If q(2y+1)/(y-2) < 1, then which of the following alternatives gives the CORRECT range of y? A. -3 < y < -2 B. -3 < y < 2 C. -2 < y < 3 D. 2 < y < 3

GATE 2011 · General Aptitude · Algebra · medium

Answer: The correct range of y is -3 < y < 2 (Option B).

1. Bring inequality to one side: (2y+1 - (y-2))/(y-2) < 0 => (2y+1-y+2)/(y-2) < 0 => (y+3)/(y-2) < 0
2. Identify critical points: Critical points: y = -3 (numerator zero) and y = 2 (denominator zero, excluded from domain).
3. Sign chart over the three intervals: y < -3: (y+3)<0, (y-2)<0 => (+) > 0. NOT satisfied. -3 < y < 2: (y+3)>0, (y-2)<0 => (-) < 0. SATISFIED. y > 2: (y+3)>0, (y-2)>0 => (+) > 0. NOT satisfied.
4. State the solution: y = -3 gives 0/(−5) = 0, which is NOT < 0, so exclude -3. y = 2 excluded by domain. Solution: -3 < y < 2.