Solve the inequality (x^2 - 4x - 12) / (x + 2) > 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -2) ∪ (6, ∞)
Principle or equation
For a rational inequality, find critical points where numerator or denominator is zero, then test intervals on a number line.
Why this answer is correct
Factor numerator: x^2 - 4x - 12 = (x - 6)(x + 2). Critical points: x = -2 (denominator zero) and x = 6 (numerator zero). Test intervals: (-∞, -2): choose -3, expression positive; (-2, 6): choose 0, expression negative; (6, ∞): choose 7, expression positive. Since inequality is > 0, solution is (-∞, -2) ∪ (6, ∞). Note x = -2 is excluded because denominator zero.
Example
For (x - 1)/(x + 2) > 0, critical points 1 and -2, solution (-∞, -2) ∪ (1, ∞).
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