Find the solution set of the inequality (x^2 - 7x + 10) / (x - 4) ≥ 0.
Show the answer and explanation
Correct answer
C. [2, 4) ∪ [5, ∞)
Principle or equation
To solve a rational inequality, find the critical points where the numerator or denominator is zero, use a sign chart, and exclude points where the denominator is zero.
Why this answer is correct
Factor the numerator: x^2 - 7x + 10 = (x - 2)(x - 5). Critical points are x = 2, 5 (numerator zero) and x = 4 (denominator zero). The sign of the expression changes at these points. Test intervals: (-∞, 2): positive; (2, 4): negative; (4, 5): positive; (5, ∞): positive. Since the inequality is ≥ 0, include x = 2 and x = 5, but not x = 4. Thus the solution is [2, 4) ∪ [5, ∞).
Example
For (x - 1)(x - 3)/(x - 2) ≥ 0, critical points 1,2,3; solution [1,2) ∪ [3,∞).
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