Solve the inequality (x^2 - 3x - 10) / (x - 2) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
B. (-∞, -2] ∪ (2, 5]
Principle or equation
For a rational inequality, find the critical points (zeros of numerator and denominator), test intervals, and exclude points where the denominator is zero.
Why this answer is correct
Factor the numerator: x^2 - 3x - 10 = (x - 5)(x + 2). The critical points are x = -2, x = 5 (zeros of numerator) and x = 2 (zero of denominator). These divide the real line into intervals: (-∞, -2), (-2, 2), (2, 5), (5, ∞). Test a point in each interval: for x = -3, the expression is positive; for x = 0, it is positive; for x = 3, it is negative; for x = 6, it is positive. The inequality is ≤ 0, so include intervals where the expression is negative or zero, but exclude x = 2 because the denominator is zero. Thus, the solution is (-∞, -2] ∪ (2, 5].
Example
Solve (x - 1)(x + 2) / (x - 3) ≤ 0. Critical points: -2, 1, 3. Test intervals: positive, negative, positive, negative. Solution: (-∞, -2] ∪ [1, 3).
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