Let's analyze this graph and the problem provided.

A) Interpretation of $f(x) < 0$:

• Solving $f(x) < 0$ means finding the $x$-values where the graph of $f(x)$ lies below the $x$-axis.
• Based on the image, you correctly chose the option "below the $x$-axis," which is appropriate for $f(x) < 0$.

B) Values of $x$ where $f(x) < 0$:

• From the graph, $f(x)$ is negative (below the $x$-axis) in the region between $x = -2$ and $x = 0$.
• Therefore, the interval for which $f(x) < 0$ is: $(-2, 0)$

C) Values of $x$ where $f(x) > 0$:

• To solve $f(x) > 0$, we look for where the graph is above the $x$-axis.
• From the graph, $f(x) > 0$ for two regions:
  1. To the left of $x = -2$, i.e., $(-\infty, -2)$,
  2. To the right of $x = 0$, i.e., $(0, 2) \cup (2, \infty)$.

Thus, the solution to $f(x) > 0$ is: $(-\infty, -2) \cup (0, 2) \cup (2, \infty)$

Would you like further details or have any specific questions?

5 Questions for Further Exploration:

1. How do we determine the behavior of the function near vertical asymptotes like $x = 2$?
2. What can be inferred about $f(x)$ at $x = -2$ and $x = 0$, where the graph crosses the $x$-axis?
3. How does the behavior of the function at $x = 2$ (the vertical asymptote) impact the intervals for $f(x) > 0$?
4. Why is it important to use interval notation when describing solutions for inequalities?
5. How would you solve $f(x) = 0$ using this graph?

Tip:

When determining intervals for $f(x) > 0$ or $f(x) < 0$, always watch for any asymptotes, as they split the function into different regions.