https://r2.erweima.ai/mathbot/image/1724596833657ps8ecppv.jpg

The expression in the image reads:

\[
\frac{x^2(x - 4)}{x^2 - 8x + 12} \leq 0
\]

We need to find the number of positive integral solutions of this inequality.

### Step 1: Factor the numerator and denominator

First, let's factorize the expression:

- Numerator: \( x^2(x - 4) \) is already factored.
- Denominator: \( x^2 - 8x + 12 \)

To factorize the denominator \( x^2 - 8x + 12 \), we look for factors of 12 that add up to -8. These factors are -6 and -2:

\[
x^2 - 8x + 12 = (x - 6)(x - 2)
\]

So, the inequality becomes:

\[
\frac{x^2(x - 4)}{(x - 6)(x - 2)} \leq 0
\]

### Step 2: Identify critical points

The critical points (where the expression is zero or undefined) are found by setting the numerator and denominator to zero:

- Numerator: \( x^2(x - 4) = 0 \)
  - \( x^2 = 0 \) gives \( x = 0 \)
  - \( x - 4 = 0 \) gives \( x = 4 \)

- Denominator: \( (x - 6)(x - 2) = 0 \)
  - \( x - 6 = 0 \) gives \( x = 6 \)
  - \( x - 2 = 0 \) gives \( x = 2 \)

So the critical points are \( x = 0 \), \( x = 2 \), \( x = 4 \), and \( x = 6 \).

### Step 3: Determine the sign of the expression in each interval

We need to determine the sign of the expression in each of the intervals defined by the critical points:

- Interval \( (-\infty, 0) \)
- Interval \( (0, 2) \)
- Interval \( (2, 4) \)
- Interval \( (4, 6) \)
- Interval \( (6, \infty) \)

Since we are interested in positive integer solutions, we only consider intervals where \( x > 0 \):

1. **Interval \( (0, 2) \)**:
   - Choose \( x = 1 \).
   - Sign of \( \frac{x^2(x - 4)}{(x - 6)(x - 2)} \) is \( \frac{1^2(1 - 4)}{(1 - 6)(1 - 2)} = \frac{1 \times (-3)}{-5 \times -1} = \frac{-3}{5} \), which is **negative**.

2. **Interval \( (2, 4) \)**:
   - Choose \( x = 3 \).
   - Sign of \( \frac{x^2(x - 4)}{(x - 6)(x - 2)} \) is \( \frac{3^2(3 - 4)}{(3 - 6)(3 - 2)} = \frac{9 \times (-1)}{-3 \times 1} = \frac{-9}{-3} \), which is **positive**.

3. **Interval \( (4, 6) \)**:
   - Choose \( x = 5 \).
   - Sign of \( \frac{x^2(x - 4)}{(x - 6)(x - 2)} \) is \( \frac{5^2(5 - 4)}{(5 - 6)(5 - 2)} = \frac{25 \times 1}{-1 \times 3} = \frac{25}{-3} \), which is **negative**.

Thus, the inequality \( \frac{x^2(x - 4)}{(x - 6)(x - 2)} \leq 0 \) is satisfied for \( x = 1 \) and \( x = 5 \).

### Step 4: Count the positive integer solutions

From the analysis above, the positive integer solutions are \( x = 1 \) and \( x = 5 \).

### Final Answer

The number of positive integral solutions is **2**.

Would you like more details or have any other questions? Here are some related questions:

1. How would the solution change if the inequality was strict (i.e., \( < \) instead of \( \leq \))?
2. Can you explain the significance of the critical points in more detail?
3. What happens to the inequality if the denominator factors were slightly different?
4. How can this problem be solved graphically?
5. What is the importance of analyzing the sign in each interval?

**Tip:** Always factorize both the numerator and denominator to find critical points when solving rational inequalities.

Learn how to solve the rational inequality (x^2(x - 4))/(x^2 - 8x + 12) <= 0 with a detailed step-by-step solution. This problem covers rational inequalities, factoring of polynomials, and sign analysis. Suitable for advanced high school students and those preparing for competitive exams.

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