Solve the inequality (2x + 1)/(x - 4) > 1. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -5) ∪ (4, ∞)
Principle or equation
For a rational inequality, bring all terms to one side, combine into a single fraction, factor numerator and denominator, find critical points, and test intervals on a number line.
Why this answer is correct
Subtract 1 from both sides: (2x+1)/(x-4) - 1 > 0 => (2x+1 - (x-4))/(x-4) > 0 => (x+5)/(x-4) > 0. Critical points: x = -5 and x = 4. Test intervals: (-∞,-5): (-)/(-) = +; (-5,4): (+)/(-) = -; (4,∞): (+)/(+) = +. Thus solution is (-∞,-5) ∪ (4,∞).
Example
Solve (x-1)/(x+2) > 0: critical points -2 and 1; solution (-∞,-2) ∪ (1,∞).
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