Find the solution set of the inequality (x - 2)/(x + 5) ≥ 0.

Correct answer

A. (-∞, -5) ∪ [2, ∞)

Principle or equation

For a rational inequality of the form (x - a)/(x - b) ≥ 0, the sign is determined by the critical points a and b. The expression is zero at x = a and undefined at x = b. Test intervals between and beyond these points to determine where the expression is nonnegative.

Why this answer is correct

The critical points are x = 2 (where numerator is zero) and x = -5 (where denominator is zero, so undefined). Test intervals: (-∞, -5): choose x = -6, (-6-2)/(-6+5) = (-8)/(-1) = 8 > 0, so included. (-5, 2): choose x = 0, (0-2)/(0+5) = -2/5 < 0, so not included. (2, ∞): choose x = 3, (3-2)/(3+5) = 1/8 > 0, so included. At x = 2, the expression equals 0, so include 2. At x = -5, undefined, so exclude -5. Thus solution is (-∞, -5) ∪ [2, ∞).

Example

Solve (x - 1)/(x + 3) ≥ 0. Critical points: 1 and -3. Intervals: (-∞, -3): positive, (-3, 1): negative, (1, ∞): positive. Include 1, exclude -3. Solution: (-∞, -3) ∪ [1, ∞).