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How To Solve system Of Inequalities Part 2: Finding Coordinate Vertices
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Feb 19, 2025
In part 2 of our section on solving system of inequalities, we'll learn how to find the coordinate of the vertices of a region formed by the graph of a system of inequalities Chapters: 00:00 Introduction 00:34 Working on finding the coordinates of the vertices of a region
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0:01
so uh we're doing the second part of
0:03
this uh section on solving systems of
0:06
inequalities by graphing right so now
0:08
we're going to learn how to find the
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vertices of an enclosed region okay so
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so far all we've done is this we trying
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to identify what a region looks like now
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we need to find the vertices okay the
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vertices of that region how you find the
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vertices of that region so to do that we
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need to just work on an example okay so
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suppose I have this uh region here I
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have these three curve uh graphs right
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one
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two and three and I want to find the
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coordinates of the vertices of the
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region formed by these three
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graphs number one we're going to
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graph each one of them two we're going
0:51
to find the solution and then three
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we're going to find the coordinate of
0:55
the vertices right so let's do that so
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I'm going to start by graphing the first
1:00
one 2x - 8 I'm going to use like
1:03
different colors so that helps us
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identify which one is
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what right so the first one is 2x - 8 so
1:12
this the Y intercept is8 so I'm going to
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go 1 2 3 four five 6 7 and eight we're
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going to go down or up to from here one
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and two and over to the right one right
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cuz that's our Y intercept that's our
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slope I mean now this is a Sol solid
1:30
line because it's more than or equal
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right so it's a solid line so I'm not
1:35
too concerned about graphing the region
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yet I just need to get my lines first so
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let's call this line one so we going to
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have one here I'm just going to put this
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here so that we we identify it the next
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one is1 over 4x + 6 right so I need to
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identify my Y intercept so this is uh I
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use the green one this
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time uh and one and two three four five
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six right six is my Y intercept and the
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slope is 1 over 4 down one over to the
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right four so 1 2 3 and four right here
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right so again this is a solid line so
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it's going to go like this right so
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we're going to call this two and in
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three we're going to use a blue
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marker so I use a blue marker
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so is anybody having trouble with this
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graphs or no are we good all right now
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number three now here before I solve it
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I before I find it I need to solve right
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for y so I'm going to divide it by what
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sh what I'm going to divide this by what
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four four
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right so we get Y is greater than or
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equal to -15 4x - 8 right so my Y
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intercept is8 so again it's here oh why
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is he have to be negative darn it you
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can just go now I have to go down 15 15
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unit down no but you can just go up to
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the
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left no I have to start from here go
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down 15 so that's like is this from the
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book yeah book pretend it's 15 you can
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go 15 up and then do four left just draw
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a random line oh yeah you guys are
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smarter than I yeah you're right so
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let's go crit for so 1 2 3 4 5 6 7 8 9
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10 11 12 13 14 15 and then to the left
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four right yeah so one 2 three and four
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you see that that's why I like you
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guys who say that you
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say huh very straight line I know you
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ain't talking you said y'all I'm not
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going to say cuz the camera is
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on yeah you better edit that one
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out so that's
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it right number three so now the
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question is asking us to find listen
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carefully right
4:00
the coordinate of the vertices right of
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the region formed by this so what's the
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region the region is Right Here Right
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the inside because if you plug it in
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right if I 0 is less th8 right Z is here
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less than is more than 8 that's true
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right so the region this is the region
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if you do it for every single one of
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them this is the region that's going to
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be your solution now make sure when you
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scribble you do not go outside of the
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region because if you scribble here you
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are technically telling me that this is
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also the solution so you have to be
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careful how you scrib the stuff right so
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the solution we're going to use a pink
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color here so that way we can pop up
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this is your solution right but now this
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is not the only thing that we are
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concerned about we also need to find the
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vertices right what what are the
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vertices what is the vertice verce what
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is
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it the verce right the veres are those
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those Corner point you can also call
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them the corner Point corner like c o r
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n e r right those are the corner point
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this one here what else this guy
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here right and then this guy here now
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which one is the easiest to find um the
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bottom one the bottom
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one the X so that's the the Y and St
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right so we already know what this one
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is what is the coordinate somebody give
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it to me please 0 a 0g8 right
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08 right we already have the first one
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we have this guy now we got to figure
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out this guy and we need to figure out
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this guy now I know you guys know how to
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do it so how would I find the
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vertices this one here how would I find
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this the coordinates there's two ways to
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do it right one way is to use what the
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graph find the intersection by going
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here and go here sometime it's not
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always obvious right we don't always
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know so if I don't know I how to do it
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graphically how can I find it
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ALB yeah we can
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what do we do this is what line line two
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and what line line one what do they do
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they intersect so when two lines
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intersect how do I find their
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coordinates I just need to set this
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equals to what that find X and then
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after that I can solve
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for y so right so two and one is going
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to give us this point so I'm going to
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set these two together right so I'm
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going to go 2X
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- 8 is equal to what 2 and 1 2 and 1 so
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equal 14 x + 6 okay so what do I do here
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to make it simple I don't want
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fractions everything by what four right
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we did that in chapter one or two so
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times everything by 4 we're going to get
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what
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8x - 32 = x + 24 and then we can solve
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for x so I'm going to add X on this side
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here
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right I hope you computer guys that are
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doing work and not just looking at some
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computer stuff I am paying attention now
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paying attention and writing is
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different two different things right so
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9x - 32 is 24 and I'm going to add
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32 so I get 9x is = to 5 six and to find
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X I just divides by 9
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right and unfortunately the book is
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ridiculous they gave us fraction I wish
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the first problem would have been this
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way so X is 59 56 over 9 right and now
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we need to find y to find y I can take
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this X and plug it in where here to
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solve for y because I need two and one I
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can plug it in here or here it doesn't
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matter which one I choose because the y
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y is going to be the same because they
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intersecting but make sure you do not
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plug the X into the third equation
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understood right because we have to put
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it in the first equation or the second
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equation right so I'm going to put plug
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in for y I'm going to go Y = 2 * 56
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9 - 8 so this this is basically 2 * 56
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is 112 over 9 - 8 find the common
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denominator is y = to this is 9 here 112
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- 72 cuz I'm timesing it and this is
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basically 50 over 9 right so my second
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ver vertices
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is 56 over 9 and 50 over9 this is this
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point right here now I need to find the
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last one which is this guy right so this
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is the intersection between what and
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what uh
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nichas three and two three and two thank
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you three and two right so we're going
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to go here and we're going to proceed
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the same way so we're going to replace
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we have this in terms of y 3 and two so
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this is 3 and
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two so we're going to go uh excuse me Mr
9:10
salami yes hi can I have Stephanie KX to
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the front office for early dismissal yes
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than
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you so we're going to have this here
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right and then we going to three and two
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= -1 4x + 6 right and then G the same
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way we did the last one we going to
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times it by what
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Jack I by what four right four wait Mr
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yes is it 112 -
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7240 oh yeah oh no 50 oh yeah yeah
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40 extra credit I no extra credit that's
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just
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simplea I'm allowed to make mistakes
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like that that's fine right so- 15 x -
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32
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=
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24 you do I I let you get away with it
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every time so
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now you always take off points I take
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off
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some right so 14x equal to 24 + 32 right
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-4x is still the same problem
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yeah so xal to what what's 14 56 over4
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is
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44 right now we have X we just need to
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find y I can plug it in here right
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because it's three and two so I'm going
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to go y = 1 4 * -4 +
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6 problem enti page
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1 + 6 y = 7 so my this vertices is going
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to be -4 and seven can you just count it
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why do we need to know this count can't
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you just go by the dot the lines you can
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use the graph but sometimes like here
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how would you figure it out it's hard to
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find right it's extremely hard to find
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so you have to solve it
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gra go over n on the homework can we use
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the graph or we have to do all this um
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all I'm saying is this you can use the
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graph but if the graph happens to be
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inaccurate now you have problems so it's
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better to know how to do it as rightly
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right so now we have three vertices now
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suppose this problem called for
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maximizing a region you could plug them
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in to see which one gives you the
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highest number yes what's the real life
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example for this construction excuse me
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real life examp construction this is
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what I just said to you right say your
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business you want to see what which one
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of these will maximize your Revenue
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instru you could plug all three in and
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you have what you call it's called
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linear uh linear uh optimization why
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don't we just take the area of each area
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you can't do it like that you have all
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these things here so you have to figure
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out which one least is going to give you
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the highest profit or the least loss so
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you use this every time that's what
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businesses use trust me it's not stupid
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I know guess what that's because you are
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a consumer and you don't know what they
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do to you because they use this I'd
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rather I'd rather yeah You' rather pay
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the money say
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over no that's what they do
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right
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so write this down okay it's an entire
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now yeah I know and I'm going to give
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you some work to do now in class holy