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How To Use The Distributive Property With numerical and Algebraic expressions
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Feb 19, 2025
Welcome to chapter 4.1 of our Pre Algebra class, we've already learned how to evaluate numerical expressions using addition and multiplication. In this section, we'll learn how to use the distributive property to write equivalent numerical expressions as well as learn how to use it to write equivalent algebraic expressions. Chapters: 00:00 Introduction 01:20 Definition 04:06 How to evaluate numerical expressions 07:11 How to evaluate algebraic expressions 10:44 How to simplify expressions with subtraction
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so we are starting a new section it's
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section
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4.1 it's called the
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distributive property right distributive
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property now this is probably one of the
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easiest property you will learn this
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this year because it's really not not
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that complicated right first we're going
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to Define what it means in terms of like
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words and I'm expecting people to write
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this down right I'm expecting you to
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write this down yes can I get it from my
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go ahead can I too
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yeah but make sure you write this
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information down you need that all right
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so here I'll do this for
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[Music]
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you our our government so all right
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ter There's an actual billboard that
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says um that has Joe Biden and um Camala
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on it I think it's like somewhere and is
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Dumb and Dumber that's not I don't know
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why they do that
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anyway all let's stay focused on that
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Gab I might have to move you there's a
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lot of commotion down
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there all right so basically to multiply
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a su or Difference by a
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number
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first multiply each term inside the
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pares hey quiet by the number hey hey
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hey hey
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right so to multiply a sumary difference
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by number you have to multiply each term
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inside the parenthesis by the number
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outside the parenthesis what does that
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mean it's a jargon right what does that
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mean though we need to understand it so
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that's why I have the symbols up here
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right so let's say I have a parentheses
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B plus C what does that mean right what
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does that mean so I have a here and I
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have inside the parenthesis B plus C
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remember what I said right when I have a
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number and there's nothing next to it
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and we have parentheses what does that
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imply a multiplication right
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multiplication so basically a
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parentheses B plus c means that I'm
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doing what I'm doing a * b + a * C so
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I'm multiplying this two this is why
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have those arrrow here a * B right plus
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a * C did you guys learn the
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distributive property in sixth grade did
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you learn that with Mrs Young probably
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probably you might have forgotten right
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so it applies to both the sum and the
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difference right here I have a sum this
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is an addition right if I have a pareses
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B plus C that means I'm doing what I'm
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doing a * b + a * C right and here if I
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have a parenthesis B minus C I'm doing
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the same thing but in this in this case
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instead of an addition I have a
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subtraction because this is a difference
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this is a * B minus a Time
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C so now we doing what we call a
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subtraction but this is called a
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distributive property because you're
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Distributing the a into B and C and
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you're also doing it here you're
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Distributing the a into B and C but the
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difference between the two is here you
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have an addition and here you have a
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subtraction yes it still work to do like
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B minus C * a you can do that but often
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times you won't
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have uh numbers or numerical expressions
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you have variables and that won't work
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for that but here you work right cuz you
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just do 4 + 5 which is what 9 9 * 3 is
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27 you could do that with numerical
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expressions it's actually advised
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because this is called the what do you
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call it again your p p butter yeah no
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pendas right P parentheses and then
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whatever is outside right so now I put
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this here on purpose example one this is
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a numerical expression right you could
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have done this both ways but in this
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chapter we're going to Use the
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distributive property right so I have
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three here and I'm doing three
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parentheses 4 + 5 I'm technically saying
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add four five and multiply that by three
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but how do you do that using the
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distributive property excuse me what's E
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I M divide add and subtract what's E uh
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parentheses exponent exponent
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parentheses exponent multiplication
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division addition and subtraction all
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right here we're not talking pen we're
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talking distributive property right so I
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have three here parentheses 4 + 5 so
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this is technically 3 * 4 right plus 3 *
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5 now I did this on purpose I put the
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plus sign in green to indicate that you
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have to respect the operation you can't
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disregard this so anytime you do the 3 *
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4 bring your sign next and then do the
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next operation so here we are
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Distributing the three into the 4 and
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the 5 so this is 3 * 4 + 3 * 5 so that
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is 12 + 5 which is 27 right so you can
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use your P to double check your answer
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to make sure you are not doing you are
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not doing any mistakes or you are not
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making any mistakes right so now on the
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next one I gave you a similar problem
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but here instead of an addition we have
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a subtraction right we doing the same
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thing we are Distributing the six into
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the three and we also Distributing the
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six into the two but then we are keeping
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the sign that's a sign to which is a
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negative right minus so we doing 6 * 3
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and then minus 6 * 2 that is 18 - 12
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that is 6 that make sense no yes or no
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yes all right right now what do you see
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here that's different from the last one
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there's more arrows there's more arrows
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but what's the difference here what's
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the main difference it's going that way
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switch not switch and what is that
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property that we talked about in chapter
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1.4 or something it's called the
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called Comm commutative right
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commutative law because you can switch
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the sign you can switch the numbers it
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does not change the outcome of the
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operation right so I'm doing the same
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thing here 10 + 6 * 2 is 10 * 2 cuz I'm
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Distributing the two again plus 2 * 6
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right which is 20 + 12 that is 32 and it
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applies for the subtraction as well you
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have the same thing here you have 5 * 7
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right because the seven is being
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distributed minus 7 * 3 right and now
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you got it gives you 14 so even if I put
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the number in front of the parentheses
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as long as the number is outside the
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parenthesis the law of distribution
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applies does that make sense yes or no
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yes yes all right now now we're going to
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talk about evaluating expressions with
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addition now what's the difference
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between this section here and what we
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just talked about here do you see that
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here I I no longer have what just
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numbers what do I have
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here variables right so how would you do
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this e you have four then x + 5 you show
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me walk me through the
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uhhuh your AR is going go where here
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first right and then where right here so
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four and x what does that give
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you 4 x 4x right and then what do I keep
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in the
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middle my sign right yeah what's the
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Sign Plus and then four
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what 4 * 5 remember it's not four 45
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it's 4 * 5 cuz I know some of you guys
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may go oh this is 45 no it's not 45 it's
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4 * 5 right the reason why I just put uh
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I put 4X is 4X means 4 * X CU this is 4
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* 5 so you have to be extremely careful
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that you you don't write 45 it's not 45
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it's 4 * 5 so now that gives us what 4x
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+ 4 * which is what's 4 * 5 20 20 so
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this is 4 x + 20 right now how would you
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do this one here I have y + 6 * 3 nor
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how would you do this here uh 3 * y all
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right 3 * 6 3 * 6 which is 3
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y+ 18 18 all right so that's how that's
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yeah a question M for the 4x + 20 yeah
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can that be 24x no because you can't add
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to
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Y they are not like terms we are we are
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not there yet but this is 4X and this is
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20 right the only way this 24 is this
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was
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20x okay 20x + 4x 24
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4
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and then you do
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the * no you can't do that because here
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you all you have is 4X right 4 * X is 4X
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and then 4 * 5 that is 20 so now you
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have 4X and 20 so it's almost I have
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four oranges right and 20 let's say $20
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can you add four oranges and $20 no but
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what if I have $4 and $20 then give you
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$24 right now if I have four oranges and
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then 20 oranges that be 24 oranges now
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here have different units so you can't
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do that that make sense yeah
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wouldn't it be 4 * x 4 * X yeah which is
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4 x what isn't
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there oh wait it's just same okay yeah
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it's the same thing four four you don't
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have to put a dot anytime you have
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nothing there it implies or
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multiplication okay same thing here we
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have 3 y + 18 we can't do we can we
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can't say that this is 21 y because
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these are not the same Expressions okay
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that make sense now all right now here
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since you just asked a question can you
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help me with this one now we are doing
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on example three evaluating an
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expression with subtraction so how would
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you do this
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here
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yeah all right
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uhhuh
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mhm subtract them
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right so 3x - what- 12 12 right 3x - 12
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so this is how you apply the
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distributive law but now this is where
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it gets really tricky here right what do
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you see here what do I have I have a
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negative number right then also I have
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another negative so we're going to learn
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how to do this the right way right all
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right so how would how would you do this
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kill help me here real
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quick
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mhm yeah
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56 right so now here's the thing you
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do5 * -6 right what is that supposed to
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give you now positive positive so
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positive means plus 30 right so this is
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where a lot of people get hung up on
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that so now I'm going to show you
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another way to do this so that way you
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don't get confused okay so you could
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have done it this
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way5 * y right and then keep your sign
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in the middle yes yeah hold on a second
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minus
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[Music]
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right5 * 6 right so you put the5 here *
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6 do you see what I'm going with it
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right so you can write this because you
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keep your sign in the middle and you
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still multiply the5 * 6 which your sign
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in the middle so now I know that this
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is5 Y and now I have what I have a
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negative and a negative right so that
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gives me what a positive so this is just
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pretty much POS 5 * 6
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so that is5 y + 30 trust me this causes
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a lot of problem to a lot of people it
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causes problems because they get hung up
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on a sign they don't know if this is
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positive this is negative so you have to
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spend more time on the one with the
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negatives okay so that that can be a
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problematic so I I want you all to
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really take your time and then really
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when you when you have negatives take
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your time with it so I'm going to give
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you some some uh examples that that you
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can work on and then we wrap up this uh
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section
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