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How To Graph Linear Inequalities
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Feb 19, 2025
In our last lesson, we learned how to describe transformation of functions and also learned how to graph them. In this current lesson ( section 2.8), we'll now learn how to grab linear inequalities. Chapters: 00:00 Introduction 00:27 Why linear inequalities? 03:37 Step by step instructions on graphing linear inequalities
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okay so we're going to talk about
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graphing
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linear and absolute value inequalities
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but today we're just going to do one
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portion which is just the linear
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equations okay we're not going to talk
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about absolute value till five tomorrow
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or maybe Friday CU I do want to give you
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a chance like absorb it and understand
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it so now why do we use
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graphing uh linear inequalities why is
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this important because I know you guys
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always ask question am I ever going to
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use use this in life sure you will use
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it right example say you want to throw a
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party right for people and then you have
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about $200 that's all you have in your
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budget right and then a large piece that
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cost you about 11 bucks and then soft
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drinks are like 225 per drink right now
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you don't want to spend more than 200
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because that's within your budget now
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you want to be able to have the kind of
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combination that will fall within this
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budget how many drinks and how many
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large pizza can you
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get right with $200 if the pizza cost
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$11 and your drinks cost 225 so this is
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the kind of stuff that we're going to be
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talking about so you can actually use
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mathematics to come up with a
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combination instead of saying well I'm
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going to figure it out but I'm going to
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try to no you say Hey listen when I was
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in a 2 I learned how to do this let me
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do this for you all and they going to
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look at you like
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wow
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amazing there are questions say how many
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people are buying food and drinks for no
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the question is that's how much you have
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and you want to see the question is how
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many pizzas and how many drinks can you
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get $200 right what if you're trying to
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feed like 200 people well obviously
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you're not going to want to Fe 200
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people with $200 that's called what
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common sense right you can feed
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so the equation is like this 11 P right
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Plus
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225 we going to call these drinks we
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want it to be less than or equal to 200
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right we going want to spend 200 so this
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is the equation that we have but then in
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this equation we have what we have
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two
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variables right we have two variables
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and then we want to be able to find the
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kind of combination so how many pizzas
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can I get and how many at the same time
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drinks can I get with
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$200 if each pizza cost me $11 and each
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drink cos me 225 so this why this
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chapter comes in handy so now we're
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going to learn how to do this right so
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this is your real life application
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introduction so that way I don't get the
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same question while I'm teaching am I am
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going to use this in life well yeah you
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are going to use this cuz you now
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have do you normally this you can graph
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it like if you going to do this for like
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an actual th would you normally graph it
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I'll graph it and I for my intersection
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point and I'm going to use that to find
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how many drinks and how many P that I
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can get instead of being like someone
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else going to be like o 200 oh no I want
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use my mathematics I'm going to with
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exact number I would just have somebody
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else do it for me who you I just have
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you money man yes sir uh what's that
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word above within his T within his
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budget combination a combo combo right
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combo all right all right so now that we
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have this formal introduction now we're
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going to learn how to do this right how
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do you
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graph linear inequalities too much Zip
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Zip down there so all right so we're
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going to learn how to graph this here
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right we're going to graph this equation
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Y is greater than -3x - 2 right so I'm
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going to show you the steps to do this
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it's super easy once you get it right so
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step one we're going to graph
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y = -3x - 2 remember we have learn how
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to graph linear equation this is
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probably the easiest function to graph
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right because we know what we know the Y
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intercept and we also know the slope
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right now this line here is going to be
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called we're going to call this line the
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boundary boundary right or the Border
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you want to use border boundary that's
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your problem right and then next
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step
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two we're going to
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find this
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solution
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right by uh graphically by graph or
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graphically if you want to use that word
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graphically that's
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fine graphically okay so now let me go
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ahead and solve this problem for you so
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the first thing we're going to do is
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we're going to graph this equation all
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right somebody help me
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here so what's my slope
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uh3 over one right and then what's my Y
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intercept so I'm going to start with -2
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right so if I'm here I'm going to go up
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how many units I'm going to go down how
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many unit how many unit I'm going to go
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down two two I've already went down two
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I have my y inste so from
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here my slope is what3 so I'm going to
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go down 1 2 3 and over to the right one
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one right so here right so normally I
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will graph this line this way
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right does anybody agree with me on that
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yes or no yes right but now we are not
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just graphing a line we are graphing
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what an inequality this here is just
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this line but what I want to graph is
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what Y is what greater right this
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indicates that this is greater greater
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right this is what we want to graph so
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how do you do this this is where now
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this is a New Concept now the first
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thing is this when whatever your
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inequality sign is either greater or
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less is not a solid L what you have is
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called a dashed line so that means this
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line has to look like this oh right I
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did it on purpose I'm sorry that on P
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right it's a dash line because this is
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more than or less than right if it's
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more than or less than you have a dash
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line because that means the line is not
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part of your solution because it's
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greater it has to be greater right so
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the line is not part anytime you have
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less than or more than alone this is a
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dashed line dashed line and if you have
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more than or equal and less than or
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equal it's called a solid line solid
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line which mean the entire line is okay
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now we're going to figure out what is
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the solution we have two regions I
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already told you that this line is
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called what it's called the boundary of
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the border right this is our boundary so
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now we have two region we're going to
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call this region region one
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and this is region
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two right now which region of this will
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satisfy this condition we don't know yet
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yes sir huh one it'ser than well we
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don't know that yet we have double check
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obviously you can see by observation but
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we don't know we have to use a way to to
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to double check that right so what I do
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is this the easiest way to do this if
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the line is not going through the the
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the point zero here's what I suggest you
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do every time you going to replace both
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Y and X by zero and you're going to see
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if this statement is true right so
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here's what we do so we're going to
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replace this is Step number three
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replace
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both Y and X by zero as long as the line
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is not going through the point zero as
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long as that's not happening you can use
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this every time as long as so I'm going
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to go 0 is greater than -3 * 0 - 2 right
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I'm going to try to see if this
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statement is true or false so 0 this is
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going to cancel out right this is going
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to cancel out because -3 * 0 is 0 so I
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get 0 is more than -2 is this a true
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statement or false statement true right
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true 0 0 is right
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here so since 0 is right here this is
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called and we say that this was true
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Therefore your solution is here
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and this is where you scratch and you
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call this
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solu so this is how you solve this
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problem okay so I'll give you a quick
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recap thanks Mr huh I was just saying
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thank you oh you're welcome I'm
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honed so so first we solve this equation
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Y is greater than -3x - 2 so the first
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thing is we graph y = -3x - 2 just a
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line reg graphing
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and because this is greater than it has
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to be a dash line because that means
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your solution cannot be on a line
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because it's greater now next you
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replace both Y and X by zero to see if
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the statement holds true or false now
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you have two region because this is your
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border line and this is your boundary
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whatever right you have two
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regions this region and this region we
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don't know which one is the solution
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since the line is not going to 0 0 you
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can replace both X and Y by 0 right you
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could have used any other point on this
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region or this region I prefer to use
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because it makes it a lot easier to work
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with so I put it in and I double check
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this 0 is greater than2 since 0 0 is
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here and this is true therefore this is
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my solution does that make sense all
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right so we're going to do a couple more
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problems like this and then we can call
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it a day and I'll give you a break all
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right
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