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Thread: Simplifying using the Commutative Property of Addition

  1. #1

    Simplifying using the Commutative Property of Addition

    Hello,

    I'm having trouble understanding how to use the commutative property of addition to simplify this expression.

    4(x + 3) - 2(x - 1)

    This is how the book solves the problem:

    4(x + 3) - 2(x - 1)
    = 4x + 12 - 2x + 2 <-- I don't understand why the sign changed from minus to plus during the distribution of 2
    = 4x - 2x + 12 + 2 <-- how on earth did they reorder the expression when there is a subtraction in there? subtraction is not commutative or associative
    = 2x + 14

    This is how I work this problem:

    4(x + 3) - 2(x - 1)
    = 4(x) + 4(3) - 2(x) - 2(1) <-- use the distrubutive property
    = 4x + 12 - 2x - 2
    = 4x + 12 + (-2x) + (-2) <-- I don't know how to deal with the subtraction unless I convert it to addition
    = 4x + (-2x) + 12 + (-2) <-- reorder using commutative property of addition
    = 2x + 10 <-- my answer is incorrect

    I tried converting the subtraction to addition as the first step but I was no nearer to getting the correct result.

    Thanks for your help.
    [X] Pre-Algebra
    [X] Beginning Algebra
    [ ] Intermediate Algebra

  2. #2
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    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by coughsyrup78
    4(x + 3) - 2(x - 1)
    = 4(x) + 4(3) - 2(x) - 2(1) <-- use the distrubutive property
    Your last term should be -2(-1):
    4(x) + 4(3) - 2(x) - 2(-1)

    and -2(-1) = +2 : quite basic stuff!
    I'm just an imagination of your figment !

  3. #3
    JeffM
    Guest

    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by coughsyrup78
    Hello,

    I'm having trouble understanding how to use the commutative property of addition to simplify this expression.

    4(x + 3) - 2(x - 1)

    This is how the book solves the problem:

    4(x + 3) - 2(x - 1)
    = 4x + 12 - 2x + 2 <-- I don't understand why the sign changed from minus to plus during the distribution of 2 Because you are not distributing 2 but (-2). And (-2) * (-1) = 2.
    = 4x - 2x + 12 + 2 <-- how on earth did they reorder the expression when there is a subtraction in there? subtraction is not commutative or associative
    Do you really not see that that 12 - 2x = 12 + (-2x) = (-2x) + 12 = - 2x + 12? If that is where your problem is, let's discuss further.

    = 2x + 14

    This is how I work this problem:

    4(x + 3) - 2(x - 1)
    = 4(x) + 4(3) - 2(x) - 2(1) <-- use the distrubutive property And how did (-1) become (+1)?
    = 4x + 12 - 2x - 2
    = 4x + 12 + (-2x) + (-2) <-- I don't know how to deal with the subtraction unless I convert it to addition
    = 4x + (-2x) + 12 + (-2) <-- reorder using commutative property of addition
    = 2x + 10 <-- my answer is incorrect

    I tried converting the subtraction to addition as the first step but I was no nearer to getting the correct result. That is because the conversion from subtraction to addition was not an error. Your error is in assigning a negative sign to the product of two negatives.
    Thanks for your help.
    Do my comments help? If not, we can take it step by step.

    PS Congratulations on showing your work. You will get MUCH better help at this site if you continue to do it.

  4. #4

    Re: Simplifying using the Commutative Property of Addition

    4(x + 3) - 2(x - 1)

    My problem is that I can't convert the subtraction to addition in my head without getting my signs all jumbled up.

    If I remove all of the subtraction at the very beginning, it makes it easier for me, although there are more steps. Something like this:

    4(x + 3) - 2(x - 1) =
    4(x + 3) + (-2)[x + (-1)] = <-- remove all subtraction signs 1st
    4(x) + 4(3) + (-2)(x) + (-2)(-1) = <-- distribute
    4x + 12 + (-2x) + 2 =
    4x + (-2x) + 12 + 2 = <-- reorder
    2x + 14 <-- seems to be correct

    Assuming I did that right, can someone explain to me why the x above didn't also become negative when I converted to addition?
    For example, why isn't it: 4(x + 3) - 2(x - 1) = 4(x + 3) + (-2)[-x + (-1)]
    [X] Pre-Algebra
    [X] Beginning Algebra
    [ ] Intermediate Algebra

  5. #5
    Elite Member
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    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by coughsyrup78
    Assuming I did that right, can someone explain to me why the x above didn't also become negative when I converted to addition?
    For example, why isn't it: 4(x + 3) - 2(x - 1) = 4(x + 3) + (-2)[-x + (-1)]
    You seem to be "inventing" questions; are you a student attending math classes?
    I'm just an imagination of your figment !

  6. #6
    JeffM
    Guest

    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by coughsyrup78
    4(x + 3) - 2(x - 1)

    My problem is that I can't convert the subtraction to addition in my head without getting my signs all jumbled up.

    If I remove all of the subtraction at the very beginning, it makes it easier for me, although there are more steps. Something like this:

    4(x + 3) - 2(x - 1) =
    4(x + 3) + (-2)[x + (-1)] = <-- remove all subtraction signs 1st
    4(x) + 4(3) + (-2)(x) + (-2)(-1) = <-- distribute
    4x + 12 + (-2x) + 2 =
    4x + (-2x) + 12 + 2 = <-- reorder
    2x + 14 <-- seems to be correct

    Assuming I did that right, can someone explain to me why the x above didn't also become negative when I converted to addition?
    For example, why isn't it: 4(x + 3) - 2(x - 1) = 4(x + 3) + (-2)[-x + (-1)]
    You did it right.

    The different uses of the + sign and - sign are a source of confusion to the beginner. + and - are used in at least three different senses in math. At a very fundamental level these senses are all coherent, but that coherence is not intuitive at first to the beginner. So be patient.

    Additive inverses: For a given number a (which may be positive or negative) there is an ADDITIVE INVERSE indicated as (- a) such that a + (- a) = 0.

    Operations: + means addition and - means subtraction.

    The rule is that (a - b) = [a + (- b)] where the first minus sign means the operation of subtraction and the second minus sign means additive inverse. So your "removing the subtraction" is CORRECT. It may add a step but if it prevents mistakes, SO WHAT?

    Now consider an expression like [a - (c * d)]. That is equal to {a + [-(c * d)]}, where [-(c * d)] is the additive inverse of (c * d). Now, in general, the additive inverse of (c * d) is NOT [(-c) * (-d)] because (c * d) + [(-c) * (-d)] = (c * d) + (c * d) = 2(c * d). The additive inverse of (c * d) = [(-c) * d] = [c * (-d)] = - (c * d).

    So when "you remove the subtraction" from 4(x + 3) - 2(x - 1) = 4(x + 3) + {- [2(x - 1)]} you have a choice.
    You can compute {- [2(x - 1)]} = (- 2) * (x - 1) = - 2x + 2, or you can compute {- [2(x - 1)]} = 2[- (x - 1)] = 2(- x + 1) = - 2x + 2.

    Clear now?

  7. #7

    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by Denis
    Quote Originally Posted by coughsyrup78
    Assuming I did that right, can someone explain to me why the x above didn't also become negative when I converted to addition?
    For example, why isn't it: 4(x + 3) - 2(x - 1) = 4(x + 3) + (-2)[-x + (-1)]
    You seem to be "inventing" questions; are you a student attending math classes?
    Is that your way of saying it was a stupid question?
    [X] Pre-Algebra
    [X] Beginning Algebra
    [ ] Intermediate Algebra

  8. #8

    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by JeffM
    You did it right.

    The different uses of the + sign and - sign are a source of confusion to the beginner. + and - are used in at least three different senses in math. At a very fundamental level these senses are all coherent, but that coherence is not intuitive at first to the beginner. So be patient.

    Additive inverses: For a given number a (which may be positive or negative) there is an ADDITIVE INVERSE indicated as (- a) such that a + (- a) = 0.

    Operations: + means addition and - means subtraction.

    The rule is that (a - b) = [a + (- b)] where the first minus sign means the operation of subtraction and the second minus sign means additive inverse. So your "removing the subtraction" is CORRECT. It may add a step but if it prevents mistakes, SO WHAT?

    Now consider an expression like [a - (c * d)]. That is equal to {a + [-(c * d)]}, where [-(c * d)] is the additive inverse of (c * d). Now, in general, the additive inverse of (c * d) is NOT [(-c) * (-d)] because (c * d) + [(-c) * (-d)] = (c * d) + (c * d) = 2(c * d). The additive inverse of (c * d) = [(-c) * d] = [c * (-d)] = - (c * d).

    So when "you remove the subtraction" from 4(x + 3) - 2(x - 1) = 4(x + 3) + {- [2(x - 1)]} you have a choice.
    You can compute {- [2(x - 1)]} = (- 2) * (x - 1) = - 2x + 2, or you can compute {- [2(x - 1)]} = 2[- (x - 1)] = 2(- x + 1) = - 2x + 2.

    Clear now?
    Thanks JeffM for your thorough explanation. The more I do this kind of problem the more I can kind of do it in my head.

    Code:
    4(x  +  3)  -  2(x  -  1)
     |       |       |      |
     1       2       3      4           <-- I look at it in four steps
     |       |       |      |
    4(x)   4(3)   -2(x)  -2(-1)        <-- 1, 2, 3, 4
     |       |       |      |
     4x   + 12  +  (-2x)   + 2          <-- stick a bunch of + signs in there
    
     2x + 14                             <-- reorder and simplify
    [X] Pre-Algebra
    [X] Beginning Algebra
    [ ] Intermediate Algebra

  9. #9
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    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by coughsyrup78
    Is that your way of saying it was a stupid question?
    Yes and no...hard for us to help "properly" if we don't know if the "student" is learning on his/her own,
    or attending math classes with a math teacher...
    I'm just an imagination of your figment !

  10. #10

    Re: Simplifying using the Commutative Property of Addition

    Quote Originally Posted by Denis
    Quote Originally Posted by coughsyrup78
    Is that your way of saying it was a stupid question?
    Yes and no...hard for us to help "properly" if we don't know if the "student" is learning on his/her own,
    or attending math classes with a math teacher...
    In that case, I'm planning on taking algebra I and I need to brush up on my pre-algebra before I do (it's been a couple years since I took pre-algebra so I've forgotten most of it).
    [X] Pre-Algebra
    [X] Beginning Algebra
    [ ] Intermediate Algebra

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