Geometry	
  	
  	
  
                                                                                                   	
                                                                                                                                                                                                                       Algebra	
  Review	
  Packet	
  
                                                                                                   You	
  are	
  expected	
  to	
  complete	
  this	
  packet.	
  	
  It	
  will	
  be	
  graded!	
  	
  This	
  packet	
  will	
  review	
  some	
  key	
  
                                                                                                   concepts	
  that	
  you	
  will	
  need	
  throughout	
  this	
  course.	
  	
  It	
  is	
  an	
  expectation	
  that	
  if	
  you	
  do	
  not	
  
                                                                                                   understand	
  or	
  do	
  not	
  know	
  how	
  to	
  do	
  any	
  of	
  these	
  concepts,	
  you	
  will	
  ask	
  for	
  help	
  or	
  seek	
  help	
  
                                                                                                   via	
  the	
  Internet.	
  
                                                                                                   	
  
                                                                                                   Each	
  section	
  of	
  the	
  review	
  may	
  contain	
  some,	
  if	
  not	
  all,	
  of	
  the	
  following:	
  
                                                                                                   	
                         •                          Definitions	
  
                                                                                                                              •                          Properties	
  
                                                                                                                              •                          Worked	
  out	
  solutions	
  	
  
                                                                                                                              •                          Web	
  address	
  and	
  a	
  corresponding	
  QR	
  Codes	
  for	
  you	
  to	
  visit	
  if	
  you	
  need	
  extra	
  explanations	
  
                                                                                                   	
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  problems	
  for	
  YOU	
  to	
  complete	
  
                                                                                                   This	
  packet	
  must	
  be	
  completed	
  by	
  	
  	
                                                                                                                                                                                                                                                                                                                                                                                                                                                  	
                                                     	
                                                    	
                                                     .	
  
                                                                                                   Please	
  complete	
  the	
  exercises	
  on	
  another	
  sheet	
  of	
  paper.	
  	
  Organize	
  your	
  work	
  by	
  the	
  sections	
  
                                                                                                   titles.	
  
                                                                                                   	
  
                                                                                                   You	
  will	
  have	
  a	
  test	
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  this	
  packet	
  on	
  	
                                                                                                                                                                                                                                                                                                                                                                                                                                       	
                                                     	
                                                    	
                                                     .	
  
                                                                                                   The	
  test	
  will	
  contain	
  questions	
  very	
  similar	
  to	
  what	
  you	
  will	
  complete	
  in	
  this	
  packet.	
  
                                                                                                   	
  
                                                                                                   We	
  will	
  spend	
  class	
  time	
  answering	
  any	
  questions	
  on	
  this	
  packet.	
  	
  You	
  MUST	
  NOT	
  wait	
  until	
  the	
  
                                                                                                   night	
  before	
  due	
  date	
  to	
  do	
  these	
  questions.	
  	
  It	
  is	
  absolutely	
  expected	
  that	
  you	
  spend	
  time	
  each	
  
                                                                                                   night	
  doing	
  this	
  packet.	
  	
  You	
  will	
  have	
  other	
  homework	
  on	
  top	
  of	
  this	
  packet,	
  so	
  you	
  MUST	
  plan	
  
                                                                                                   your	
  time	
  wisely!	
  
                                                                                                   	
  
                                                                                                   As	
  you	
  complete	
  sections,	
  you	
  are	
  more	
  than	
  welcomed	
  to	
  ask	
  me	
  to	
  see	
  the	
  answer	
  key,	
  so	
  that	
  
                                                                                                   you	
  can	
  check	
  you	
  work.	
  	
  Please	
  do	
  this	
  at	
  the	
  end	
  of	
  the	
  period	
  or	
  before	
  school	
  starts.	
  
                                                                                                   	
  
                                                                                                   Good	
  Luck!	
  	
  If	
  you	
  need	
  help,	
  please	
  don't	
  hesitate	
  to	
  ask	
  me,	
  email	
  me,	
  text	
  my	
  home	
  phone	
  
                                                                                                   number,	
  call	
  me	
  at	
  home,	
  and	
  go	
  to	
  the	
  tutoring	
  center.	
  	
  	
  	
  
                                                                                                   	
                                                 Without	
  the	
  skills	
  included	
  in	
  this	
  packet,	
  you	
  will	
  struggle	
  
                                                                                                                                                                                                                                                                                                 throughout	
  the	
  year	
  in	
  Geometry.	
  
                                                                                                   	
                                                                                                                                                                                                                                                                                                                                                                                 1	
  
                  Section	
  1-­	
  Order	
  of	
  Operations-­	
  PEMDAS	
  
                  	
  
                  PEMDAS	
  is	
  a	
  standard	
  used	
  for	
  combining	
  Real	
  numbers.	
  	
  It	
  is	
  the	
  order	
  in	
  which	
  you	
  combine	
  
                  numbers.	
  
                  	
  
                  What	
  does	
  PEMDAS	
  stand	
  for?	
  	
  	
  
                  P-­parenthesis	
  
                  E-­exponents	
  
                  (MD)-­	
  Multiplication	
  and	
  Division	
  
                  (AS)-­	
  Addition	
  and	
  Subtraction	
  
                  	
  
                  You	
  will	
  always	
  do	
  parenthesis	
  first,	
  then	
  exponents,	
  but	
  for	
  multiplication	
  and	
  division,	
  it	
  is	
  
                  done	
  left	
  to	
  right,	
  as	
  well	
  as,	
  addition	
  and	
  subtraction.	
  
                  	
  
                  Examples:	
  
                  	
  
                  1.      100 ÷ 2⋅3÷(8 + 2)                                 2.      6 ÷ 2 ⋅ 3− 3(3+ 2)−(−25 + 5)
                           100 ÷ 2 ⋅ 3 ÷ 10                                          6 ÷ 2 ⋅ 3− 3(5)− (−20)
                                         (    )                                      3⋅ 3− 3(5)−(−20)
                               50 ⋅ 3 ÷ 10        	
    	
        	
                                                      	
  
                                      (    )                                               9 −15 + 20
                                 150 ÷10                                                     − 6 + 20
                                      15                                                          14
                  	
  
                  Need	
  Additional	
  Help?	
  
                  	
  
                  http://www.khanacademy.org/math/algebra/	
  
                  solving-­‐linear-­‐equations/v/order-­‐of-­‐operations-­‐example	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  	
  
                  Your	
  Turn-­	
  These	
  must	
  be	
  completed	
  on	
  another	
  sheet	
  of	
  paper,	
  making	
  sure	
  you	
  write	
  
                  the	
  problem	
  as	
  well.	
  
                  	
                                                                                      2             2
                  1.     50 ÷ 2 + 5⋅10 ÷ 2 + 7                                                      2.      − 3 + 12 ÷ 3  ⋅ 2 − 5
                                                                                                              (        )
                  3.     ⎛ 18 ÷ 6 +12⎞2                                                                 4.       − 10 ÷ ⎛ 9⎞ ÷ 2
                         ⎜              ⎟                                                                      ⎜  ⎟
                         ⎝      3       ⎠                                                                5     ⎝ 4⎠
                  	
                                                               2	
  
                                                                                                        Section	
  2-­	
  Combining	
  Like	
  Terms	
  
                                                                                                        	
  
                                                                                                        To	
  combine	
  like	
  terms,	
  you	
  must	
  first	
  make	
  sure	
  the	
  expression	
  contains	
  no	
  parenthesis.	
  	
  Get	
  rid	
  
                                                                                                        the	
  parenthesis	
  by	
  using	
  the	
  distributive	
  property.	
  	
  Then,	
  you	
  combine	
  the	
  same	
  "family"	
  of	
  
                                                                                                        terms	
  by	
  adding	
  or	
  subtracting	
  the	
  coefficients.	
  	
  	
  
                                                                                                        	
  
                                                                                                        Distributive	
  property-­	
  a(b + c) = ab + ac	
  
                                                                                                        	
  
                                                                                                        A	
  Coefficient	
  is	
  a	
  real	
  number	
  that	
  is	
  multiplied	
  to	
  a	
  variable.	
  	
  If	
  there	
  is	
  no	
  number	
  visible	
  with	
  
                                                                                                        a	
  variable,	
  then	
  the	
  coefficient	
  is	
  assumed	
  to	
  be	
  a	
  1.	
  
                                                                                                        	
  
                                                                                                        A	
  "family"	
  must	
  have	
  the	
  same	
  variable(s)	
  and	
  the	
  same	
  exponent(s).	
  	
  Look	
  below	
  to	
  see	
  
                                                                                                        examples	
  of	
  families	
  and	
  examples	
  that	
  aren't	
  families.	
  	
  
                                                                                                        	
                                                                                                                                                                                                                                                                                                            Families                                                                                                 Not Families
                                                                                                                                                                                                                                                                                                                                                                                                                   3x,x,−5x                                                                                                                           3x,−x2                                                                                    	
  
                                                                                                                                                                                                                                                                                                                                                                                                  3xy,5xy,−7xy 	
                                                                                                                                          3x,3y
                                                                                                        Examples	
  of	
  Combining	
  Like	
  Terms:	
  
                                                                                                        	
                                                        1.     6x − 5x + 3x − x                                                                                                                                                                                                                                                                                                                                                                  2.     − 2x + 3x2 + 9x − 6x2
                                                                                                                                                                             (6 − 5 + 3−1)x                                                                                                                                                                                                                                                                                                                                                                          (−2 + 9)x +(3− 6)x2
                                                                                                        	
                                                                                                                                                                                                                              	
   	
                                                       	
                                                       	
                                                       	
                                                       	
                                                        	
                                                                                2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            7x + −3 x
                                                                                                                                                                                        3x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     (                          )
                                                                                                        	
                                                                                                                                                                                                                                                                                                                                                                                                                                                                     7x − 3x2
                                                                                                        	
                                                       	
                                                       	
                                                       	
                                                                                                                                                                       2
                                                                                                                                                                                                                                                                                      3.      4 2x − 5 − 2 x − 3 + 5x x −1
                                                                                                                                                                                                                                                                                                                                              (                                                       )                                   (                                                     )                                            (                                          )
                                                                                                                                                                                                                                                                                                  8x − 20 − 2x2 + 6 + 5x2 − 5x
                                                                                                        	
                                                       	
                                                       	
                                                                  (−2 + 5)x2 + (8 − 5)x +(−20 + 6) 	
  
                                                                                                                                                                                                                                                                                                            3x2 + 3x + (−14)
                                                                                                        	
                                                                                                                                                                                                      3x2 + 3x −14
                                                                                                        	
  Need	
  Additional	
  Help?	
  	
  	
  
                                                                                                        	
  
                                                                                                        http://www.khanacademy.org/math/arithmetic/	
  
                                                                                                        number-­‐properties/v/distributive-­‐property-­‐example-­‐1	
  	
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    	
  
                                                                                                        	
  
                                                                                                        	
  
                                                                                                        Your	
  Turn-­	
  These	
  must	
  be	
  completed	
  on	
  another	
  sheet	
  of	
  paper,	
  	
  
                                                                                                        making	
  sure	
  you	
  write	
  the	
  problem	
  as	
  well.	
  
                                                                                                        	
             1.     10x − 6y + 3x − 7x + 9y
                                                                                                                         2.     2 2x + 3 − 4 5x − 6
                                                                                                                                                                          (                                                       )                                   (                                                        )
                                                                                                                         3.     6x(2x − 3y)+ 5y(2x + y)                                                                                                                                                                                                                                                                                                                                                      3	
  
                                                                                                        	
  
                                                                                                   Section	
  3-­	
  Solving	
  Equations	
  
                                                                                                   	
  
                                                                                                   To	
  solve	
  a	
  linear	
  equation,	
  use	
  the	
  properties	
  of	
  equality	
  and	
  properties	
  of	
  real	
  numbers	
  to	
  find	
  
                                                                                                   the	
  value	
  of	
  the	
  variable	
  that	
  satisfies	
  the	
  equation.	
  	
  In	
  the	
  case	
  of	
  literal	
  equations	
  (equations	
  
                                                                                                   that	
  have	
  more	
  than	
  one	
  variable),	
  you	
  solve	
  for	
  the	
  variable	
  being	
  asked,	
  and	
  your	
  answer	
  will	
  
                                                                                                   be	
  another	
  expression.	
  
                                                                                                   	
  
                                                                                                   Remember	
  to	
  always	
  use	
  the	
  Golden	
  Rule	
  of	
  Equality-­‐	
  what	
  you	
  do	
  to	
  one	
  side	
  of	
  the	
  equation,	
  
                                                                                                   must	
  be	
  done	
  to	
  the	
  other	
  side.	
  
                                                                                                   	
  
                                                                                                   You	
  can	
  ALWAYS	
  check	
  your	
  answer	
  for	
  equations.	
  	
  Substitute	
  the	
  answer	
  value	
  back	
  into	
  the	
  
                                                                                                   original	
  equation,	
  and	
  you	
  get	
  a	
  TRUE	
  statement,	
  then	
  the	
  answer	
  is	
  correct.	
  
                                                                                                   	
  
                                                                                                   Examples:	
  
                                                                                                   	
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             3.      Solve for l
                                                                                                    1.      5x − 3 = 2                                                                                                                                                                                                             2.      1− 2 x +1 = x + 6
                                                                                                                                                                                                                                                                                                                                                                                                               (                                        )
                                                                                                             5x − 3+ 3= 2+ 3                                                                                                                                                                                                                   1− 2x − 2 = x + 6                                                                                                                                                                                                                                                                             P = 2(l + w)
                                                                                                                     5x = 5                                                                                                                           	
  	
           	
                                                    	
                                                     	
                                                                                                                                                                                                                                                     P = 2l + 2w
                                                                                                                                                                                                                                                                                                                                                   − 1− 2x = x + 6
                                                                                                                        x = 1                                                                                                                                                                                                                    1−1− 2x = x + 6 −1                                                                                                                                                                                                                                                               P−2w=2l+2w−2w
                                                                                                   	
                                                                                                                                                                                                                                                    − 2x = x + 5                                                                                                                                                                                                                                                             P−2w=2l
                                                                                                   	
                                                                                                                                                                                                                                              − 2x − x = −x + x + 5                                                                                                                                                                                                                                                            P−2w
                                                                                                   	
                                                                                                                                                                                                                                                    − 3x = 5                                                                                                                                                                                                                                                                                       2                                    =l
                                                                                                   	
  
                                                                                                   	
                                                                                                                                                                                                                                                        x = − 5
                                                                                                   	
                                                                                                                                                                                                                                                                                                                                                                                           3
                                                                                                   	
  
                                                                                                   	
  
                                                                                                   	
                                                                                                                                                                                                                                                             1                                                                                                             2
                                                                                                                                                                                                                                                                                                                      4.                                                              x − 4 + 6 =                                                                                                  2x−1 +2
                                                                                                                                                                                                                                                                                                                                                                   3(                                                      )                                                    3(                                                             )
                                                                                                                                                                                                                                                                                                                           3⎛ 1 x − 4 + 6 = 2 2x −1 + 2⎞
                                                                                                                                                                                                                                                                                                                                                        ⎜                    (                                            )                                                                (                                                  )                                ⎟
                                                                                                                                                                                                                                                                                                                                                        ⎝ 3                                                                                                                     3                                                                                              ⎠
                                                                                                                                                                                                                                                                                                                               1 x − 4 +18 = 2 2x −1 + 6
                                                                                                                                                                                                                                                                                                                                                                        (                                           )                                                                     (                                                  )
                                                                                                                                                                                                                                                                                                                                    x − 4 +18 = 4x − 2 + 6
                                                                                                                                                                                                                                                                                                                                          x + 14 = 4x + 4
                                                                                                                                                                                                                                                                                                                                    x +14 − 4 = 4x + 4 − 4
                                                                                                                                                                                                                                                                                                                                          x + 10 = 4x
                                                                                                                                                                                                                                                                                                                                 − x + x +10 = 4x − x
                                                                                                                                                                                                                                                                                                                                                10 = 3x
                                                                                                                                                                                                                                                                                                                                                10 = x
                                                                                                                                                                                                                                                                                                                                                                                                                                                3
                                                                                                   	
                                                                                                                                                                                                                                                                                                                                                                                 4