Find the solution set of the inequality (x^2 - 6x + 8) / (x - 5) ≤ 0.
Show the answer and explanation
Correct answer
A. (-∞, 2] ∪ [4, 5)
Principle or equation
For a rational inequality, find critical points where numerator or denominator is zero, then test intervals on the real number line. The denominator cannot be zero.
Why this answer is correct
Factor numerator: x^2 - 6x + 8 = (x - 2)(x - 4). Critical points are x = 2, 4, 5. Test intervals: (-∞,2): choose 0 -> (positive)/(negative) = negative, so ≤0 true. (2,4): choose 3 -> (negative)/(negative) = positive, false. (4,5): choose 4.5 -> (positive)/(negative) = negative, true. (5,∞): choose 6 -> (positive)/(positive) = positive, false. Include x=2 and 4 (numerator zero), exclude x=5 (denominator zero). Thus solution is (-∞,2] ∪ [4,5).
Example
Solve (x-1)(x-3)/(x-2) ≤ 0. Critical points 1,2,3. Test intervals: (-∞,1): (-)/(-) = +, false; (1,2): (+)/(-) = -, true; (2,3): (+)/(+) = +, false; (3,∞): (+)/(+) = +, false. Include 1 and 3, exclude 2. Solution: [1,2) ∪ [3,∞).
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