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How To Solve System Of Inequalities Graphically Part 1
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Feb 19, 2025
Section 3.2-Solve System Of Inequalities In Section 3.1, we learned how to solve systems of linear equations graphically and algebraically. Now we're going to move onto section 3.2 and learn how to solve system of inequalities in part 1 of this section. Chapters: 00:00 Introduction 00:53 Why learn this? 04:23 Solving a system with intersecting regions 08:54 Solving a system with separated regions
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0:00
so now we are talking about solving
0:02
system of inequalities right in the last
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chapter we talked about solving system
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of equations and I remember we talked
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about solving
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equations by using what a graph where we
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found the intersection and we talk about
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consistent system independent we also
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talk about inconsistencies and we also
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talk about solving inequalities right no
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no we didn't do inequalities just
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equations and we talk about how to solve
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it using what the substitution method or
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the addition method we talked about that
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before we went on our break so now we're
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going to talk about solving inequalities
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as a system not just two equ two
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equations but two inequalities right so
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how do you solve it then why because I'm
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now anticipating questions like why do
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we ever going to need this I'll tell you
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why before we even start so that way I
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don't get that question again now assume
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this Mr and Mrs Callis right I use your
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last name I don't know why it just came
1:00
M are driving across the country with
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their two kids right and then they plan
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on driving a Max of 10 hours each day
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now Mr khx wants to drive at least 4
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hours a day but not more than 8 hours a
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day and Mrs khx can drive between two
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and 5 hours per day now the the thing is
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is how do you solve this how do you
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solve these things like you want to find
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a order PA that
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satisfy this system right now if you
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were to figure that out how would you do
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you proba can argue with your husband
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like I ain't
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driving you know you can do that or you
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canot say let's try to find a way to
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solve this mathematically right so this
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is why this lecture here comes into play
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do you do that with your wife no I don't
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they pull out the paper I just drive I
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know it's going to solve all problems
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right I'm saying that's the solution I'm
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saying C that don't know how to discuss
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things for you you can solve it
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mathematically but I know how to say
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I'll drive cuz I'm a gentleman so I'll
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do it yes sir he that guys so they want
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to drive 10 hours well they they want to
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drive 10 hours right not more than 10
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hours a day so excuse me can't they both
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just drive 5 hours they can do that that
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seems EAS there's different uh options
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but the thing is this you can solve this
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using inequalities that's the point
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my what my point is this somebody may
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say well I don't want to do five and
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five I don't want to do this so what
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could work for
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he
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says Mr says that I want to drive at
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least four hours a day but not more than
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eight right but she says what she wants
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to do between two and five right so then
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we can solve this system this is called
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this is how you solve system of
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inequalities right now first of all the
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definition right I gave you the reasons
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why we used this now we're going to talk
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about the definition so what is the
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definition what what is the system of
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equalities it means finding an order
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pair that satisfies the conditions right
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solving this is meaning finding an
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ordered prayer that will solve or that
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will satisfy the conditions we can find
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an ordered prayer that will satisfy this
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condition which means okay Mr College
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you can do six your wife can do four Mr
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College you can do seven your wife can
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do three or cuz she doesn't want to do
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more than two and five right so she's in
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this anal here so there's different
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option you can tell her honey I'm tired
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today you can do three I'll do seven or
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you can do both five and five right if
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you really want to be like 5050 because
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women say 5050 right or you can say all
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right I'm going to be nice I do eight
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and then you do two right okay
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so so you get my point there's different
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options right you can have an order that
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will satisfy this because there's
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several options here right and we use
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this system of inequalities to do that
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so that's basically you guys are talking
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and then while you're talking I'm
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sitting there Computing this in my I'm
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thinking all right I can use my
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mathematics knowledge to do this right
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all right so now the first example is
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this we're going to talk about solving
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uh intersecting regions solving uh A
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system that gives us intersecting
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regions right intersecting regions so
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how do I solve this here I have two
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inequalities right the first one says
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what Y is greater than what 2x - 4 and
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the second one says Y is less than or
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equal to what 0.5x + 3 and I want to
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solve this so how do I solve
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this I'm going to graph it and figure
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out the solution right remember before
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we left we learned how to graph what
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inequalities but it was just one
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inequality now you have two of them it's
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not just one you have two inequalities
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that you have to put on the set of axles
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right so now let's go ahead and let's do
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that so let me draw
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my uh set of axes right so my first
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equation Y is more than 2x - 4 I already
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know that my solution is going to be
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what it's a dotted line or a solid line
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here dotted line right so my Y intercept
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is what4 so I'm going to start with4 1 2
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3 4 and I'm going to go up how many unit
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two two and over to the right one so 1
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two one right here right so I'm going to
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my do and then I'm going to have this
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here right so now I need to figure out
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which region is the solution but before
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before I get there I also need to graph
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this guy here okay so that's what we're
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going to do we're going to not graph
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this now 0.5 because this is a decimal
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number I can turn this into a what
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fraction fraction thank you very
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much Nick so this going to be1
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2x + 3 right that's what I have now
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what's my Y
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intercept three right so I'm going to go
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1 2 3 and I'm going to go down what down
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one and then over two right one two
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right down one over two so then I'm
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going to draw this is a solid line now I
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need to figure out what my solution is
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right I have two system of equations so
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again remember what we said we always
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use what zero and zero right and double
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check the statement so I'm going to
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start with my first line here so I have
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0 is less than 2 * 0 - 4 0 is more than4
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two or
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fours true right this is
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true and then 0 0 is here right so
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because this is true am I going to shade
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here I'm I going to shade this side I'm
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shade the side right I'm going to shade
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the 0 side so I'm going to go bing bing
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bing bing bing sh side is true right
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yeah the true side is what you shade
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right for the first line now for the
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second line
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I'm going to do the same thing right I'm
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going to replace this by zero and this
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by zero I get 0 is less than or equal to
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three true or false true true right so
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therefore I'm going to use the pink pen
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the pink marker now so this is 0 0 again
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right I'm going to go boom boom boom
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boom boom boom boom right so now the
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solution is where they both
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intersect let me say that again the
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solution is where they both do what
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intersect what do the inter is it here
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is it here no where is it right here
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because this Closs right you see that's
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where the intersect why you laughing so
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this is the solution the solution is
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right here oh is it where they're shaded
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yeah well they both shaded right they're
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both here I have a green I have a pink
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here so that's my solution so do you
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Circle it I Circle that right I put
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solution and I I put the big Sol Circle
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and put solution right so that's my
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solution this is where want to be
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because that's where both regions
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intersect right let me say that again
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regions it's not just two lines are
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intersecting but region right this
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region and this region so my
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intersection is right here so that's
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where they intersect and this is going
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to be my solution does that make sense
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are we good on that yeah all right okay
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so these are called intersecting regions
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the regions are what overlapping right
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this is where the me so that's the
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intersecting region now the next thing
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we're going to do is we're going to
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we're going to also have another example
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where now we have what you call separate
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regions
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yeah all
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right all right so now we have like
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separate regions here right separate
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regions so we're going to see a case
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where we have separate regions see how
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that look looks like so now I'm going to
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erase this again the same process I have
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my my uh set of axis now here it say Y
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is more than or equal to 1 x + 5 right
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so 1 over 1 so my that's my slope is one
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over one so if I'm at five here right 1
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2 3 4 5 I'm going to go up one and over
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to the right one right again this time
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is going to be a solid line that's my
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first graph right solid line again I
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don't worry about like drawing my my
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solution yet I need to wait and then the
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next graph is uh x - 4 the slope is the
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same again one over one so I'm going to
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start at -4 1 2 3 4 I'm going to go up
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one over to the right one so right
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here what do you already notice here par
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line right something is going to happen
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but I need to I need to double check
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this I'm going to replace this by zero
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and this by zero right so if I do that I
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get 0 is more than or equal to five true
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or false false false right so 0 Z is
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here so where's my solution here or here
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up out up out right here
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right now watch this and now
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here 0 is less than4 true or
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false false false right your here so am
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I Shing here or out here outside
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outside now what do you think the
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solution to this system
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is no solution because there's no
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intersection right these guys way here
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the wife that complains a lot and these
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guys here the husband that say I'm not
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going to talk today I'm going to be
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quiet right so therefore there's no
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intersection they're not overlapping so
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therefore there is
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no
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right solution hey hey we need a real
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world
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example example
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veryy she's out
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here he goes you know what I'm not going
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to talk I'm just going to
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Beav no solution right no solution
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because they don't they don't overlap
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now here we have a solution I can use a
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real example the wife is talking and he
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goes you know what for the sake of just
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being like a gentleman I'm going to
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abide by your law so be call that kind
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of man what a sucker
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cuz he like yes ma'am you know like
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where did you get this examp
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