Solve the inequality (x^2 - 4x - 12) / (x + 3) < 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -3) ∪ (-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 = -3 (denominator zero), x = -2, x = 6. Test intervals: (-∞, -3): choose -4, (-)(-)/(-) = negative? Actually (-4-6)(-4+2)=(-10)(-2)=20, denominator -1, quotient -20 0. (-2,6): choose 0, (-6)(2)=-12, denominator 3, quotient -4 0. So solution: (-∞,-3) ∪ (-2,6).
Example
Solve (x-1)/(x+2) < 0: critical -2,1; test intervals gives (-2,1).
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