The image you provided involves solving the inequality $x^2 - 2x - 3 > 0$ and visually representing its solution using points and line segments on a number line.

Step-by-step solution:

1. Factor the quadratic expression: $x^2 - 2x - 3 > 0$ This can be factored as: $(x - 3)(x + 1) > 0$ Now, we have the factored inequality: $(x - 3)(x + 1) > 0$

2. Find the critical points: Set each factor equal to 0 to find the critical points: $x - 3 = 0 \quad \Rightarrow \quad x = 3$ $x + 1 = 0 \quad \Rightarrow \quad x = -1$ The critical points are $x = -1$ and $x = 3$.

3. Determine the intervals: These critical points divide the number line into three intervals:

   • $(-\infty, -1)$
   • $(-1, 3)$
   • $(3, \infty)$

4. Test each interval: We need to check the sign of the product $(x - 3)(x + 1)$ in each interval:

   • Interval $(- \infty, -1)$: Choose $x = -2$, $(-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 0$ So, the inequality holds in this interval.

   • Interval $(-1, 3)$: Choose $x = 0$, $(0 - 3)(0 + 1) = (-3)(1) = -3 < 0$ So, the inequality does not hold in this interval.

   • Interval $(3, \infty)$: Choose $x = 4$, $(4 - 3)(4 + 1) = (1)(5) = 5 > 0$ So, the inequality holds in this interval.

5. Conclusion: The solution to the inequality is: $(-\infty, -1) \cup (3, \infty)$

Visual Representation:

On the number line:

• There is an open circle at $x = -1$ and $x = 3$ (since these points are excluded because the inequality is strict).
• The solution is the intervals to the left of $-1$ and to the right of $3$.

Would you like further details or have any specific questions? Here are some related questions to explore this topic:

1. What are the differences between solving quadratic equations and inequalities?
2. How would the solution change if the inequality were $x^2 - 2x - 3 \geq 0$?
3. How can you solve quadratic inequalities using a sign chart?
4. Why are the critical points excluded from the solution in this inequality?
5. How would you graphically represent the solution on the number line using closed or open circles?

Tip: When solving inequalities, always test points in each interval to determine where the inequality holds.