Let's solve the given compound inequality:

$x + 7 < 3 \quad \text{or} \quad x - 5 \geq -1$

1. Solve the first inequality:

$x + 7 < 3$ Subtract 7 from both sides: $x < 3 - 7$ $x < -4$

2. Solve the second inequality:

$x - 5 \geq -1$ Add 5 to both sides: $x \geq -1 + 5$ $x \geq 4$

3. Combine the solution:

The compound inequality is: $x < -4 \quad \text{or} \quad x \geq 4$ This means that the solution includes all values of $x$ that are less than $-4$ or greater than or equal to $4$.

4. Identify the graph:

From the available options, the graph that corresponds to this solution is Option C: $x < -4$ or $x \geq 4$, represented by the arrows extending to the left of $-4$ and from $4$ onward to the right.

Would you like further details on this solution or have any other questions?

Here are five related questions to explore further:

1. How would the solution change if the inequalities were connected by "and" instead of "or"?
2. Can compound inequalities have both a solution involving "and" and "or" at the same time?
3. What happens if you reverse the inequality signs in both inequalities?
4. How would you graph an inequality involving absolute values?
5. What real-world situations might be modeled by compound inequalities?

Tip: When solving inequalities, always pay attention to the direction of the inequality sign, especially when multiplying or dividing by a negative number—it reverses the sign.