Solve the inequality (x^2 - 3x - 10) / (x - 2) < 0. Which of the following is the solution set?

Correct answer

A. (-∞, -2) ∪ (2, 5)

Principle or equation

To solve a rational inequality, factor the numerator and denominator, find critical points where either is zero, then test intervals on a number line. The solution set is the union of intervals where the inequality holds, excluding points where the denominator is zero.

Why this answer is correct

Factor numerator: x^2 - 3x - 10 = (x - 5)(x + 2). Critical points: x = -2, 5 (numerator zero) and x = 2 (denominator zero). Test intervals: (-∞, -2): choose -3, expression = (-8)(-1)/(-5) = -8/5 < 0, so included. (-2, 2): choose 0, expression = (-5)(2)/(-2) = 5 > 0, excluded. (2, 5): choose 3, expression = (-2)(5)/(1) = -10 < 0, included. (5, ∞): choose 6, expression = (1)(8)/(4) = 2 > 0, excluded. Also x = -2 and 5 make numerator zero, but the inequality is strict (<0), so they are not included. x = 2 is excluded because denominator zero. Thus solution is (-∞, -2) ∪ (2, 5).

Example

For (x-1)(x+2)/(x-3) < 0, critical points -2,1,3; testing gives (-∞,-2) ∪ (1,3).