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How To Solve Compound Inequalities & Graph The Solution
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Feb 19, 2025
Welcome back to today's session Algebra 2, in our last lesson, we learned how to solve one step and multi step inequalities. Today we'll go a step further and learn how to solve compound inequalities involving the words "and" and "or". Chapters 00:00 Introduction 00:44 Defining Compound Inequalities 01:44 "AND" Compound Inequalities 04:45 Graphing "and" compound inequalities 07:13 "OR" Compound Inequalities 09:28 Graphing "or" compound inequalities
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0:00
So today we're going to talk about
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solving
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compound inequalities and absolute value
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right as you can see yes what does it
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say under imp I'll get to that I'll get
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to that in a minute right so we're going
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to talk about compound inequalities
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we've already talked about regular
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inequalities didn't we right we talked
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about less than more than we learn
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graphic can we do 18 together I don't
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know no not right now after fin this we
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going to talk about me sir I can't see
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what's behind you I'm getting today
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you're not supposed to be like right
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right now you got to wait give me a
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second right first you need to
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understand the definition right what's
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the definition so when you hear the word
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compound what do you think about
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compound not just two compound is what
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two things two more things put together
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together right it's more complex first
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one we just did Regular and then now the
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compound inequality basically consist of
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two inequalities joined together by the
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word and or by the word or so when you
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have two equations that are
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um that are joined together by those two
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words you have what you call a compound
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inequality all right now to solve this
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you must solve each part of the
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inequality you can't just solve one and
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be like I'm done no you got to do both
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of them right so that's what we're going
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to learn this chapter and then we're
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going to move on to absolute values
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which are again compound inequalities
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but for the first thing we're going to
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learn how to do this kind
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now the first example is going to deal
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with the end word okay A and D end okay
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so end implies intersection is that what
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you were looking for yeah right uh
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Stephanie is that the way you looking
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for intersection this where where were
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you solve a
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example two what's behind you oh we
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going to talk yeah I get to that so now
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when you hear end this type of
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inequalities you are thinking
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intersection what is intersection
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intersection implies that both
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inequalities must be two at the same
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time not just one both of them have to
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be true at the same time does that make
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sense you can't have one to be true and
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the other one to be known true if that
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happens you have what you call empty set
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no Solutions does that make sense
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both of them have to be true at the same
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time how is it true we're going to see
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that right so basically if I can't say
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that okay she is a girl and at the same
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time she is a man that's not
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true I can say that she is a girl and
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she is 15 that's the true statement
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right you can't have two things that
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like one is untrue and one is true that
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does not make a compound inequality that
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is false when you use the word end it
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has to be either or to for that to make
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sense right for so when you use the word
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end the inequalities have to be true
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both of them have to be true okay so our
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first example is this I have two
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statement that are made here right X is
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what x is greater than or equal to
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ne4 and then I I just say or I say and X
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is also less than three right X has to
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be at the same time more than -4 and X
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has to be less than three if I choose an
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example let's say I
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choose -5 is that true
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for5 no why wait which one X I want X to
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be more than or equal to4 and X to be
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less than three at the same time it's
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wrong it's is not greater and then you
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changing no no I'm not changing anything
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I'm just choosing a random number conf
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right I'm just choosing a random number
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to see that that is that number going to
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satisfy this equation this inequality
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what is what are the two inequalities
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say it again what are the two
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inequalities X is more than equal to4
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and X is less than three these are two
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inequalities right and those two
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inequalities are going to constitute a
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solution we're going to find that
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solution but the thing is I'm giving you
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an example I'm saying if I choose X to
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be5 is this going to satisfy this
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condition Co you think it's going to no
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because 5 is no more than equal to yes
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sir
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implies implies
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intersection intersection right so now
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how do you graph this solution so this
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is what we're going to learn how to do
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here right uh if x is more than or equal
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to4 what does that mean sh that means
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I'm going from what here to what three
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no not to just three I just want to plot
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this first okay4 to all the way to
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Infinity right so which I did I put the
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red one there on purpose because I
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wanted to differentiate between the blue
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that's my red marker
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don't then you speak pink yeah you have
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my pink one you I need my pink here can
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I borrow you for a minute uhhuh thank
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you yeah all right so I'm going to do it
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here again right I want X to be more
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than or equal to -4 so I'm going to
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start here is it going to be open circle
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or closed Circle close close close and
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I'm going to go to what Infinity right
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and then now I also want X to be less
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than three I also want that to happen
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right so if three is here where am I
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going am I open or close Circle and then
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where am I going all the way here right
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I'm supposed to work there but for this
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to be a solution I have to be between
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what4 and three that's where the
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intersection is they're intersecting
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somewhere they're intersecting here and
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they also inter intersecting here so my
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solution is -4 to 3 and this is why this
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is aimed because both conditions have to
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be satisfied simultaneously does that
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make sense now you see why do you have
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to have a bracket and a okay I have a
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bracket here because this is closed okay
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so the bracket means that4 is included
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the parenthesis means four and three is
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not included that make sense yeah all
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right yeah so if you gave us this
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problem our answer would be 43 we
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wouldn't have to draw a number line or
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anything I mean if you can do without
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number line more part to you I like to
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use the number line like I want to
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visualize what I'm doing you know like
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some for some of us it's like okay I
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can't figure out what is the solution
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but when you draw the number line it
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clearly shows you you can visualize it
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because I know that they're intersecting
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here and intersecting here outside of
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here they're not intersecting so I want
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to be between -4 and three right and
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then the key word is what how do I know
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hey thank you very much and is a key
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word this is a key word if you don't see
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the this word you cannot use the
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intersection so that takes me to the
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next one and or or and and the next word
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is what or or now or in this case it
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implies what a union right you're
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uniting uniting okay so with this type
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of
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inequality if it's it doesn't have to be
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true for both of them does that make
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sense Patrick
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for or I
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wasn't all right so I'm saying for what
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this one is clear for all for this one
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for these type of inequalities when you
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use the word when I use the word or it
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implies a union now what is the
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difference between union and
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intersection how would you like in real
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life how would you describe the distance
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between an intersection and a union is
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it the same thing what's a union a un
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you take one you take another they come
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together right intersection is where
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they meet what do they have in common is
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intersection a union is you take one you
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take another and you want to put them
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together you want to unite them right
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here with this type of inequality it is
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true if one or more of the inequalities
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is true so they both don't have to be
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true at the same time right as long as
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one is true you're good to go now let's
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try to solve this for this one using the
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word or and I show you exactly how to do
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it okay so I have two inequalities here
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I have x + 6 is less than -4 and I also
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have 3K is greater than or equal to 12
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so let me try to solve this so
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um Jack how do I solve
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here - 6 - 6 right so X is going to be
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less than what -10 and then make sure
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you don't forget this keyword right or
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in this case what would I
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do is X is not supposed to be K I don't
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know why I put K there yeah divide both
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sides by three three right so X has to
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be more than equal to four right so now
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if I want to draw the solution here I
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have to always look at my keyword I did
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not say n here I said what or so now I'm
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going to draw my line again and I'm
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going to put my 10 so - 10 somewhere
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here and then four is going to be here
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right so I I want X to be what less than
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-10 so is going to be open circle or
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closed Circle open open and I'm going
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where left left right here I want X to
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be more than or equal to four closed and
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I'm going where here so my solution is
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what negative Infinity to what -10
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right comma or if you want to say
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Union
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four to positive Infinity does that make
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sense does that make sense to everybody
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all right sort of where where's
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confusing I want X to be
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what why do you have to write both it's
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problem because it's both is or either
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one it's not just one solution is X has
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to be either less than -10 which is from
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-10 all the way here or X has to be more
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than or equal four it's four here so I
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have to have both wait we don't have
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homework today right not yet I'm still
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processing that
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information now let me ask you guys the
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question right P what if I change
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here what if I change it into n what do
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you think the solution would be if you
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were to let's say we changed
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it right do you think you will have a
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solution for this why no why because
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they have to meet in the middle they
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don't and they don't meet anywhere no
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you see what I'm saying did you see that
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they don't meet anywhere good because if
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you this one is going here this one is
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going here there's no way to meet so
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what would be the solution then no
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solution no solution you can put the
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empty set or say no solution right and
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last you no pollution you were there you
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use that no pollution right no solution
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so there's no solution because there's
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no intersection so if I was to use the
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word end that would be no solution does
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that make sense right and here here this
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is we use or so the solution is this to
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this and that to that so I just wanted
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to make the distinction between the two
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I think I want to stop here for for now
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I don't want to go to absolute value
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we're going to do that maybe tomorrow
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because I don't want to put too much in
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your head today I think we need to stop
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here and then do some more practice work
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and then tomorrow we can go to Absol
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value what do you all think unless you
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guys want to I think that's good you
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want to move on to the next thing or no
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no no all right so let's let's do that