To solve the inequalities based on the graph of $y = f(x)$, we need to find where the function is positive or negative on the $y$-axis and express the solution sets in interval notation.

Step-by-step analysis:

1. Identify where the graph is above the x-axis:

   • The function $f(x) > 0$ (positive values of $f(x)$) occurs when the graph is above the x-axis.
   • From the graph, we observe that the function is positive in the following intervals:
     • From $x \approx -6$ to $x \approx -3$
     • From $x \approx 2$ to $x \approx 6$

2. Identify where the graph is below the x-axis:

   • The function $f(x) < 0$ (negative values of $f(x)$) occurs when the graph is below the x-axis.
   • From the graph, we observe that the function is negative in the following intervals:
     • From $x \approx -3$ to $x \approx 2$
     • For $x > 6$

3. Identify the points where the function crosses the x-axis:

   • The function crosses the x-axis (where $f(x) = 0$) at approximately $x = -6$, $x = -3$, $x = 2$, and $x = 6$.

Solution sets in interval notation:

• For $f(x) > 0$ (where the function is positive): $(-6, -3) \cup (2, 6)$

• For $f(x) < 0$ (where the function is negative): $(-3, 2) \cup (6, \infty)$

Let me know if you'd like further clarifications or details!

Here are 5 related questions:

1. How can we identify the critical points of $f(x)$ from the graph?
2. What is the significance of the intervals where $f(x) = 0$?
3. How would the solution change if the inequalities were $f(x) \geq 0$ or $f(x) \leq 0$?
4. What are the turning points of the function based on the graph?
5. How can we determine whether a graph represents a polynomial function?

Tip:

Always pay attention to the points where the graph intersects the x-axis, as these are crucial for determining the solution to inequalities involving $f(x)$.