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How To Solve Linear Inequalities with Two variables & Graph Them
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Feb 19, 2025
We'll be discussing how to solve linear inequalities with two variables and graph them. This is an important topic for high school students to understand, as it will be useful in a variety of math courses. We'll go over the steps needed to solve these equations and provide some helpful tips along the way. By the end of this video , you should have a clear understanding of how to tackle these types of problems. Let's get started!
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0:06
all right so that in chapter two we
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learn how to just grab regular functions
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so basically to graph the three presets
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in regular linear equations okay
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you're gonna treat this
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as if you had something
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two eggs
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plus y equals four right so treat this
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like this like a regular equation okay
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so if you have two X Plus y equals four
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and then you have to graph it how would
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you do it you're going to put it in
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terms of what y m x plus b we've been
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doing that so you guys know the song
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right now right so I'm going to subtract
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one yeah 2X
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so I get y equals what
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negative two negative two X plus plus
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four so that's my y-intercept that's my
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slope right
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this muscle
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and then this is my why
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is
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right so now this is the most important
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portion of this thing once you graph the
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function
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now we need to learn how to shade a
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region this is our without what we're
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talking about R1 and R2 yeah oh I got it
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now
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oh smarty I guess you're
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dirty all right so come on yeah right
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yeah we did wait
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[Music]
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the first step is we're gonna graph this
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equation regularly and then try to shade
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the solution okay so negative 2x plus 4
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I need to find my what my Y intercept
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okay so one two three four
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so now I'm going I'm gonna go down two
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units and to the right
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one unit so
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one two and then one
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now this is where things get really
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crazy so we have two regions right we
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call this region one and we're gonna
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call this region two okay now one of
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this region is the solution to the
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system but before I can go further
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there's one thing that we need to keep
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in mind
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because this is less than or equal you
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have a solid line okay so this was like
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less than four you'd be a dash five
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you'll be like this
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right that's what you have that means
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that this slide itself is not included
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but because this is less than or equal
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we're gonna have a solid line just like
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what we did before when it's a less than
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or equal when it's like a little line
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yeah
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it's an equal line then it's solid when
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you say equal line when it's less than
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or equal or more than or equal you have
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a solid line
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and when there's like less than or more
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than it's a dashed slide any questions
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on that no no or no
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food line and when it's just less than
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strictly less it's a dashed line okay
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dashed potato
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that when he's less than
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I don't know
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or more
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okay and then you have a solid line
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one is less than or equal
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or more than
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or equal
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all right now the question is this now
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why don't you wait till this because
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this is a crucial portion of it yeah um
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since you're adding oh oh actually never
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mind I'm sorry okay now I want you guys
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to watch what's gonna happen next okay
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now we have two sides okay this is like
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a fence this is like a fence now you
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have two Porsches you have portion one
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and Portion two now which one of these
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region is going to be the solution to
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the system and this is how we figure it
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out okay I'll show you the technique
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here real quick
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so I'm gonna take
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I'm gonna pick it like a random point on
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the on the line okay and then the best
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one to use is zero zero okay
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it's right here
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now zero zero velocity region two or
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region one
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region two right region one is here
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trying to division two so zero zero
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belongs to region two so what we're
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going to do next is this we're going to
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replace both X and Y by zero and see if
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this statement holds now if he holds
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this region will be the solution if it
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doesn't hold then we reject this we're
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gonna shade this solution this this
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other region okay so here's what I'm
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going to do I'm gonna replace
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foreign
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X and Y by zero zero this is my test
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point so 2 times 0 is what zero right
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and zero plus zero is zero so
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zero is less than or equal to four is it
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true or false
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is zero less than or equal to four two
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or four it's a true statement true
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statement right
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so since this is a true statement my
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solution will be here and you shade this
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region
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so we've got to find we gotta converted
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to y equals that we're supposed to be
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yeah from there graph it and then look
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at the given like coordinates and figure
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out if it's in the first or second one
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sure that's it that's what you're doing
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right now if I have changes right if I
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had this
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future okay let's let's do something
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else if I have two X Plus Y is greater
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than four you do the same thing right
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let me erase this real quick would it be
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a dash line you get Dodge line you sound
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good yeah that's that's awesome yeah
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last night because it's modern right
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and then one now we have we still have
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two regions we have region one and
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region two so I'm gonna do the same
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process I'm gonna choose the best point
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to use the zero zero most for the most
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part you always want to use zero zero
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that's what I've learned when I was in
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like high school I'll run zero zero is
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the best one to use yes
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the other side so we're trying to do
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this right now okay let's say now
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I changed my inequality now I have two X
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Plus Y is greater than four I'm going to
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do the same thing okay I'm going to
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refresh X and Y by zero zero so I'm
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gonna have two
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foreign
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just one equation that's when we have
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systems in this case it's just one
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equation yes and we like if we do a
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question are we dividend coordinates of
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ethograph or do we have to like because
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you said take zero zero zero no use zero
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zero for the most part
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okay I'm not gonna give you the
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coordinate you're just gonna have to use
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some test point because you have a
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region you have two regions so that's
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what I'm saying the best thing to do is
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to always use zero zero that would be
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the best one to use
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you could have used one and zero you
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could have used anything that's right
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okay you could use as long as you choose
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a point in one of the regions and test
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it does that make sense so I can take
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like six and six into yeah yeah let's do
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that let's do six and six so let's put
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six and six here right so if you put six
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and six
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let's see if this is true right two and
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six is what 12 right yeah
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so we'll be 12 plus 6.
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so 12 plus 6 is 18. it's 18 more than
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four yes or no true right six and six is
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where let's go one two three four five
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six one two three four five so six and
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six is here you see
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you see
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um well it's right here and now it's a
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solution so yes six and six will work
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because it's in the right region if you
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shake this portion so you don't have to
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necessarily use zero and zero you could
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use whatever you want well zero zero is
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a lot easier it's probably easier to use
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because it's like you know it's gonna
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just cancel out but you don't have to
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necessarily choose zero and zero does
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that make sense no any questions
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yeah well that's what we just did you
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choose a point right on each week in
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each one of the region and then you test
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let's work on another problem and we're
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going to do the same thing so let's work
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on another one
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all right
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are you gonna tell me what region you're
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gonna shade so let's say I have
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3x right
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negative 3X
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3X minus 2y is less than four right
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random problem 3x minus two y the first
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thing is what put this in terms of opens
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up all right so what's that going to be
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minus 3x right
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so that cancels out so we have negative
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2i is less than four negative 3x always
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make sure you put this first all right
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and then divide by what
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negative two now be careful when we
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divide by negative to what happens it is
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right
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it won't really hurt us but that's fine
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we just do it for the sake of keeping
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our integrity so that becomes three
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minus two so that's the equation that we
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have to deal with right do I get it yeah
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yeah you good all right so now let's go
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and graph it
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so what's my wire intercept
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it's negative two
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the slope is three halves so I'm gonna
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go one two three and into the right two
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unit right one two wait will it be
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negative three
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up to two negatives what happened uh
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right
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so is it gonna be a solid line or a dash
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line
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what kind of line is it is
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right yeah
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we have two regions we have region one
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and then region two right they are
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divided by this border now which one is
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the solution sometimes pick a random
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questions on each of the flames right
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pick a random Point give me a point with
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like two points
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two three all right two three
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so two three is here right so two
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uh one two three so one two three so two
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somewhere here two three right now we're
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gonna see if this is gonna hold we're
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gonna replace yes okay so you go
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through the wires up
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you go down to then you go back up
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through the right two that's about three
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or two yeah put it on one except for the
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slope I mean yes
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more than we're not equal
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they have to be more than or equal
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when you have a line under that's what
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you saw it when you don't have any line
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it's uh docked okay it doesn't matter
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what it's less or more as long as you
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don't have at least under that it's a
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solid line
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okay now you chose two and three so
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let's plug it in here this is our
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initial found three x minus two Y is
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less than four
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I'm going to press X by 2 right
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and I work this y by three okay and then
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we're gonna verify these six
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three and two is six right and then six
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six between six minus six
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minus six and zero is zero less than
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four yes or no yes true so since this is
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true
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is here I'm gonna say
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situation because this is true
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of course if a student
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does that make sense okay that's it any
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questions
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that's it all right that's simple so all
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you have to do just load up the graph
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the functions decide whether this is a
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solid line or a dashed line and then
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choose your point at random point I like
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to choose zero zero because it's easier
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and then you just boom get out of here
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any questions no no all right so that's
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that's good so that's a new chapter and
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then remember next week we're going to
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talk about we're going
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