ is negative or zero (touching the x-axis).

• From the graph, $f(x) \leq 0$ on:
  • $(-\infty, -3]$ and $[0, 2]$.

Solution Set: $(-\infty, -3] \cup [0, 2]$.

(c) $f(x) > 0$:

This represents where the graph is above the x-axis.

• From the graph, $f(x) > 0$ on:
  • $(-3, 0)$ and $(2, \infty)$.

Solution Set: $(-3, 0) \cup (2, \infty)$.

(d) $f(x) \geq 0$:

This includes where $f(x)$ is positive or zero.

• From the graph, $f(x) \geq 0$ on:
  • $[-3, 0]$ and $[2, \infty)$.

Solution Set: $[-3, 0] \cup [2, \infty)$.

Would you like a detailed explanation of how I derived these intervals? Let me know!

Related Questions:

1. How do you determine the intervals where a function is positive or negative?
2. What is the significance of using brackets ($[ ]$) vs. parentheses ($( )$) in interval notation?
3. How do we check the behavior of a function around critical points like $x = -3, 0, 2$?
4. What changes if the inequality includes or excludes the equals sign ($\leq, <$)?
5. Can you explain how to solve such inequalities algebraically if given $f(x)$ explicitly?

Tip: Always check the graph thoroughly for key points where $f(x) = 0$, as these divide the intervals for inequalities.