Find the solution set of the inequality (x^2 - 2x - 15)/(x + 4) < 0.
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Correct answer
C. (-∞, -4) ∪ (-3, 5)
Principle or equation
A rational inequality is solved by finding critical points where numerator or denominator is zero, then testing intervals on the real line.
Why this answer is correct
Factor the numerator: x^2 - 2x - 15 = (x - 5)(x + 3). Critical points are x = -4 (denominator zero), x = -3, and x = 5. These divide the real line into intervals: (-∞, -4), (-4, -3), (-3, 5), (5, ∞). Test a point in each interval. For x = -5, the expression is positive; for x = -3.5, negative; for x = 0, negative; for x = 6, positive. Since the inequality is strict (<0), include intervals where the expression is negative: (-4, -3) and (-3, 5). Combine them as (-4, 5) excluding -3, but note that -3 makes the numerator zero, so the expression is 0, not allowed. Thus the solution is (-4, -3) ∪ (-3, 5).
Example
Solve (x^2 - 1)/(x - 2) < 0. Critical points: x = -1, 1, 2. Test intervals: (-∞,-1): positive; (-1,1): negative; (1,2): negative; (2,∞): positive. Solution: (-1,1) ∪ (1,2).
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