Solve the inequality (x^2 - 5x + 4) / (x - 2) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, 1] ∪ (2, 4]
Principle or equation
To solve a rational inequality, find critical points (zeros of numerator and denominator), test intervals, and exclude points where denominator is zero.
Why this answer is correct
Factor numerator: x^2 - 5x + 4 = (x-1)(x-4). Critical points: x=1, x=4 (zeros of numerator), x=2 (zero of denominator). Test intervals: (-∞,1): choose 0, (0-1)(0-4)/(0-2) = (-1)(-4)/(-2)=4/-2=-2 ≤0 true. (1,2): choose 1.5, (0.5)(-2.5)/(-0.5)=(-1.25)/(-0.5)=2.5 >0 false. (2,4): choose 3, (2)(-1)/(1)= -2 ≤0 true. (4,∞): choose 5, (4)(1)/(3)=4/3 >0 false. Include x=1 and x=4 because inequality is ≤0, exclude x=2. So solution: (-∞,1] ∪ (2,4].
Example
Solve (x-3)/(x+1) ≤ 0: critical points 3 and -1, test intervals gives (-1,3].
This published item includes a stored explanation and passed the platform’s publication workflow. It is independent preparation material, not a claim of an official or recalled examination question.