Question
Solve the inequality: (x − 1)/(x + 2) > 2. Which of the following represents the solution set? (
- A x < −5 (
- B −5 < x < −2 (
- C −2 < x < 1 (
- D x > 1
Solution
ATQ, x ≠ −2. (x − 1)/(x + 2) > 2 ⇒ (x − 1)/(x + 2) − 2 > 0 ⇒ (x − 1 − 2x − 4)/(x + 2) > 0 ⇒ (−x − 5)/(x + 2) > 0 Multiply numerator and denominator by −1 (reverses inequality): (x + 5)/(x + 2) < 0. Critical points: x = −5, x = −2. Sign analysis: For x < −5: numerator < 0, denominator < 0 ⇒ ratio > 0 (reject). For −5 < x < −2: numerator > 0, denominator < 0 ⇒ ratio < 0 (accept). For x > −2: numerator > 0, denominator > 0 ⇒ ratio > 0 (reject). Solution: −5 < x < −2.
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