The problem consists of solving quadratic inequalities graphically, as shown in the images. Let me go through each case and explain their solutions and solution sets:

a) $2(x - 2)(x - 3) < 0$

• Graph Analysis: The graph shows a parabola opening upwards. The inequality is $f(x) < 0$, which means we are looking for the x-values where the graph lies below the x-axis.

• Solution: The parabola crosses the x-axis at $x = 2$ and $x = 3$. Since the graph is below the x-axis between these points: $\text{Solution: } x \in (2, 3)$

b) $-3(x + 5)^2 > 0$

• Graph Analysis: The equation represents a downward-facing parabola (since the coefficient of $x^2$ is negative). The inequality is $f(x) > 0$, which means we are looking for x-values where the graph is above the x-axis.

• Solution: The vertex of the parabola is at $x = -5$, and the graph does not cross the x-axis (it is completely below or touching the x-axis). Hence, there are no x-values where $f(x) > 0$. $\text{Solution: } \emptyset \; (\text{No solutions})$

d) $-3x^2 - \frac{3}{2}x + \frac{9}{16} < 0$

• Graph Analysis: This represents a downward-opening parabola. The inequality $f(x) < 0$ means we are looking for x-values where the graph lies below the x-axis.

• Solution: The graph crosses the x-axis at approximately $x = -1$ and $x = \frac{1}{2}$. Between these points, the graph is below the x-axis: $\text{Solution: } x \in \left(-1, \frac{1}{2}\right)$

e) $5x^2 + 3 \geq 0$

• Graph Analysis: The equation represents an upward-opening parabola. The inequality $f(x) \geq 0$ means we are looking for x-values where the graph is on or above the x-axis.

• Solution: Since the parabola does not cross the x-axis (it is entirely above the x-axis), the inequality is satisfied for all $x$: $\text{Solution: } x \in \mathbb{R} \; (\text{All real numbers})$

Would you like a detailed step-by-step explanation for any specific part? Or perhaps graphs for better visualization? Here are some follow-up questions to extend your understanding:

1. How do we algebraically solve quadratic inequalities without graphing?
2. Why is it important to check intervals when solving quadratic inequalities graphically?
3. How can we verify the solution using test points from each interval?
4. Can inequalities of higher degrees (e.g., cubic) also be solved graphically?
5. What is the significance of the vertex of a parabola in solving inequalities?

Tip: When solving inequalities graphically, always pay attention to the direction of the parabola and its intersection points with the x-axis!