MAC College Algebra Fall Dr Yiannis Vourtsanis Examination II

							                      MAC 1105 College Algebra
                                Fall 2007
                        Dr. Yiannis Vourtsanis
                  Examination II (Sample Problems)
           Equations, Inequalities, Systems, and Applications

I.   Equations, Inequalities and Systems

"Þ   Solve the following 1st degree (linear) equation:

     5a#B b "b b %aB b &b œ #aB c "b c $a%B b $b
     "!B b & b %B b #! œ #B c # c "#B c *
     "%B b #& œ c "!B c ""
     #%B b #& œ c ""
     #%B œ c $'
     B œ c$' œ c #
          #%
                   $



#Þ   Solve the following 2nd degree (quadratic) equation:

     &B# œ #"aB c "b b "(
     &B# œ #"B c #" b "(
     &B# œ #"B c %
     &B# c #"B b % œ !
     a&B c "baB c %b œ !
     &B c " œ !       SV             Bc%œ!
         "
     Bœ &                     Bœ%
     B œ ˜ " ß %™
           &


$Þ   Solve the following 2nd degree (quadratic) equation:

     "'B# b )B œ c "
     "'B# b )B b " œ !
     a%B b "ba%B b "b œ !
     a%B b "b# œ !
     %B b " œ !
     Bœ c"  %
%Þ   Solve the following 2nd degree (quadratic) equation:

     B# œ #B c &!
     B# c #B b &! œ !
     B# c #B b " œ c &! b "
     aB c "b# œ c %*
     ÉaB c "b# œ „ È c %*
     B c " œ „ (3
     B œ " „ (3

&Þ   Solve the following 3rd degree (cubic) equation:

     #B$ b B# c #B c " œ !
     B# a#B b "b c "a#B b "b œ !
     aB# c "ba#B b "b œ !
     aB b "baB c "ba#B b "b œ !
     B œ ˜ c "ß "ß c " ™
                      #


'Þ   Solve the following 4th degree (quartic) equation:

     %B% œ &aB# c #b b *
     %B% œ &B# c "! b *
     %B% œ &B# c "
     %B% c &B# b " œ !
     a%B# c "baB# c "b œ !
     %B# c " œ !       SV            B# c " œ !
      #   "
     B œ %                           B# œ "
     Bœ „"  #                        Bœ „"

     B œ ˜ „ " ß „ "™
             #
(Þ   Solve the following radical equation:

     ÈB b * c ÈB œ "
     ˆÈ B b * ‰ œ ˆ" b È B ‰#
               #

     B b * œ " b #È B b B
     B b * œ B b " b #È B
     ) œ #È B
     % œ ÈB
     "' œ B

)Þ   Solve the following exponential equation:
                   #
       #       )B bB
     #B bB œ     '%
       #         B# bB
               )
     #B bB œ     8#
       #
                 B# bBc#
     #B bB œ )
     #B bB œ a#$ b
       #           B# bBc#
       #        #
     #B bB œ #$B b$Bc'
     B# b B œ $B# b $B c '
     ! œ #B# b #B c '
     ! œ B# b B c $
             È             È
     B œ c"„ # "b"# œ c"„# "$
*Þ    Solve the following exponential equation:

      #Bb" œ &Bb#
      #Bb" œ &Bb#
      aB b "blna#b œ aB b #blna&b
      Blna#b b lna#b œ Blna&b b #lna&b
      Balna#b c lna&bb œ #lna&b c lna#b
      Blnˆ # ‰ œ lnˆ #& ‰
           &         #
           lnˆ #& ‰
            lnˆ # ‰
      Bœ        #


      B œ log # ˆ #& ‰
                &

              &   #


      If you did not combine the logs into one, also acceptable is À
      B œ #lnaa#&bbclnaa&#bb , and in place of ln you can have a logarithm of any
            ln cln

      base.

"!Þ   Solve the following logarithmic equation:

      logaB b "b b logaB b #b œ logaB b $b b logaB b %b
      logaaB b "baB b #bb œ logaaB b $baB b %bb
      aB b "baB b #b œ aB b $baB b %b
      B# b $B b # œ B# b (B b "#
      ! œ %B b "!
      B œ #Þ&

      However, as logaB b "b œ loga c "Þ&b does not exist, but the right
      hand side has all logs existing, we have no solution.

      Bœg
""Þ   Solve the following logarithmic equation:

      log# aB c "b b log# aB b $b œ &
      log# aaB c "baB b $bb œ &
      #log# aaBc"baBb$bb œ #&
      aB c "baB b $b œ $#
      B# b #B c $ œ $#
      B# b #B c $& œ !
      aB b (baB c &b œ !
      B œ c (ß &

      B œ c ( causes log# aB c "b œ log# a c )b to not exist, so À
      B œ &.

"#Þ   Solve the following 1st degree (linear) inequality:

      #aB c "b b &aB c "b  "
      #B c # b &B c &  "
      (B c (  "
      (B  )
      B )(


"$    Solve the following 2nd degree (quadratic) inequality:

      B# b (  )BSolve the equality, then check points in each region.
      B# c )B b ( œ !
      aB c (baB c "b œ !
      B œ (ß "

      If B œ !ß (  ! is false, so the range B  " is not in the solution.
      If B œ #ß ""  "' is true, so the range "  B  ( is in the solution.
      If B œ "!ß "!(  )! is false, so the range B € ( is not in the solution.

      eB À "  B  (f or in interval notation, B − a"ß (b
"%Þ   Solve the following rational inequality:
      &Bb"
      %Bb)  c " Solve the equality and then check ranges as before.
      &B b " œ c %B c )
      *B  c *
      B c"
      However, we cannot have a division by zero so there will be another
      breakpoint at B œ c #.

      If B œ !ß "  c " is false, so the range B € c " is not in the
                )
      solution.

      If B œ c $, c"% œ
                    c%
                              (
                              #    c " is false, so the range B  c # is not in
      the solution.

      If B œ c$ ,
              #
                    c"$
                     %     c " is true, so the range c #  B  c " is in the
      solution.

      c#B c"

"&Þ   Solve the following # ‚ # linear system:


      œ #B b &C            Êœ
        %B b C        œ)      %B b C                œ)
                      œ ##    c %B c "!C            œ c %%

      Adding the two equations yields À
      c *C œ c $'
      Cœ%

      Plugging C œ % into the first equation yields À
      %B b % œ )
      %B œ %
      Bœ"

      aBß Cb œ a"ß %b
"'Þ   Solve the following $ ‚ $ linear system:

       BbCbD œ#
      #B b C c D œ c "
       BcCbD œ%

      Adding the last two equations gives À $B œ $
      So À                                  Bœ"

      Now that we know B let us eliminate B from every equation.

      "bCbD œ#
      #bCcD œ c"
      "cCbD œ%

      CbD œ"
      CcD œ c$
      cCbD œ$

      Multiplying the last equation by a c " makes it a copy of the second,
      so remove it.

      CbD œ"
      CcD œ c$

      Adding these two equations gives us À
      #C œ c #
      C œ c"

      Plugging into any of the original equations will give us À
      D œ #.

      aBß Cß D b œ a"ß c "ß #b
II.   Applications

"(Þ   Two vehicles are travelling in the same direction on a straight
      freeway at constant speeds. Find the time they will meet, given:

3Ñ    At 2:00 PM the locations were at the 50th mile and 85th mile
      respectively
33Ñ   Their speeds are 70mph and 50mph respectively.

      (!> b &! œ &!> b )&
      #!> œ $&
      > œ $& œ ( hr after 2:00 PM, or 3:45 PM
          #!   %


")Þ   Let X be a right triangle. Find the (acute) angles of X given that their
      difference is &# degrees.

      Let these two angles be Bß C, and let B € C À
      B b C b *! œ ")!
      B c C œ &#

      B b C œ *!
      B c C œ &#

      #B œ "%#
      B œ ("
      C œ *! c (" œ "*.

      The angles are "* and (" degrees.
"*Þ   Let V be a rectangle. Find its dimensions given that:

3Ñ    The perimeter is 17.
33Ñ   The area is %.

      T œ #6 b #A œ "(
      E œ 6A œ %
      Aœ % 6     Since an equation is non-linear, solve it for one of 6ß A.

      #6 b #A œ #6 b #ˆ % ‰
      ˆ#6 b ) œ "(‰6
                         6
             6
        #
      #6 b ) œ "(6
      #6# c "(6 b ) œ !
      a#6 c "ba6 c )b œ !
      6 œ "ß )
          #


      The rectangle is ) ‚ " .
                           #



#!Þ   Two individuals; A, B can do a certain job in % hours while A alone
      can do the same job in ( hours. How long will it take B to do the
      same job alone?
                  >+=5
      A can do ("29?<=
                  >+=5
      B can do B"29?<=
                               " >+=5
      Together they can do    % 29?<=

      "   "   "
      ( bB œ %
      "   "   "
      B œ %c(
      "    $
      B œ #)
      B œ #) œ * hours #! minutes
           $