Step 1: Understanding the Concept:
To solve rational inequalities, move all terms to one side: \( \frac{x^2 - 4x + 7}{x^2 - 7x + 12} - \frac{2}{3} \le 0 \).
Step 2: Detailed Explanation:
\( \frac{3(x^2 - 4x + 7) - 2(x^2 - 7x + 12)}{3(x^2 - 7x + 12)} \le 0 \)
\( \frac{3x^2 - 12x + 21 - 2x^2 + 14x - 24}{3(x-3)(x-4)} \le 0 \)
\( \frac{x^2 + 2x - 3}{3(x-3)(x-4)} \le 0 \implies \frac{(x+3)(x-1)}{3(x-3)(x-4)} \le 0 \).
Using the sign-scheme method (wavy curve):
Critical points are \( -3, 1, 3, 4 \).
The expression is \( \le 0 \) in intervals \( [-3, 1] \) and \( (3, 4) \).
Step 3: Final Answer:
The solution is \( x \in [-3, 1] \cup (3, 4) \), matching (A) and (D).