Solve the inequality: (x − 1)/(x + 2) > 2.

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.