Solve the inequality (x^2 - 2x - 8) / (x - 1) > 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-2, 1) ∪ (4, ∞)
Principle or equation
To solve a rational inequality, find critical points where numerator or denominator is zero, test intervals, and exclude values where the denominator is zero.
Why this answer is correct
Factor the numerator: x^2 - 2x - 8 = (x - 4)(x + 2). Critical points: x = -2, 4 (numerator zero) and x = 1 (denominator zero). Test intervals: (-∞, -2): pick -3, (-)(-)/(-) = negative; (-2, 1): pick 0, (-)(+)/(-) = positive; (1, 4): pick 2, (-)(+)/(+) = negative; (4, ∞): pick 5, (+)(+)/(+) = positive. Since the inequality is > 0, the solution is (-2, 1) ∪ (4, ∞). Note x = 1 is excluded.
Example
For (x - 2)/(x + 3) > 0, critical points 2 and -3, solution (-∞, -3) ∪ (2, ∞).
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