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									 Lesson 4.7

Double Operators
                    Compare

12 + [5] = 17 with                12 + [5 + 3] = 20
  Increase of 5                     The increase of 5 has
                                    been increased by 3.

                   When an increase is increased,
17  20              the effect is an increase.

 “If you receive more, you will have more.”

                  + (+ a) = + a
                    Compare

12 + [5] = 17 with                12 + [5 – 3] = 14
  Increase of 5                     The increase of 5 has
                                    been decreased by 3.

                   When an increase is decreased,
17  14              the effect is a decrease.

  “If you receive less, you will have less.”
                  + (– a) = – a
                    Compare

12 – [5] = 7         with         12 – [5 + 3] = 4
  Decrease of 5                     The decrease of 5 has
                                    been increased by 3.

                    When a decrease is increased,
 7  4                the effect is a decrease.

“If you give more away, you will have less left.”
                  – (+ a) = – a
                    Compare

12 – [5] = 7         with         12 – [5 – 3] = 10
  Decrease of 5                     The decrease of 5 has
                                    been decreased by 3.

                   When a decrease is decreased,
 7  10              the effect is an increase.

“If you give less away, you will have more left.”
                  – (– a) = + a
            Summary of the Sign Operators

+ ( ) changes a number into an increase: + (b) = + b
– ( ) changes a number into a decrease: – (b) = – b

+ ( ) leaves an increase or a decrease the same.
            + ( + b ) = + b and       +(–b) = –b
– ( ) changes an increase into a decrease and a decrease into an
increase.
            – ( + b ) = – b and – ( – b ) = + b
            Recall:   – ( ) is called the opposite operator.

What happens when a sign operator is applied to an expansion or a
contraction?
When a sign operator is applied to an expansion or a contraction, we
get a double operator.

First      Second      Double
Operator   Operator    Operator

 5( )        +( )        + 5( )        an increase of 5 times something

 5( )        –( )        – 5( )        a decrease of 5 times something

  ( )                      ( )
             +( )        +             an increase something divided by 2
   2                        2
  ( )                        ( )
             –( )        –             a decrease something divided by 2
   2                          2

        The + ( ) and the – ( ) determine the kind of change,
                                   ( )
        while the 5( ) and the         effect the size of the change.
                                    2
The sign operators distribute over sums and differences.


                     28 + (6 – 4) = 28 + 6 – 4

                     28 – (6 – 4) = 28 – 6 + 4

                     28 – (6 + 4) = 28 – 6 – 4

Double operators also distribute over sums and differences.


                   28 + 3(6 – 4) = 28 + 18 – 12

                   28 – 3(6 – 4) = 28 – 18 + 12

								
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