Solving rational equations.

Here are a few questions he needs help with: 

1. 2 / (x + 2) - (1/x) = (-4) / (x(x + 2)) 



        2        1         -4

    _______  -  ___  = __________  Multiply each term by the LCD x(x+2)

     x + 2       x      x(x + 2)





        2                 1                 -4

    _______ (x(x+2)) -   ___ (x(x+2)) = __________  (x(x+2))

     x + 2                x              x(x + 2)



                        2x - 1(x + 2) = -4

                           2x - x - 2 = -4

                                    x -2 = -4

                           x - 2 +  2 = -4 + 2



                                    x = -2

      

    



2. c - (c/3) + (c/5) = 26 



         c       c

    c - ___  +  ___ = 26   Multiply each term by the LCD = 15

         3       5



              c           c

    c (15) - ___ (15) +  ___ (15) = 26  (15) 

              3           5



    15c - 5c + 3c = 390    Combine like terms

              13c = 390    Divide 13 both sides



                c = 30







3. (1/8) + ((5x) / (x +2)) = (5/2) 



        1         5x       5

      _____ +  _______ =  ___  Mulitply each term by the LCD = 8(x+2)

        8       x + 2      2



     1                  5x                  5

   _____ (8(x+2)) +  _______ (8(x+2)) =  ___  (8(x+2)) 

     8                 x + 2                2



    1(x+2) + 5x(8) = 5(4(x+2))       Multiply (Distribution Property)

           x + 2 + 40x = 20x + 40    Combine like terms

              41x + 2 = 20x + 40     

                 21x = 38

                        38

                   x = ____   

                        21

       

4. (k /(k + 1)) + (k/(k - 2)) = 2



     k          k

   _____  +  ______  = 2 . LCD is (k + 1)(k - 2)

   k + 1      k - 2



    k                       k

   _____ (k + 1)(k - 2)+  ______ (k + 1)(k - 2) = 2 (k + 1)(k - 2)

   k + 1                  k - 2



        k(k - 2) + k (k + 1) = 2   Mulitply



             Combine like terms



            



                    k = -4





5. (10/(2y +8)) - ((7y + 8)/(y^(2) - 16)) = (-8 / (2y - 8)) 



       10        7y + 8      -8

     ______ -  _________ = ______

     2y + 8     y^2 - 16   2y - 8





        10         7y + 8        -8

     ________ -  __________ = ________ LCD = 2 (y-4)(y+4)

     2(y + 4)    (y-4)(y+4)   2(y - 4)





   10                      7y + 8                     -8

________ (2(y-4)(y+4))-  __________ (2(y-4)(y+4))= ________ (2 (y-4)(y+4))

2(y + 4)                 (y-4)(y+4)               2(y - 4)





           10(y - 4) - 2(7y + 8) = -8 (y + 4)

           10y - 40 - (14y + 16) = -8y - 32

          10y - 40 - 1(14y + 16) = -8y - 32

             10y - 40 + 14y - 16 = -8y - 32  Combine like terms

                         4y - 56 = -8y - 32

                             12y = 24        divide both sides by 12

                               y = 2





6. (5/(x+2)) = (-1/(x^(2) +7x+10)) + (3/(-x-5)) 



     5            -1              3

   _____ = ______________  +  ________

   x + 2    x^2 + 7x + 10      -x - 5



    5            -1               3

   _____ = ______________  +  _________   LCD = -1(x + 5)(x + 2)

   x + 2   (x + 5)(x + 2)     -1(x + 5)





     5                            -1                                  3

   _____ (-1(x + 5)(x + 2)) = ______________ (-1(x + 5)(x + 2)) +  _________ (-1(x + 5)(x + 2))

   x + 2                      (x + 5)(x + 2)                       -1(x + 5)





              5 (-1(x + 5)) = -1 (-1) + 3 (x + 2)



             -5x - 25 = 1 + 3x + 6



             -5x - 25 = 3x + 7



                 -32 = 8x



                  -4 = x  



 

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