Solve the inequality (x^2 - 5x + 6) / (x - 4) < 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, 2) ∪ (3, 4)
Principle or equation
For a rational inequality, find critical points where numerator or denominator is zero, then test intervals on the number line.
Why this answer is correct
Factor numerator: x^2 - 5x + 6 = (x-2)(x-3). Critical points: x=2,3,4. Test intervals: (-∞,2): (-)(-)/(-) = - (negative); (2,3): (+)(-)/(-) = + (positive); (3,4): (+)(+)/(-) = - (negative); (4,∞): (+)(+)/(+) = +. Since inequality is < 0, solution is (-∞,2) ∪ (3,4).
Example
Solve (x-1)(x-2)/(x-3) < 0. Critical 1,2,3. Intervals: (-∞,1): -; (1,2): +; (2,3): -; (3,∞): +. Solution: (-∞,1) ∪ (2,3).
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