<p>Let Z be the set of integers. If
A = {x ∈Z : 2(x+2)(x2−5x+6) = 1}
and
B = {x ∈Z : −3 < 2x −1 < 9},
then the number of subsets of A × B is:</p>
Step-by-Step Solution
Key Concept: 2f (x) = 1 \Rightarrow f (x) = 0. Solve (x + 2)(x2 -5x + 6) = 0. Number of subsets of a set with n elements
= 2n.
<p><strong>Step 1</strong>: Find A: need (x + 2)(x2 -5x + 6) = 0.</p><br>• x + 2 = 0 \Rightarrow x = -2<br>• x2 -5x + 6 = (x -2)(x -3) = 0 \Rightarrow x = 2 or x = 3<br>Hence A = {-2, 2, 3}, so |A| = 3.<p><strong>Step 2</strong>: Find B: solve -3 < 2x -1 < 9. Adding 1: -2 < 2x < 10, dividing by 2: -1 < x < 5. Integer solutions:</p><br>x \in {0, 1, 2, 3, 4}. Check x = -1: 2(-1) -1 = -3, which is not strictly > -3. Excluded. Hence B = {0, 1, 2, 3, 4},<br>|B| = 5.<p><strong>Step 3</strong>: |A \times B| = 3 \times 5 = 15. Number of subsets = 215.</p>
Correct Answer: 3