Solve the rational inequality: (x - 2) / (x + 4) >= 0.

Answer and explanation

Correct answer: C. (-infinity, -4) U [2, infinity)

  1. Step 1: Find critical points: numerator is zero at x = 2; denominator is zero at x = -4.
  2. Step 2: Test sign of quotient across intervals: x < -4 (positive), -4 < x < 2 (negative), x > 2 (positive).
  3. Step 3: At x = 2, quotient is 0 (included because inequality is >=).
  4. Step 4: At x = -4, denominator is 0 (must be strictly excluded with parenthesis).
  5. Step 5: Solution is (-infinity, -4) U [2, infinity).
Answer
(-infinity, -4) U [2, infinity).
Why not the others: '[-4, 2]' gives the interval where the quotient is negative; '(-4, 2]' reverses the inequality direction; '(-infinity, -4] U [2, infinity)' incorrectly includes the undefined point x = -4 with a square bracket.

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