simplify the rational expression

Question

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

 

Answer 1

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

 

Can't divide by zero, so x ≠ -1, -2, -3.

 

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

 

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

+ ((x+3)(x+1))/((x+3)(x+1))

+ (x+1)/((x+1)(x+3)), x ≠ -1, -2, -3

 

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

 

E = (x+3 + x^2+4x+3 + x+1)/((x+1)(x+3)), x ≠ -1, -2, -3

 

E = (x^2+6x+7)/((x+1)(x+3)), x ≠ -1, -2, -3

 

Can't factor numerator over the integers.

 

E = (x^2+6x+7)/(x^2+4x+3), x ≠ -1, -2, -3

 

Answer 2

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

 

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

 

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

Answer 3

Simplify numerator first

1/(x+1)  + 1/(x+2)

 

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

 

Simplify denominator

 

1/(x+2)  + 1/(x+3)

 

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

(2x+5)/(x+2)(x+3)

 

Now the expression becomes

 

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

(x+2) cancells off

 

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

:)