Solve the inequality (x^2 + 2x - 8) / (x - 1) ≥ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
C. [-4, 1) ∪ [2, ∞)
Principle or equation
For a rational inequality, find critical points where numerator or denominator is zero, test intervals, and exclude points where denominator is zero.
Why this answer is correct
Factor numerator: x^2 + 2x - 8 = (x + 4)(x - 2). Critical points: x = -4, x = 2, x = 1 (denominator zero). Test intervals: (-∞, -4): choose -5, (-)(-)/(-) = -; (-4,1): choose 0, (+)(-)/(-) = +; (1,2): choose 1.5, (+)(-)/(+) = -; (2,∞): choose 3, (+)(+)/(+) = +. Include -4 and 2 because inequality is ≥0, exclude 1. Solution: [-4,1) ∪ [2,∞).
Example
For (x^2 - 1)/(x - 2) ≥ 0, critical points -1,1,2; solution [-1,1] ∪ (2,∞).
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