Solve the inequality (2x^2 + 3x - 2) / (x^2 - 4x + 3) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
B. [-2, 1/2] ∪ (1, 3)
Principle or equation
Solve rational inequalities by finding critical points (zeros of numerator and denominator), testing intervals, and excluding points where denominator is zero.
Why this answer is correct
Factor numerator: 2x^2 + 3x - 2 = (2x - 1)(x + 2). Denominator: x^2 - 4x + 3 = (x - 1)(x - 3). Critical points: x = -2, 1/2, 1, 3. Denominator zero at x = 1, 3 (excluded). Test intervals: (-∞, -2): positive? Choose -3: numerator = ( -7)(-1)=7 positive, denominator = (-4)(-6)=24 positive, so positive. (-2, 1/2): choose 0: numerator = (-1)(2)=-2 negative, denominator = (-1)(-3)=3 positive, so negative. (1/2, 1): choose 0.75: numerator = (0.5)(2.75)=1.375 positive, denominator = (-0.25)(-2.25)=0.5625 positive, so positive. (1, 3): choose 2: numerator = (3)(4)=12 positive, denominator = (1)(-1)=-1 negative, so negative. (3, ∞): choose 4: numerator = (7)(6)=42 positive, denominator = (3)(1)=3 positive, so positive. Include zeros of numerator: -2 and 1/2. Thus solution: [-2, 1/2] ∪ (1, 3).
Example
Solve (x-1)/(x-2) ≤ 0: critical points 1, 2; test intervals gives [1, 2).
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