Solve the inequality (x^2 - 1) / (x - 3) > 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
C. (-1, 1) ∪ (3, ∞)
Principle or equation
For a rational inequality, determine the sign of the expression by analyzing the zeros of the numerator and denominator. The expression changes sign at these critical points, and the denominator cannot be zero.
Why this answer is correct
Factor the numerator: x^2 - 1 = (x - 1)(x + 1). Critical points are x = -1, x = 1 (zeros of numerator) and x = 3 (zero of denominator, excluded). Test intervals: (-∞, -1): choose x = -2, expression = (4-1)/(-5) = 3/(-5) < 0. (-1, 1): choose x = 0, expression = (-1)/(-3) = 1/3 > 0. (1, 3): choose x = 2, expression = (3)/(-1) = -3 < 0. (3, ∞): choose x = 4, expression = (15)/(1) = 15 > 0. The inequality > 0 holds on (-1, 1) and (3, ∞).
Example
Solve (x^2 - 4)/(x - 1) > 0. Critical points: -2, 2, 1. Test intervals: (-∞,-2): negative, (-2,1): positive, (1,2): negative, (2,∞): positive. Solution: (-2,1) ∪ (2,∞).
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