https://file.aiquickdraw.com/mathbot/file/1729573183274gsmkiaq9.jpg

The inequality provided in the image is:

\[
(x + 4)(x - 8) \geq 0
\]

### Step-by-Step Solution:

1. **Find the critical points**: 
   The expression \((x + 4)(x - 8) = 0\) gives two critical points:
   - \(x + 4 = 0 \Rightarrow x = -4\)
   - \(x - 8 = 0 \Rightarrow x = 8\)

2. **Intervals to test**:
   The critical points divide the number line into three intervals:
   - \( (-\infty, -4) \)
   - \( (-4, 8) \)
   - \( (8, \infty) \)

3. **Test each interval**:
   - For \(x \in (-\infty, -4)\), choose \(x = -5\):
     \[
     (x + 4)(x - 8) = (-5 + 4)(-5 - 8) = (-1)(-13) = 13 \geq 0
     \]
     This interval satisfies the inequality.

   - For \(x \in (-4, 8)\), choose \(x = 0\):
     \[
     (x + 4)(x - 8) = (0 + 4)(0 - 8) = (4)(-8) = -32 \not\geq 0
     \]
     This interval does not satisfy the inequality.

   - For \(x \in (8, \infty)\), choose \(x = 9\):
     \[
     (x + 4)(x - 8) = (9 + 4)(9 - 8) = (13)(1) = 13 \geq 0
     \]
     This interval satisfies the inequality.

4. **Include the critical points**:
   Since the inequality is \(\geq 0\), we include the points \(x = -4\) and \(x = 8\), because at these points the expression equals 0.

5. **Solution in interval notation**:
   \[
   (-\infty, -4] \cup [8, \infty)
   \]

### Graphing the Solution:

- For \(x = -4\) and \(x = 8\), plot solid dots (because these points are included).
- Shade the regions to the left of \(-4\) and to the right of \(8\) on the number line, as these intervals satisfy the inequality.

Would you like further details, or assistance with graphing?

Here are 5 questions you might explore:
1. What happens if the inequality were \((x+4)(x-8) > 0\)?
2. How would the solution change if the inequality was \(\leq 0\)?
3. Can you graph inequalities involving more than two factors?
4. What if the inequality had a negative sign in front, such as \(-(x+4)(x-8) \geq 0\)?
5. How do we handle inequalities with quadratic terms like \(x^2 - x - 12 \geq 0\)?

**Tip**: Always test intervals between critical points to check whether they satisfy the inequality!

Solve the inequality (x + 4)(x - 8) ≥ 0, write the solution in interval notation, and graph the solution.

Learn how to solve and graph the quadratic inequality (x + 4)(x - 8) ≥ 0. This solution includes critical points, interval testing, and interval notation for a detailed understanding. Ideal for students in Grades 9-12.

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