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Section 2-5
Multiply and Divide Variable Expressions
Essential Questions

 How are variable expressions simplified?
 How are variable expressions evaluated?


 Where you’ll see this:
   Part-time job, weather, engineering, spreadsheets
Vocabulary
1. Property of the Opposite of a Sum:




2. Distributive Property:
Vocabulary
1. Property of the Opposite of a Sum: The negative
    outside the parentheses makes everything inside
    it opposite


2. Distributive Property:
Vocabulary
1. Property of the Opposite of a Sum: The negative
    outside the parentheses makes everything inside
    it opposite


2. Distributive Property: Multiply each term inside the
    parentheses by the term outside
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)




c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)




c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab −9ac
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab −9ac
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab −9ac                      8x   2
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab −9ac                      8x   2
Example 1
                  Simplify.

a. − 2(n − 5)                 b. .3( x + .7)
    −2n +10                      .3x +.21


c. -(2ab + 9ac)               d. 2x(4 x + 7y )
   −2ab −9ac                      8 x +14 xy
                                     2
Dividing Variable
Expressions
Dividing Variable
Expressions
 Divide each term in numerator by denominator
Dividing Variable
Expressions
 Divide each term in numerator by denominator
 Rewrite as distribution
Dividing Variable
Expressions
 Divide each term in numerator by denominator
 Rewrite as distribution
         3− x
          2
Dividing Variable
Expressions
 Divide each term in numerator by denominator
 Rewrite as distribution
         3− x         1 3− x
                       i
          2           2 1
Dividing Variable
Expressions
 Divide each term in numerator by denominator
 Rewrite as distribution
         3− x         1 3− x
                       i
                                   1
                                   2
                                       (3 − x)
          2           2 1
Dividing Variable
Expressions
 Divide each term in numerator by denominator
 Rewrite as distribution
         3− x         1 3− x
                       i
                                   1
                                   2
                                       (3 − x)
          2           2 1

            3
            2
                − x
                 1
                 2
Dividing Variable
Expressions
 Divide each term in numerator by denominator
 Rewrite as distribution
         3− x         1 3− x
                       i
                                      1
                                      2
                                          (3 − x)
          2           2 1

            3
            2
                − x
                 1
                 2
                                − x+
                                  1
                                  2
                                             3
                                             2
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.

     6x + 3                           10 y − 5
  a.                               b.
       3                                 2
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.

     6x + 3                           10 y − 5
  a.                               b.
       3                                 2

    6x 3
      +
    3 3
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.

     6x + 3                           10 y − 5
  a.                               b.
       3                                 2

    6x 3
      +
    3 3
    2x + 1
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.

     6x + 3                           10 y − 5
  a.                               b.
       3                                 2

    6x 3
      +
                                    1
                                    2
                                        (10 y − 5)
    3 3
    2x + 1
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.

     6x + 3                           10 y − 5
  a.                               b.
       3                                 2

    6x 3
      +
                                    1
                                    2
                                        (10 y − 5)
    3 3
    2x + 1                              5y −   5
                                               2
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.
     1.6 − .8 z                       9x + 5y
  c.                               d.
        −8                               7
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.
     1.6 − .8 z                       9x + 5y
  c.                               d.
        −8                               7
    1.6 .8 z
       −
    −8 −8
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.
     1.6 − .8 z                       9x + 5y
  c.                               d.
        −8                               7
    1.6 .8 z
       −
    −8 −8
    −.2 + .1z
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.
     1.6 − .8 z                       9x + 5y
  c.                               d.
        −8                               7
    1.6 .8 z
       −
    −8 −8
    −.2 + .1z
    .1z − .2
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.
     1.6 − .8 z                       9x + 5y
  c.                               d.
        −8                               7
    1.6 .8 z
       −                            1
                                        (9 x + 5 y )
    −8 −8                           7


    −.2 + .1z
    .1z − .2
Example 2
Simplify. Practice both methods of division to see
               which one you prefer.
     1.6 − .8 z                       9x + 5y
  c.                               d.
        −8                               7
    1.6 .8 z
       −                            1
                                        (9 x + 5 y )
    −8 −8                           7


    −.2 + .1z
                                        9
                                            x+ y
                                              5
    .1z − .2                            7     7
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
          2
                            b. − 4 6 − y
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
            2
                            b. − 4 6 − y

   −3x − 3
        2
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2


   −3(2) − 3
         2
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2


   −3(2) − 3
         2


    −3(4) − 3
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2


   −3(2) − 3
         2


    −3(4) − 3
    −12 − 3
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2


   −3(2) − 3
         2


    −3(4) − 3
    −12 − 3
     −15
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2
                            −4 6 − (−4)
   −3(2) − 3
         2


    −3(4) − 3
    −12 − 3
     −15
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2
                            −4 6 − (−4)
   −3(2) − 3
         2
                              −4 6 + 4
    −3(4) − 3
    −12 − 3
     −15
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2
                            −4 6 − (−4)
   −3(2) − 3
         2
                              −4 6 + 4
    −3(4) − 3                  −4 10
    −12 − 3
     −15
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2
                            −4 6 − (−4)
   −3(2) − 3
         2
                              −4 6 + 4
    −3(4) − 3                  −4 10
    −12 − 3                    −4(10)
     −15
Example 3
      Evaluate when x = 2 and y = -4.
  a. − 3( x + 1)
             2
                            b. − 4 6 − y

   −3x − 3
        2
                            −4 6 − (−4)
   −3(2) − 3
         2
                              −4 6 + 4
    −3(4) − 3                  −4 10
    −12 − 3                    −4(10)
     −15                        −40
Example 3
    Evaluate when x = 2 and y = -4.
            y-x
       c.
            x-y
Example 3
    Evaluate when x = 2 and y = -4.
             y-x
       c.
             x-y

            −4 − 2
            2 − (−4)
Example 3
    Evaluate when x = 2 and y = -4.
             y-x
       c.
             x-y

            −4 − 2
            2 − (−4)

              −6
               6
Example 3
    Evaluate when x = 2 and y = -4.
             y-x
       c.                    6
             x-y
                             6
            −4 − 2
            2 − (−4)

              −6
               6
Example 3
    Evaluate when x = 2 and y = -4.
             y-x
       c.                    6
             x-y
                             6
            −4 − 2
            2 − (−4)         1
              −6
               6
Problem Set
Problem Set


                p. 74 #1-47 odd




“Nobody got anywhere in the world by simply being
           content.” - Louis L'Amour

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Int Math 2 Section 2-5 1011

  • 1. Section 2-5 Multiply and Divide Variable Expressions
  • 2. Essential Questions How are variable expressions simplified? How are variable expressions evaluated? Where you’ll see this: Part-time job, weather, engineering, spreadsheets
  • 3. Vocabulary 1. Property of the Opposite of a Sum: 2. Distributive Property:
  • 4. Vocabulary 1. Property of the Opposite of a Sum: The negative outside the parentheses makes everything inside it opposite 2. Distributive Property:
  • 5. Vocabulary 1. Property of the Opposite of a Sum: The negative outside the parentheses makes everything inside it opposite 2. Distributive Property: Multiply each term inside the parentheses by the term outside
  • 6. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 7. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 8. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 9. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 10. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 11. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 12. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 13. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 14. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 15. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y )
  • 16. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab
  • 17. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab
  • 18. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab −9ac
  • 19. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab −9ac
  • 20. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab −9ac 8x 2
  • 21. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab −9ac 8x 2
  • 22. Example 1 Simplify. a. − 2(n − 5) b. .3( x + .7) −2n +10 .3x +.21 c. -(2ab + 9ac) d. 2x(4 x + 7y ) −2ab −9ac 8 x +14 xy 2
  • 24. Dividing Variable Expressions Divide each term in numerator by denominator
  • 25. Dividing Variable Expressions Divide each term in numerator by denominator Rewrite as distribution
  • 26. Dividing Variable Expressions Divide each term in numerator by denominator Rewrite as distribution 3− x 2
  • 27. Dividing Variable Expressions Divide each term in numerator by denominator Rewrite as distribution 3− x 1 3− x i 2 2 1
  • 28. Dividing Variable Expressions Divide each term in numerator by denominator Rewrite as distribution 3− x 1 3− x i 1 2 (3 − x) 2 2 1
  • 29. Dividing Variable Expressions Divide each term in numerator by denominator Rewrite as distribution 3− x 1 3− x i 1 2 (3 − x) 2 2 1 3 2 − x 1 2
  • 30. Dividing Variable Expressions Divide each term in numerator by denominator Rewrite as distribution 3− x 1 3− x i 1 2 (3 − x) 2 2 1 3 2 − x 1 2 − x+ 1 2 3 2
  • 31. Example 2 Simplify. Practice both methods of division to see which one you prefer. 6x + 3 10 y − 5 a. b. 3 2
  • 32. Example 2 Simplify. Practice both methods of division to see which one you prefer. 6x + 3 10 y − 5 a. b. 3 2 6x 3 + 3 3
  • 33. Example 2 Simplify. Practice both methods of division to see which one you prefer. 6x + 3 10 y − 5 a. b. 3 2 6x 3 + 3 3 2x + 1
  • 34. Example 2 Simplify. Practice both methods of division to see which one you prefer. 6x + 3 10 y − 5 a. b. 3 2 6x 3 + 1 2 (10 y − 5) 3 3 2x + 1
  • 35. Example 2 Simplify. Practice both methods of division to see which one you prefer. 6x + 3 10 y − 5 a. b. 3 2 6x 3 + 1 2 (10 y − 5) 3 3 2x + 1 5y − 5 2
  • 36. Example 2 Simplify. Practice both methods of division to see which one you prefer. 1.6 − .8 z 9x + 5y c. d. −8 7
  • 37. Example 2 Simplify. Practice both methods of division to see which one you prefer. 1.6 − .8 z 9x + 5y c. d. −8 7 1.6 .8 z − −8 −8
  • 38. Example 2 Simplify. Practice both methods of division to see which one you prefer. 1.6 − .8 z 9x + 5y c. d. −8 7 1.6 .8 z − −8 −8 −.2 + .1z
  • 39. Example 2 Simplify. Practice both methods of division to see which one you prefer. 1.6 − .8 z 9x + 5y c. d. −8 7 1.6 .8 z − −8 −8 −.2 + .1z .1z − .2
  • 40. Example 2 Simplify. Practice both methods of division to see which one you prefer. 1.6 − .8 z 9x + 5y c. d. −8 7 1.6 .8 z − 1 (9 x + 5 y ) −8 −8 7 −.2 + .1z .1z − .2
  • 41. Example 2 Simplify. Practice both methods of division to see which one you prefer. 1.6 − .8 z 9x + 5y c. d. −8 7 1.6 .8 z − 1 (9 x + 5 y ) −8 −8 7 −.2 + .1z 9 x+ y 5 .1z − .2 7 7
  • 42. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y
  • 43. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2
  • 44. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −3(2) − 3 2
  • 45. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −3(2) − 3 2 −3(4) − 3
  • 46. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −3(2) − 3 2 −3(4) − 3 −12 − 3
  • 47. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −3(2) − 3 2 −3(4) − 3 −12 − 3 −15
  • 48. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −4 6 − (−4) −3(2) − 3 2 −3(4) − 3 −12 − 3 −15
  • 49. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −4 6 − (−4) −3(2) − 3 2 −4 6 + 4 −3(4) − 3 −12 − 3 −15
  • 50. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −4 6 − (−4) −3(2) − 3 2 −4 6 + 4 −3(4) − 3 −4 10 −12 − 3 −15
  • 51. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −4 6 − (−4) −3(2) − 3 2 −4 6 + 4 −3(4) − 3 −4 10 −12 − 3 −4(10) −15
  • 52. Example 3 Evaluate when x = 2 and y = -4. a. − 3( x + 1) 2 b. − 4 6 − y −3x − 3 2 −4 6 − (−4) −3(2) − 3 2 −4 6 + 4 −3(4) − 3 −4 10 −12 − 3 −4(10) −15 −40
  • 53. Example 3 Evaluate when x = 2 and y = -4. y-x c. x-y
  • 54. Example 3 Evaluate when x = 2 and y = -4. y-x c. x-y −4 − 2 2 − (−4)
  • 55. Example 3 Evaluate when x = 2 and y = -4. y-x c. x-y −4 − 2 2 − (−4) −6 6
  • 56. Example 3 Evaluate when x = 2 and y = -4. y-x c. 6 x-y 6 −4 − 2 2 − (−4) −6 6
  • 57. Example 3 Evaluate when x = 2 and y = -4. y-x c. 6 x-y 6 −4 − 2 2 − (−4) 1 −6 6
  • 59. Problem Set p. 74 #1-47 odd “Nobody got anywhere in the world by simply being content.” - Louis L'Amour