Selected Answers to Math 390 Review Exam 4



1.  Write in standard form.

     a.  - (-7/2) = 7/2     b.  (x - 3)/(-2) = ( -(x - 3) )/2    c.  - ( (2-y)/3 ) = ( -(2 - y) )/3

2.  Reduce.

     a.  16/28 = 4/7     b.  32a³ b/36a² b³ = (8a)/(9b² ) 

     c.  (4x²  + 20x + 25)/(4x²  - 25) = (2x + 5)² /( (2x + 5)(2x - 5) ) = (2x + 5)/(2x - 5)

3.  Express as an equivalent fraction with indicated denominator.

     a.  (x - 1)/(x + 2) = ?/(x²  - 4) 

         (x - 1)/(x + 2) = ?/( (x + 2)(x - 2) )  Building Factor: x - 2

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

     b.  (x + 7)/(x - 2) = ?/(x²  + x - 6)

         (x + 7)/(x - 2) = ?/( (x + 3)(x - 2) )  Building Factor: x + 3

         (x + 7)/(x - 2) = ( (x + 7)(x + 3) )/( (x - 2)(x + 3) )

4.  Perform the indicated operations.  Be sure to reduce final answer to lowest terms.

     a.  (2x)/( 5(x + 3) ) + 6/( 5(x + 3) ) = ( 2x + 6 )/( 5(x + 3) ) = ( 2(x + 3) )/ ( 5(x + 3) ) = 2/5

     b.  x/(x²  - 9) - 3/(x²  - 9) = (x - 3)/(x² - 9) = (x - 3)/( (x + 3)(x - 3) ) = 1/(x + 3)

     c.  (4x)/(3y² ) + x/(6y) = (4x)/(3y² ) + x/(2(3)y) =  ( (4x)(2) )/( 2(3)y² ) + (xy) /(2(3)y² )

         = ( 8x + xy )/(6y² ) = ( x(8 + y) )/(6y² )

     d.  (5x)/(x²  + 8x + 16) - (3x)/(x²  + 6x + 8) = (5x)/( (x + 4)² - (3x)/( (x + 4)(x + 2)

         =  ( 5x(x + 2) )/( (x + 4)² (x + 2) ) - ( 3x(x + 4) )/( (x + 4)² (x + 2) )

         = ( 5x²  + 10x )/( (x + 4)² (x + 2) ) - ( 3x²  + 12x )/( (x + 4)² (x + 2) )

         = ( ( 5x²  + 10x ) - (3x²  + 12x) )/( (x + 4)² (x + 2) )

         = ( 5x²  + 10x - 3x²  - 12x )/( (x + 4)² (x + 2) )

         = ( 2x² - 2x )/( (x + 4)² (x + 2) )

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

     e.  (2x - 1)/(x²  + 7x + 10) - (x + 1)/(x²  + 8x + 15) = (2x - 1)/( (x + 5)(x + 2) ) - (x + 1)/( (x + 5)(x + 3) )

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

         = ( (2x²  + 5x - 3) - (x²  + 3x + 2) )/( (x + 2)(x + 3)(x + 5) )

         = ( 2x² + 5x - 3 - x²  - 3x - 2)/( (x + 2)(x + 3)(x + 5) )

         = ( x² + 2x - 5)/( (x + 2)(X + 3)(x + 5) )

     f.  (4a² )/(9b² c)(6bc² )/(5a) = (8ac)/(15b)

     g.  ( (15a³ b)/(7c) ) / ( (5a) /(14c² ) ) = ( (15a³ b)/(7c) )((14c² )/(5a) ) = 6a² bc

     h.  ( (2x + 14)/(x²  + 2x - 15) )/( (x²  + 8x + 7)/(x²  - 2x - 3) )

         = ( 2(x + 7)/( (x + 5)(x - 3) )( (x - 3)(x + 1)/( (x + 7)(x + 1) ) )

         = 2/(x + 5)

     i.  ( (2x - 10)/(x²  - 1) )( ( x²  + 6x + 5)/(4x² ) )( (4)/(x² - 25) ) 

         = [ ( 2(x - 5) )/( (x + 1)(x - 1) ) ][ ( (x + 5)(x + 1) )/ (4x² ) ][ 4/( (x + 5)(X - 5) ) ]

         = 2/(x² (x - 1))

5.  Write as a simple fraction in lowest terms.  The main fraction bar is the one in bold face type.

     a.  (2/3)/(7/5)  = (2/3)(5/7) = 10/21

     b.  ((4x)/3)/((2x)/5) = ( (4x)/3 )( 5/(2x) ) = 10/3

     c.  (x - 1/x)/(x + 1/x) = ( x² /x - 1/x )/( x² /x + 1/x )

          = ( (x²  - 1)/x )/( (x²  + 1)/x )

          = ( (x²  - 1)/x)( x/(x²  + 1) )

          = (x²  - 1)/(x²  + 1)

6.  Use long division technique to divide each of the following.

     a.  (2x³  + x²  - 5)/(x + 3) = (2x²  - 5x + 15) + (-50)/(x + 3)

     b.  (2x²  - 3x + 4)/(x - 2) =  (2x + 1) + 6/(x - 2)

7.  Solve for x.

     a.  (5x² )/8 - (3x)/4 = 1

         8( (5x² )/8 ) - 8( (3x)/4 ) = 8(1)

         5x² - 6x = 8

         5x² - 6x - 8 = 0

         (5x + 4)(x - 2) = 0

         Either 5x + 4 = 0 or x - 2 = 0

         5x = -4 or x = 2

         x = -4/5 or x = 2

     b.  4/(x - 4) + 3/(x + 1) = 20/(x²  - 3x - 4)

           [(x - 4)(x + 1) ][ 4/(x - 4) ] + [(x - 4)(x + 1) ][ 3/(x + 1) ] = [(x - 4)(x + 1)][ 20/( (x - 4)(x + 1) )]

           (x + 1)4 + (x - 4)3 = 20

           4x + 4 + 3x - 12 = 20

           7x - 8 = 20

           7x = 28

           x = 4

8.  Solve each of the pollowing stated problems.

     a.  Antoine traveled 36 miles downstream by motorboat to visit his parents and then returned
          home.  Antoine spent 5 hours traveling, and the speed of the current is 3 miles per hour.
          How fast does the motorboat travel in still water?

          Let x denote speed of boat in still water.

                                 rate       time               distance

           downstream     x + 3      36/(x + 3)       36

           upstream         x - 3       36/(x - 3)        36

 
          36/(x + 3) + 36/(x - 3) = 5

          [ (x + 3)(x - 3) ][ 36/(x + 3) ] + [ (x + 3)(x - 3) ][ 36/(x - 3) ] = [ (x + 3)(x - 3) ](5)

          (x - 3)(36) + (x + 3)(36) = (x²  - 9)5

          36x - 108 + 36x + 108 = 5x²  - 45

          72x = 5x²  - 45

          0 = 5x²  - 72x - 45

          0 = (5x + 3)(x - 15)

          Either 5x - 3 = 0 or x - 15 = 0

          Either 5x = 3 or x = 15

          Either x = -3/5  or x = 15

          Since speed in still water can't be negative, the speed is 15 miles per hour.

     b.  It takes Professor Gilbert 3 hours to grade a set of homework papers, and it takes her 
          teaching assistant 5 hours to grade a set.  If they work together, how long will it take
          them to grade a set of homework papers?

          Let x denote the time in hours it takes them to do job if they work together.

                            rate of doing work     time work    fractional part of job done         

          Gilbert         1/3 job per hour        x hours       x/3

          Assistant      1/5 job per hour        x hours      x/5

          x/3 + x/5 = 1

          15(x/3) + 15(x/5) = 15(1)

          5x + 3x = 15

          8x = 15

          x = 15/8

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