Solve the inequality (x - 5)/(x + 2) ≤ 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
B. (-2, 5]
Principle or equation
For a rational inequality, find the critical points where the numerator or denominator is zero. Test intervals between these points. The denominator cannot be zero, so exclude that point.
Why this answer is correct
Critical points: x = 5 (numerator zero) and x = -2 (denominator zero). Test intervals: (-∞, -2): choose x = -3, (-3-5)/(-3+2) = (-8)/(-1) = 8 > 0, not ≤0. (-2, 5]: choose x = 0, (0-5)/(0+2) = -5/2 ≤ 0, works. [5, ∞): choose x = 6, (6-5)/(6+2) = 1/8 > 0, not ≤0. At x = 5, expression is 0, included. At x = -2, undefined, excluded. So solution is (-2, 5].
Example
Solve (x-1)/(x+3) < 0: critical points 1 and -3; test intervals gives (-3,1).
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