Solve the inequality (x^2 - 3x - 4) / (x - 5) ≥ 0. Which of the following is the solution set?

Correct answer

A. [-1, 4] ∪ (5, ∞)

Principle or equation

For rational inequalities, find critical points from numerator and denominator, test intervals on a number line, and exclude values where the denominator is zero.

Why this answer is correct

Factor numerator: x^2 - 3x - 4 = (x - 4)(x + 1). Critical points: x = -1, 4, and 5 (denominator zero). Test intervals: (-∞,-1): numerator positive, denominator negative → negative; [-1,4]: numerator negative or zero, denominator negative → nonnegative; (4,5): numerator positive, denominator negative → negative; (5,∞): numerator positive, denominator positive → positive. Include endpoints -1 and 4 because numerator zero gives 0 ≥ 0, but exclude 5 because denominator undefined. Solution: [-1,4] ∪ (5,∞).

Example

For (x-1)/(x-2) ≥ 0, critical points 1 and 2; solution (-∞,1] ∪ (2,∞).