Solve the inequality (x^2 - 2x - 8) / (x + 1) < 0. Which of the following is the solution set?
Show the answer and explanation
Correct answer
A. (-∞, -2) ∪ (-1, 4)
Principle or equation
To solve a rational inequality, factor the numerator and denominator, find the critical points where either is zero, then test intervals on the number line. The inequality is strict (<0), so critical points are excluded.
Why this answer is correct
Factor the numerator: x^2 - 2x - 8 = (x - 4)(x + 2). The critical points are x = -2, x = 4, and x = -1 (where denominator is zero). These divide the real line into intervals: (-∞, -2), (-2, -1), (-1, 4), (4, ∞). Test a point in each interval: For x = -3, the expression is positive; for x = -1.5, it is negative; for x = 0, it is negative; for x = 5, it is positive. Since we need <0, the solution is (-∞, -2) ∪ (-1, 4).
Example
For (x - 1)/(x - 2) < 0, critical points 1 and 2; test intervals gives solution (1, 2).
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