The least positive integer x, which satisfies the inequality `log_(log(x/2)) (x^2-10x+22) > 0` is equal to

Options

A. `3`
B. `4`
C. `7`
D. `8`

Answer

To solve the inequality \( \log_{\log(x/2)}(x^2 - 10x + 22) > 0 \), we will follow these steps:

### Step 1: Understand the Logarithmic Inequality
The inequality \( \log_a(b) > 0 \) implies that \( b > 1 \) when \( a > 1 \) or \( 0 < b < 1 \) when \( 0 < a < 1 \). Here, \( a = \log(x/2) \) and \( b = x^2 - 10x + 22 \).

### Step 2: Determine Conditions for \( a \)
We need \( \log(x/2) > 0 \):
\[ \log(x/2) > 0 \implies \frac{x}{2} > 1 \implies x > 2. \]

### Step 3: Determine Conditions for \( b \)
Next, we need \( x^2 - 10x + 22 > 1 \):
\[ x^2 - 10x + 21 > 0. \]
Factoring gives:
\[ (x - 3)(x - 7) > 0. \]

### Step 4: Analyze the Quadratic Inequality
The critical points from the factorization are \( x = 3 \) and \( x = 7 \). We will test intervals around these points:
- For \( x < 3 \): Choose \( x = 2 \) → \( (2 - 3)(2 - 7) = (-1)(-5) = 5 > 0 \) (True)
- For \( 3 < x < 7 \): Choose \( x = 5 \) → \( (5 - 3)(5 - 7) = (2)(-2) = -4 < 0 \) (False)
- For \( x > 7 \): Choose \( x = 8 \) → \( (8 - 3)(8 - 7) = (5)(1) = 5 > 0 \) (True)

Thus, the solution for \( x^2 - 10x + 21 > 0 \) is:
\[ x < 3 \quad \text{or} \quad x > 7. \]

### Step 5: Combine Conditions
From Step 2, we have \( x > 2 \), and from Step 4, we have \( x < 3 \) or \( x > 7 \). Therefore, we combine these:
1. \( 2 < x < 3 \)
2. \( x > 7 \)

### Step 6: Identify the Least Positive Integer
The intervals give us:
- From \( 2 < x < 3 \): No integers are included.
- From \( x > 7 \): The smallest integer is \( 8 \).

### Conclusion
The least positive integer \( x \) that satisfies the inequality is:
\[ \boxed{8}. \]