The graph of $y = f(x)$ is given, and we are tasked with solving inequalities based on this graph. Specifically, we want to determine where the function $f(x)$ is positive (above the x-axis) and negative (below the x-axis). The solutions should be written in interval notation.

Step-by-Step Breakdown:

1. Find the x-intercepts (zeros of the function): These are the points where $f(x) = 0$, meaning the function crosses the x-axis. From the graph, the approximate x-intercepts are:

   • $x = -4$
   • $x = 2$
   • $x = 5$

2. Identify intervals where $f(x)$ is positive or negative:

   • The function is positive where the graph is above the x-axis.
   • The function is negative where the graph is below the x-axis.

3. Intervals of positivity ($f(x) > 0$):

   • From $x \in (-4, 2)$, the graph is below the x-axis, so there is no positivity in this region.
   • The function is positive between $x = 2$ and $x = 5$, meaning $f(x) > 0$ on the interval: $(2, 5)$

4. Intervals of negativity ($f(x) < 0$):

   • For $x \in (-\infty, -4)$, the function is negative (graph is below the x-axis).
   • Between $x = -4$ and $x = 2$, the function remains negative.
   • Finally, for $x \in (5, \infty)$, the graph again dips below the x-axis.

   So, the function is negative in the intervals: $(-\infty, -4) \cup (-4, 2) \cup (5, \infty)$

Solution Sets in Interval Notation:

• $f(x) > 0$ on the interval $(2, 5)$.
• $f(x) < 0$ on the intervals $(-\infty, -4) \cup (-4, 2) \cup (5, \infty)$.

Do you have any questions about these inequalities or need more details?

Here are 5 related questions that may help you expand your understanding:

1. How do you determine if a function is positive or negative based solely on its graph?
2. What happens at the x-intercepts of a function in terms of positivity or negativity?
3. Can a function have more than one interval where it is positive or negative?
4. How would the solution change if the graph touched the x-axis but didn’t cross it at certain points?
5. How would you find the exact values of the x-intercepts algebraically?

Tip: Always pay attention to the behavior of the function around the x-intercepts when solving inequalities; the function switches sign at these points.