6
algebra

Solve: x+1x2>0\frac{x+1}{x-2} > 0 (Non-linear preview)

A

x<1x < -1 or x>2x > 2

B

1<x<2-1 < x < 2

C

x>2x > 2

D

x<1x < -1

Correct Answer: A

Choice A is the correct answer. Sign analysis.

  1. Critical Points: Numerator 0 at x=1x=-1, Denominator 0 at x=2x=2.
  2. Test Intervals:
    • x<1x < -1 (e.g., -2): Neg/Neg = Pos (>0>0) ??n - 1<x<2-1 < x < 2 (e.g., 0): Pos/Neg = Neg (<0<0)
    • x>2x > 2 (e.g., 3): Pos/Pos = Pos (>0>0) ??n3. Result: x<1x < -1 or x>2x > 2

?�� Strategic Tip: A quotient is positive if both parts have the SAME sign.

Choice B is incorrect because it represents the negative region. Choice C is incorrect because it misses the negative x solution. Choice D is incorrect because it misses the positive x solution.