Solve the inequality (x^2 - 7x + 12) / (x + 1) > 0. Which of the following is the solution set?
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Correct answer
A. (-1, 3) ∪ (4, ∞)
Principle or equation
To solve a rational inequality, find the critical points where the numerator or denominator is zero, test intervals between these points, and include or exclude endpoints based on the inequality sign and domain restrictions.
Why this answer is correct
Factor the numerator: x^2 - 7x + 12 = (x - 3)(x - 4). Critical points: x = -1 (denominator zero), x = 3, x = 4. Test intervals: (-∞, -1): choose x = -2, expression = (4*6)/(-1) = -24 < 0; (-1, 3): choose x = 0, expression = (12)/(1) = 12 > 0; (3, 4): choose x = 3.5, expression = (0.5*-0.5)/(4.5) = -0.055... < 0; (4, ∞): choose x = 5, expression = (2*1)/(6) = 1/3 > 0. The inequality is > 0, so the solution is (-1, 3) ∪ (4, ∞).
Example
For (x - 2)/(x + 1) > 0, critical points -1 and 2, solution (-∞, -1) ∪ (2, ∞).
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