Question

solution of inequality 2x²-x-3/x²-x-6<0

Answer 1

(2x²-x-3)/(x²-x-6)<0,

(2x-3)(x+1)/[(x-3)(x+2)]<0.

The quotient can only be negative is the numerator and denominator have different signs.

Therefore we need to solve for x when:

(1) (2x-3)(x+1)<0 and (x-3)(x+2)>0; or:

(2) (2x-3)(x+1)>0 and (x-3)(x+2)<0.

(1) (2x-3)(x+1)<0 when -1<x<1.5; (x-3)(x+2)>0 when x<-2 or x>3. So these conditions cannot be satisfied.

(2) (2x-3)(x+1)>0 when x<-1 or x>1.5; (x-3)(x+2)<0 when -2<x<3. This can be satisfied when -2<x<-1 or when 1.5<x<3.

CHECK: 

When x=-1.5, the expression = (-6)(-0.5)/[(-4.5)(0.5)]=3/(-2.25)<0 OK.

When x=2, the expression = (1)(3)/[(-1)(4)]<0 OK.