Solve the rational inequality (2x - 3) / (x + 2) ≤ 1. Which of the following is the solution set?
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Correct answer
B. (-2, 5]
Principle or equation
To solve a rational inequality, bring all terms to one side, combine into a single fraction, find critical points (zeros and undefined points), test intervals, and include or exclude endpoints based on the inequality sign.
Why this answer is correct
Start with (2x - 3)/(x + 2) ≤ 1. Subtract 1: (2x - 3)/(x + 2) - 1 ≤ 0 → (2x - 3 - x - 2)/(x + 2) ≤ 0 → (x - 5)/(x + 2) ≤ 0. Critical points: x = 5 (zero) and x = -2 (undefined). Test intervals: (-∞, -2): choose -3 → (-8)/(-1)=8 >0; (-2,5): choose 0 → (-5)/(2)=-2.5 ≤0; (5,∞): choose 6 → (1)/(8)=0.125 >0. The inequality is ≤0, so include x=5, exclude x=-2. Solution: (-2, 5].
Example
Solve (x - 1)/(x + 3) ≤ 0. Critical points: 1 and -3. Test: (-∞,-3): positive; (-3,1): negative; (1,∞): positive. Solution: (-3, 1].
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