simplify the rational expression

Question

((1/x+1)+(1/x+2))/((1/x+2)+(1/x+3))= ()/()

 

Answer 1

f(x) = (1/(x+1)+1/(x+2)) / (1/(x+2)+1/(x+3))

x ≠ -1,-2,-3

f(x) = (1/(x+1)+1/(x+2))((x+1)(x+2)(x+3))

/ (((x+1)(x+2)(x+3))(1/(x+2)+1/(x+3)))

f(x) = ((x+2)(x+3)+(x+1)(x+3))

/ ((x+1)(x+3)+(x+1)(x+2))

f(x) = ((x+3)(x+2+x+1)) / ((x+1)(x+3+x+2))

f(x) = ((x+3)(2x+3)) / ((x+1)(2x+5)), x ≠ -1,-2,-5/2,-3

f(x) = (2x^2+3x+6x+9) / (2x^2+5x+2x+5), x ≠ -1,-2,-5/2,-3

f(x) = (2x^2+9x+9) / (2x^2+7x+5), x ≠ -1,-2,-5/2,-3

Horizontal asymptote at y = 1,
Vertical asymptotes at x = -1 and x = -5/2,
Zeros at x = -3 and x = -3/2, except
Holes at x = -2 and x = -3, (so no zero at x = -3),
y-intercept at y = ((3)(3)) / ((1)(5)) = 9/5.

See GeoGebra sketch at http://www.wyzant.com/resources/files/259912/hard_rational_function.

Answer 2

 1/ (X+1 )  + 1/ ( X +2)   /   1/( X+2) + 1/ ( X +3) =

 

 

1 / ( X + 1 ) + 1 / ( X +2 ) =  (2X +3) / (X + 1 ) ( X + 2) 

 

1/ ( X +2 ) + 1/ ( X +3 )   =  ( 2X +5 ) / ( X +2 ) ( X+3)

 

   ( 2X +3 )  / ( X+ 1) ( X +2 )  / ( 2X +5 ) /( X + 2 ) ( X+3)

 

     ( 2X + 3) / ( X +1 ) ( X +2 )  /  ( 2X +5 ) / ( X +2 ) ( X +3 )

 

     ( 2X + 3 ) / ( X+1 ) ( X +2 )* ( ( X+2) ( X +3) / ( 2X + 5)

 

     (2 X + 3 ) * ( X+3) / ( X +1 ) * ( 2X + 5) 

 

      (2X^2  + 9x + 9 ) / ( 2X^2 + 7X + 5 )