Solve the inequality (x^2 - 2x - 15) / (x + 1) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -3] ∪ (-1, 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 - 2x - 15 = (x - 5)(x + 3). Critical points: x = -3, x = 5 (numerator zero), x = -1 (denominator zero). Test intervals: (-∞, -3): choose -4, (-)(-)/(-) = - (negative) → satisfies ≤0. (-3, -1): choose -2, (-)(-)/(-) = + (positive) → not. (-1, 5): choose 0, (-)(+)/(+) = - (negative) → satisfies. (5, ∞): choose 6, (+)(+)/(+) = + → not. Include -3 and 5 (numerator zero), exclude -1. Solution: (-∞, -3] ∪ (-1, 5].
Example
Solve (x^2 - 1)/(x - 2) ≤ 0. Critical: -1, 1, 2. Test: (-∞,-1] ∪ [1,2).
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