Solve the inequality (x^2 - 3x - 18) / (x - 2) < 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -3) ∪ (2, 6)
Principle or equation
To solve a rational inequality, find the critical points where the numerator or denominator equals zero, then test intervals between these points. The inequality is strict, so endpoints are excluded.
Why this answer is correct
Factor the numerator: x^2 - 3x - 18 = (x - 6)(x + 3). Critical points are x = -3, 2, 6. Test intervals: (-∞, -3): choose -4, numerator positive, denominator negative -> negative, so true. (-3, 2): choose 0, numerator negative, denominator negative -> positive, false. (2, 6): choose 4, numerator negative, denominator positive -> negative, true. (6, ∞): choose 7, numerator positive, denominator positive -> positive, false. Thus solution is (-∞, -3) ∪ (2, 6).
Example
Solve (x - 1)/(x + 2) > 0. Critical points 1 and -2. Test intervals: (-∞, -2) positive, (-2, 1) negative, (1, ∞) positive. Solution: (-∞, -2) ∪ (1, ∞).
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