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Let's go through each part of the problem step by step.

### Given functions:
- \( f(x) = 5x - 22 \)
- \( g(x) = -5x - 2 \)

---

### Part (a) - Calculate the intersection point \( S \) of the graphs of \( f(x) \) and \( g(x) \).

To find the intersection, we set \( f(x) = g(x) \):
\[
5x - 22 = -5x - 2
\]
Solving for \( x \):
\[
5x + 5x = 22 - 2
\]
\[
10x = 20 \quad \Rightarrow \quad x = 2
\]

Now substitute \( x = 2 \) into either function to find \( y \):
\[
f(2) = 5(2) - 22 = 10 - 22 = -12
\]

Thus, the intersection point is:
\[
S = (2, -12)
\]

This matches the answer provided.

---

### Part (b) - For what value(s) of \( x \) is \( f(x) > g(x) \)?

We need to solve the inequality \( f(x) > g(x) \), which becomes:
\[
5x - 22 > -5x - 2
\]
Solving:
\[
5x + 5x > 22 - 2
\]
\[
10x > 20 \quad \Rightarrow \quad x > 2
\]

So the solution in interval notation is:
\[
x \in (2, \infty)
\]

This also matches the provided answer.

---

### Part (c) - For what value(s) of \( x \) is \( g(x) \geq f(x) \)?

Now, we solve the inequality \( g(x) \geq f(x) \), which becomes:
\[
-5x - 2 \geq 5x - 22
\]
Solving:
\[
-5x - 5x \geq -22 + 2
\]
\[
-10x \geq -20 \quad \Rightarrow \quad x \leq 2
\]

So the solution in interval notation is:
\[
x \in (-\infty, 2]
\]

This matches the provided answer.

---

### Summary:
- Intersection point: \( S = (2, -12) \)
- \( f(x) > g(x) \) for \( x \in (2, \infty) \)
- \( g(x) \geq f(x) \) for \( x \in (-\infty, 2] \)

Would you like more details or further explanations on any part of the solution?

Here are five related questions:
1. What does the graph of \( f(x) \) and \( g(x) \) look like?
2. How do we find the intersection point of two nonlinear functions?
3. Can we generalize the process for solving inequalities between two functions?
4. How would the solution change if the functions were quadratic instead of linear?
5. What role does slope play in comparing two linear functions?

**Tip**: Always check the solution by substituting the intersection point into both equations to ensure accuracy!

Given the two functions f(x) = 5x - 22 and g(x) = -5x - 2:

(a) Calculate the intersection point S of the graphs of f(x) and g(x). Provide the answer in the form S = (a, b).

(b) For which value(s) of x does f(x) > g(x)? Provide the answer in interval notation.

(c) For which value(s) of x does g(x) ≥ f(x)? Provide the answer in interval notation.

Learn how to find the intersection of two linear functions and solve inequalities involving them. This example uses f(x) = 5x - 22 and g(x) = -5x - 2, covering concepts such as graph intersection and interval notation. Ideal for students in Grades 10-12.

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