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To solve the inequality:

\[
(x - 7)(4 - x)(x + 1) \geq 0
\]

we'll proceed by analyzing the sign changes across the intervals defined by the roots of the inequality. Here are the steps:

### Step 1: Identify the roots
The roots (or zeros) of the inequality are the values of \( x \) for which each factor is zero:
1. \( x - 7 = 0 \Rightarrow x = 7 \)
2. \( 4 - x = 0 \Rightarrow x = 4 \)
3. \( x + 1 = 0 \Rightarrow x = -1 \)

These roots divide the real line into four intervals: \( (-\infty, -1) \), \( (-1, 4) \), \( (4, 7) \), and \( (7, \infty) \).

### Step 2: Test each interval
We will test each interval by selecting a point within it and checking the sign of each factor:

1. **Interval \( (-\infty, -1) \):** Choose \( x = -2 \)
   - \( x - 7 < 0 \)
   - \( 4 - x > 0 \)
   - \( x + 1 < 0 \)
   - Product: Negative

2. **Interval \( (-1, 4) \):** Choose \( x = 0 \)
   - \( x - 7 < 0 \)
   - \( 4 - x > 0 \)
   - \( x + 1 > 0 \)
   - Product: Positive

3. **Interval \( (4, 7) \):** Choose \( x = 5 \)
   - \( x - 7 < 0 \)
   - \( 4 - x < 0 \)
   - \( x + 1 > 0 \)
   - Product: Negative

4. **Interval \( (7, \infty) \):** Choose \( x = 8 \)
   - \( x - 7 > 0 \)
   - \( 4 - x < 0 \)
   - \( x + 1 > 0 \)
   - Product: Positive

### Step 3: Determine intervals where the product is non-negative
We want the product to be greater than or equal to zero:
- Positive intervals: \( (-1, 4) \) and \( (7, \infty) \)
- Check endpoints: Since the inequality is \( \geq 0 \), include points where any factor is zero:
  - \( x = -1 \), \( x = 4 \), and \( x = 7 \) are included.

### Final Answer
The solution in interval notation is:

\[
[-1, 4] \cup [7, \infty)
\]

Would you like a more detailed explanation, or do you have any questions?

---

1. How do we find the roots of more complex polynomial inequalities?
2. Why does each root split the number line into different intervals?
3. How can we handle inequalities where factors are squared?
4. What if the inequality sign was \( > \) instead of \( \geq \)?
5. How do we know when to include or exclude endpoints in inequalities?

**Tip:** When solving polynomial inequalities, always test points within each interval to determine the sign of the product. This method works efficiently for inequalities with multiple factors.

Learn how to solve the polynomial inequality (x - 7)(4 - x)(x + 1) ≥ 0. This solution covers concepts like roots, sign testing, and interval analysis, perfect for high school math students.

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