From the provided image, we can see the graph of $y = f(x)$ and a series of inequalities asking for the intervals where $f(x)$ is less than, less than or equal to, greater than, and greater than or equal to zero.

Step-by-Step Solution

Let's analyze each inequality based on the graph.

1. Identify the x-intercepts: The x-intercepts are where $f(x) = 0$. By observing the graph, the x-intercepts appear to be at approximately:

   • $x = -3$
   • $x = -1$
   • $x = 2$

   These are the points where the graph crosses the x-axis.

2. Determine intervals for $f(x) < 0$, $f(x) \leq 0$, $f(x) > 0$, and $f(x) \geq 0$:

   • For $f(x) < 0$ (Part 1): Find the intervals where the graph is below the x-axis.

     • The graph is below the x-axis from $x = -3$ to $x = -1$ and from $x = 2$ to $\infty$.
     • Therefore, the solution is: $(-3, -1) \cup (2, \infty)$.

   • For $f(x) \leq 0$ (Part 2): Find the intervals where the graph is on or below the x-axis.

     • This includes the intervals $(-3, -1)$ and $(2, \infty)$, as well as the points where $f(x) = 0$ at $x = -3$, $x = -1$, and $x = 2$.
     • Thus, the solution is: $[-3, -1] \cup [2, \infty)$.

   • For $f(x) > 0$ (Part 3): Find the intervals where the graph is above the x-axis.

     • The graph is above the x-axis in the interval $(-1, 2)$.
     • The solution is: $(-1, 2)$.

   • For $f(x) \geq 0$ (Part 4): Find the intervals where the graph is on or above the x-axis.

     • This includes the interval $(-1, 2)$ as well as the points $x = -1$ and $x = 2$, where $f(x) = 0$.
     • The solution is: $[-1, 2]$.

Would you like further explanation on any part?

Related Questions:

1. How can we determine intervals where a function is positive or negative without a graph?
2. What are the implications of strict inequalities (e.g., $f(x) < 0$) vs. non-strict (e.g., $f(x) \leq 0$) in interval notation?
3. How does interval notation differ when a function includes discontinuities?
4. How can we solve inequalities for more complex functions that involve multiple x-intercepts?
5. How do the zeros of a function impact its inequality solutions?

Tip:

When interpreting inequalities on graphs, pay close attention to the x-intercepts and the intervals where the graph lies above or below the x-axis. These indicate where the function values are positive or negative.