Solve the rational inequality: (x - 2) / (x + 3) <= 2.

Answer and explanation

Correct answer: B. x <= -8 or x > -3

  1. Step 1: Subtract 2 from both sides: (x - 2)/(x + 3) - 2 <= 0.
  2. Step 2: Combine into a single fraction: (x - 2 - 2*x - 6)/(x + 3) = (-x - 8)/(x + 3) <= 0.
  3. Step 3: Multiply by -1 and reverse inequality: (x + 8)/(x + 3) >= 0.
  4. Step 4: The quotient is >= 0 when numerator and denominator have the same sign (or numerator is 0).
  5. Step 5: Critical points are x = -8 and x = -3. Testing intervals: for x <= -8 (both negative, quotient >= 0); for -8 < x < -3 (numerator positive, denominator negative, quotient < 0); for x > -3 (both positive, quotient > 0).
  6. Step 6: The solution set is x <= -8 or x > -3.
Answer
x <= -8 or x > -3.
Why not the others: -8 <= x < -3 is where the quotient is negative (< 0); x >= -8 includes the invalid interval between -8 and -3; x < -3 omits x > -3.

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