Solve the inequality (x^2 - 2x - 3) / (x + 2) ≥ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
C. (-2, -1] ∪ [3, ∞)
Principle or equation
For a rational inequality, find critical points (zeros of numerator and denominator), test intervals, and exclude points where denominator is zero.
Why this answer is correct
Factor numerator: x^2 - 2x - 3 = (x - 3)(x + 1). Critical points: x = -1, 3 (numerator zero), x = -2 (denominator zero). Intervals: (-∞, -2), (-2, -1], [-1, 3], [3, ∞). Test x = -3: (-)(-)/( - ) = - → not ≥0. Test x = -1.5: (-)(-)/( + ) = + → include. Test x = 0: (-)(+)/(+) = - → exclude. Test x = 4: (+)(+)/(+) = + → include. Denominator zero at x = -2 excluded. Thus solution: (-2, -1] ∪ [3, ∞).
Example
Solve (x^2 - 1)/(x - 2) ≥ 0. Critical: x = -1, 1, 2. Test intervals: (-∞,-1] +, [-1,1] -, [1,2) +, (2,∞) +. Solution: (-∞,-1] ∪ [1,2) ∪ (2,∞).
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