MODERATE EXERCISES

(2) Solve; graph the solution set on the real number line.

 (a) (x - 1)(x + 2)
     -------------- < 0 
         x - 4

 (b) 5x - 3
     ------ > 1
     2x + 1

 (c) 3 + x2  4x

 (d) x3 + 6x > 5x2

 (e)    2x + 5
     ------------  0
     x2 + 2x - 35

 (a) (x - 1)(x + 2)
     -------------- < 0 
         x - 4
     (x-1)    - - -           - - -              + + +          + + +
     (x+2)    - - -           + + +              + + +          + + +
     (x-4)    - - -           - - -              - - -          + + + 
          ____________o__________________o__________________o__________
                     -2                  1                  4
     
  (x-1)(x+2)
  ----------  - - -           + + +              - - -          + + +  
    (x-4)  

          ____________o__ _ _ _ _ _ _ _ _o__________________o_ _ _ _ _ _
                     -2                  1                  4


 (b) 5x - 3
     ------ > 1
     2x + 1

     5x - 3
     ------ - 1 > 0
     2x + 1

     5x -3 - (2x + 1)
     ---------------- > 0
          2x + 1

     3x - 4
     ------ >0
     2x + 1

     (3x-4)    - - -           - - -            + + +          
     (2x+1)    - - -           + + +            + + + 
          ____________o__________________o_______________
                    -1/2                4/3                  
     
    3x - 4
    ------     + + +           - - -            + + +  
    2x + 1

          ____________o__ _ _ _ _ _ _ _ _o_______________
                    -1/2                4/3                  


 (c) 3 + x2  4x

     x2 - 4x + 3  0

     (x-3)(x-1)  0

     x-3     - - -       - - -         + + + 
     x-1     - - -       + + +         + + +
         ____________o____________o____________
                     1            3
 (x-3)(x-1)  + + +       - - -         + + + 

          _ _ _ _ _ ______________ _ _ _ _ _ 


 (d) x3 + 6x > 5x2

     x3 - 5x2 + 6x > 0

     x(x-2)(x-3) > 0

     x       - - -        + + +        + + +      + + +
     x-2     - - -        - - -        + + +      + + + 
     x-3     - - -        - - -        - - -      + + + 
         ___________o________________o________o__________
                    0                2        3
 x(x-2)(x-3) - - -        + + +        - - -      + + + 

         _ _ _ _ _ _o________________o_ _ _ __o__________


 (e)    2x + 5
     ------------  0
     x2 + 2x - 35

       2x + 5 
     ----------  0
     (x-5)(x+7)


     2x+5       - - -    - - -         + + +        + + +
     x-5        - - -    - - -         - - -        + + +
     x+7        - - -    + + +         + + +        + + +
          ____________o_________o_______________o________
                     -7       -5/2              5
       2x + 5
     ---------- - - -    + + +         - - -        + + +
     (x-5)(x+7)

         _ _ _ _ _ _ ___________ _ _ _ _ _ _ _________
                     -7       -5/2              5