The graph of $f(x) = ax^2 + bx + c$ is shown in the image. Based on the visual of the parabola and the inequality conditions:

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

The inequality $f(x) > 0$ refers to the intervals where the graph of the parabola lies above the x-axis.

From the graph:

• The parabola intersects the x-axis at two points (let's call them $x_1$ and $x_2$).
• The graph is above the x-axis between $x_1$ and $x_2$.

Thus, the solution is the open interval: $(x_1, x_2)$ where $x_1$ and $x_2$ are the x-intercepts of the graph.

If you identify the approximate values of $x_1$ and $x_2$ from the graph, you can express the solution more specifically. Based on the provided options:

• A is the correct choice: "There is a solution, and it is not all real numbers."

Part (b) Solve $f(x) < 0$:

The inequality $f(x) < 0$ refers to the intervals where the graph of the parabola lies below the x-axis.

From the graph:

• The parabola is below the x-axis for $x < x_1$ and $x > x_2$.

Thus, the solution is: $(-\infty, x_1) \cup (x_2, \infty)$

If you'd like a step-by-step breakdown of how to solve it further or need to interpret the exact intercepts $x_1$ and $x_2$, let me know!

Follow-up Questions:

1. What is the general form of a quadratic inequality solution?
2. How do you identify the intervals of positivity or negativity for a parabola?
3. What do the roots $x_1$ and $x_2$ represent in this graph?
4. What is the role of the vertex in determining the maximum/minimum of the parabola?
5. How does the coefficient $a$ in $ax^2$ affect the parabola's shape and direction?

Tip:

Always analyze where the parabola intersects the x-axis to break inequalities into appropriate intervals!