Question 72801
<pre>
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 {{{(k^2 - k -2)}}}  Mulitply

        {{{k^2 - 2k + k^2 + k = 2k^2 - 2k - 4}}}     Combine like terms

            {{{2k^2 - k = 2k^2 - 2k - 4}}}

                    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