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The inequality to solve is:

\[
(x - 6)(x + 5) \geq 0
\]

### Step 1: Identify the critical points
The two critical points are the values of \(x\) that make either factor equal to zero:

1. \(x - 6 = 0 \Rightarrow x = 6\)
2. \(x + 5 = 0 \Rightarrow x = -5\)

So, the critical points are \(x = -5\) and \(x = 6\).

### Step 2: Test intervals
The real number line is divided into three intervals by the critical points:
- \((-\infty, -5)\)
- \((-5, 6)\)
- \((6, \infty)\)

We will test points within each of these intervals to determine where the product \((x - 6)(x + 5)\) is non-negative.

1. **For \(x \in (-\infty, -5)\), test with \(x = -6\):**
   \[
   (-6 - 6)(-6 + 5) = (-12)(-1) = 12 \geq 0
   \]
   The product is positive, so the inequality holds in this interval.

2. **For \(x \in (-5, 6)\), test with \(x = 0\):**
   \[
   (0 - 6)(0 + 5) = (-6)(5) = -30 < 0
   \]
   The product is negative, so the inequality does not hold in this interval.

3. **For \(x \in (6, \infty)\), test with \(x = 7\):**
   \[
   (7 - 6)(7 + 5) = (1)(12) = 12 \geq 0
   \]
   The product is positive, so the inequality holds in this interval.

### Step 3: Analyze the critical points
At \(x = -5\):
\[
(-5 - 6)(-5 + 5) = (-11)(0) = 0
\]
At \(x = 6\):
\[
(6 - 6)(6 + 5) = (0)(11) = 0
\]
Both points make the product equal to zero, and since the inequality is \(\geq 0\), the inequality holds at these points.

### Step 4: Conclusion
The solution is:
\[
(-\infty, -5] \cup [6, \infty)
\]
This corresponds to **Option B**.

---

Would you like further details on any part of this solution? Here are 5 related questions to expand on this concept:

1. How do we solve inequalities that are strictly greater than \(0\) (instead of \(\geq 0\))?
2. What happens if the inequality had been less than or equal to zero?
3. How do we solve more complex polynomial inequalities with higher degrees?
4. What role do critical points play in graphing polynomial inequalities?
5. How does the number of intervals change with more than two factors?

**Tip:** Always check the sign of each interval separately when solving polynomial inequalities.

Solve the inequality (x - 6)(x + 5) ≥ 0.

Learn how to solve the quadratic inequality (x - 6)(x + 5) ≥ 0 using critical points and interval testing. This step-by-step solution includes analyzing signs across different intervals and finding the solution set. Suitable for high school students in Grades 10-12.

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