Solve the inequality (x^2 - 3x - 18) / (x - 1) ≥ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. [-3, 1) ∪ [6, ∞)
Principle or equation
To solve a rational inequality, find the zeros of the numerator and the denominator, use these to partition the real line, and test each interval. The denominator cannot be zero.
Why this answer is correct
Factor the numerator: x^2 - 3x - 18 = (x - 6)(x + 3). The critical points are x = -3, x = 6 (zeros of numerator) and x = 1 (zero of denominator). Test intervals: (-∞, -3): choose -4, expression positive? (-)(-)/(-) = negative? Actually compute: (-4-6)(-4+3)/(-4-1) = (-10)(-1)/(-5) = 10/(-5) = -2 < 0. (-3,1): choose 0, ( -6)(3)/(-1) = 18 > 0. (1,6): choose 2, (-4)(5)/(1) = -20 < 0. (6,∞): choose 7, (1)(10)/(6) = 10/6 > 0. Also include x = -3 and x = 6 because numerator zero gives 0 ≥ 0. Exclude x = 1. Thus solution: [-3,1) ∪ [6,∞).
Example
For (x-2)/(x+1) ≥ 0, critical points -1 and 2, test intervals gives (-∞,-1) ∪ [2,∞).
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