Solve the inequality (x^2 - 4x - 5) / (x - 3) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -1] ∪ (3, 5]
Principle or equation
For a rational inequality, find critical points where numerator or denominator is zero, then test intervals on the number line. The denominator cannot be zero.
Why this answer is correct
Factor numerator: x^2 - 4x - 5 = (x - 5)(x + 1). Critical points: x = -1, 5 (numerator zeros) and x = 3 (denominator zero, excluded). Test intervals: (-∞, -1): choose x=-2, expression = (+)(-)/(-) = +, not ≤0. (-1,3): choose x=0, expression = (-)(+)/(-) = +, not ≤0. (3,5): choose x=4, expression = (-)(+)/(+) = -, ≤0. (5,∞): choose x=6, expression = (+)(+)/(+) = +. Include endpoints where numerator zero: -1 and 5. Exclude 3. So solution: (-∞, -1] ∪ (3,5].
Example
Solve (x-2)/(x+1) ≤ 0: critical points 2 and -1. Test intervals gives (-1,2].
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