Solve the inequality (2x^2 - 5x - 3) / (x - 2) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -1/2] ∪ (2, 3]
Principle or equation
To solve a rational inequality, find critical points (zeros of numerator and denominator), test intervals on a number line, and include endpoints where the expression equals zero and the denominator is not zero.
Why this answer is correct
Factor the numerator: 2x^2 - 5x - 3 = (2x + 1)(x - 3). Critical points: x = -1/2, x = 3 (zeros) and x = 2 (undefined). Test intervals: (-∞, -1/2): choose x = -1, expression = (2(-1)+1)(-1-3)/(-1-2) = (-1)(-4)/(-3) = -4/3 < 0. (-1/2, 2): choose x = 0, expression = (1)(-3)/(-2) = 3/2 > 0. (2, 3): choose x = 2.5, expression = (6)(-0.5)/(0.5) = -6 < 0. (3, ∞): choose x = 4, expression = (9)(1)/(2) = 4.5 > 0. The inequality is ≤ 0, so include intervals where negative and points where zero: (-∞, -1/2] and (2, 3]. Note x = 2 is excluded because denominator zero.
Example
Solve (x^2 - 1)/(x - 2) ≤ 0. Critical points: -1, 1, 2. Intervals: (-∞, -1] ∪ [1, 2).
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