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simplify the rational expression

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3 Answers

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
 
( 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)    

Comments

My answer has been checked.
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)
:)