Let R ( x ) = x − 3 / x − 6 . Solve for R ( x ) ≤ 0 .

Answer 1

(x-3)/(x-6) < 0

(x-3)/(x-6) = 0

x =3

or in general, the numerator and denominator have opposite signs for (x-3)/(x-6) to be less than 0

x-3>0 and x-6<0 or x-3<0 and x-6>0

x>3 and x<6 or x<3 and x>6 which is impossible

the solution is

3<x<6 or x=3

so

3< x <6

x is an element of the interval [3, 6)

x can =3 but never= 6 or more and never less than 3

UNLESS you really did mean to write x - 3/x +6

then multiply by x

x^2 +6x -3 = 0

x = -6/2 + or - (1/2)sqr(36-12)

x = -3 + or - .5sqr24

x =-3 + or - sqr6

x = about -5.45 or -0.54

and

x^2 +6x -3 <0

(x+5.45)(x+0.54) <0

to be less than zero, the two factors must have opposite signs

either x<-5.45 and x>-0.54 or x>-5.45 and x<-0.54

the first has no solution, for the 2nd -5.45 < x < -0.54

full solution is about: -5.45 < x < -0.54

x is an element of the interval [-5.45, -0.54], approximately

or the exact interval: -3-sqr(6) < x < -3+sqr(6)

Odds are though, you really meant (x-3)/(x-6) and not x- (3/x) -6, even though technically it's written the 2nd way