<p>The complete solution set of inequality \(\dfrac{(x-5)^{1005}(x+8)^{1008}(x-1)}{x^{1006}(x-2)^3(x-3)^5(x-6)(x+9)^{10}} \leq 0\) is</p>
<p>\((-\infty, -9) \cup (-8, 0) \cup (0, 1) \cup (2, 3) \cup [5, 6)\)</p>
<p>\((-\infty, -9) \cup (-9, 0) \cup (0, 1) \cup (2, 3) \cup (5, 6)\)</p>
<p>\((-\infty, -9) \cup (-9, 0) \cup (0, 1] \cup (2, 3) \cup [5, 6)\)</p>
<p>\((-\infty, 0) \cup (0, 1] \cup (2, 3) \cup [5, 6)\)</p>
Step-by-Step Solution
Key Concept: For rational inequalities with even/odd powers, even powers don't change sign across roots (they create 'touch' points), while odd powers cause sign flips. Identify roots, determine multiplicity parity, perform sign analysis on intervals, and include roots where the expression equals zero.
<p><strong>Step 1: Identify all critical points and their multiplicities:</strong></p><p>Numerator roots: x=5 (odd: 1005), x=-8 (even: 1008), x=1 (odd: 1)</p><p>Denominator roots: x=0 (odd: 1006), x=2 (odd: 3), x=3 (odd: 5), x=6 (odd: 1), x=-9 (even: 10)</p><p><strong>Step 2: Order critical points:</strong> -9, -8, 0, 1, 2, 3, 5, 6</p><p><strong>Step 3: Analyze sign in each interval:</strong></p><p>Test x→+∞: All factors positive in numerator/denominator → expression is positive</p><p>Working backward through intervals, flip sign at ODD multiplicity roots, DON'T flip at EVEN roots:</p><ul><li>x∈(5,6): negative ✓</li><li>x=5: equals 0 ✓ (included)</li><li>x∈(3,5): positive ✗</li><li>x∈(2,3): negative ✓</li><li>x∈(1,2): positive ✗</li><li>x=1: equals 0 ✓ (included)</li><li>x∈(0,1): negative ✓</li><li>x=-8: equals 0 ✓ (included, even power touches axis)</li><li>x∈(-9,-8): negative ✓</li></ul><p><strong>Step 4: Exclude all denominator roots (undefined points):</strong> x ≠ -9, 0, 2, 3, 6</p><p><strong>∴ Answer: C is</strong> <strong>[-9,-8]∪(-8,0)∪(0,1]∪(1,2)∪(2,3)∪(3,5]∪(5,6)</strong> or equivalent notation depending on options given</p>
Correct Answer: C