To solve the inequalities based on the graph of the function $f(x)$, we need to analyze where the function is positive or negative.

(a) Solve $f(x) > 0$:

The inequality $f(x) > 0$ asks us to find where the function is above the $x$-axis (positive values of $f(x)$).

From the graph:

• The function is positive between $x = -2$ and $x = -1$, and again between $x = 1$ and $x = 3$.

Thus, the solution for $f(x) > 0$ in interval notation is: $(-2, -1) \cup (1, 3)$

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

Now, we need to find where the function is either zero or negative (on or below the $x$-axis).

From the graph:

• The function is negative or zero between $x = -\infty$ and $x = -2$, at $x = -1$, and between $x = 3$ and $x = \infty$.

Thus, the solution for $f(x) \leq 0$ in interval notation is: $(-\infty, -2] \cup [-1, 1] \cup [3, \infty)$

Would you like to see more detailed steps or explanations for these intervals? Here are some related questions to explore:

1. How do we handle inequalities involving functions that cross the $x$-axis?
2. What is the difference between strict inequalities and non-strict inequalities in terms of interval notation?
3. How can we use the first and second derivative tests to confirm where the function is positive or negative?
4. Can we solve inequalities without a graph? What are alternative methods?
5. How does the domain of a function affect the solution of an inequality?

Tip: Always check if your intervals should include or exclude the boundaries by considering strict ($>$) versus non-strict ($\geq$) inequalities.