Solve the inequality (x + 2)/(x - 3) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. [-2, 3)
Principle or equation
For a rational inequality, find critical points where numerator or denominator is zero: x = -2 and x = 3. Test intervals determined by these points. The denominator cannot be zero, so x = 3 is excluded.
Why this answer is correct
Critical points are x = -2 (numerator zero) and x = 3 (denominator zero). Test intervals: (-∞, -2): choose -3, fraction = (-1)/(-6) = 1/6 > 0. (-2, 3): choose 0, fraction = 2/(-3) = -2/3 < 0. (3, ∞): choose 4, fraction = 6/1 = 6 > 0. The inequality is ≤ 0, so the solution is [-2, 3).
Example
Solve (x - 1)/(x + 2) ≥ 0: critical points 1 and -2; test intervals give (-∞, -2) ∪ [1, ∞).
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