<p>A number <i>x</i> is chosen at random from the set \(\{1, 2, 3, 4, \ldots, 100\}\). Define the event: \(A\) = the chosen number <i>x</i> satisfies \(\dfrac{(x-10)(x-50)}{(x-30)} \geq 0\). Then \(P(A)\) is</p>
Step-by-Step Solution
Key Concept: Solve the rational inequality by finding critical points (10, 30, 50) and using sign analysis on each interval, remembering that x ≠ 30 (denominator zero) and checking boundary conditions carefully.
<p><strong>Step 1:</strong> Identify critical points where numerator or denominator equals zero.</p><p>Numerator: (x-10)(x-50) = 0 → x = 10 or x = 50</p><p>Denominator: (x-30) = 0 → x = 30 (excluded from domain)</p><p><strong>Step 2:</strong> Create sign chart for intervals: (-∞,10), (10,30), (30,50), (50,∞)</p><p><strong>Step 3:</strong> Test signs in each interval:</p><ul><li>(10,30): (−)(−)/(−) = (−) ✗</li><li>(30,50): (−)(+)/(+) = (−) ✗</li><li>x < 10: (+)(−)/(−) = (+) ✓</li><li>x > 50: (+)(+)/(+) = (+) ✓</li></ul><p><strong>Step 4:</strong> Check boundary points: x=10 gives 0 ✓ and x=50 gives 0 ✓</p><p><strong>Step 5:</strong> Solution is x ∈ [1,10] ∪ [50,100] within the set {1,2,...,100}</p><p>Favorable outcomes: 10 + 51 = 61</p><p><strong>Step 6:</strong> P(A) = 61/100</p><p>∴ Answer: A</p>
Correct Answer: A