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How To Solve Compound Absolute Value Inequalities
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Feb 19, 2025
In the second part of our lecture on compound inequalities, we'll discuss how to solve absolute value compound inequalities. Chapters: 00:00 Introduction 00:36 "Less Than" Compound Absolute Value Inequalities 02:30 " More Than" Compound Absolute Value Inequalities 03:10 Working on Examples 05:03 Multi Step Absolute Value Inequalities 06:28 Conclusion
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0:01
all right so
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basically
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basically we're talking about absolute
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value inequalities right yesterday we
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talked about two days ago we talked
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about what the compound inequalities
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where we had to split them and I wanted
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to conjoin these and the other one but I
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figured it would not be a good idea to
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do that together so I wanted to split
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this right so now we're going to talk
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about this but then what we're going to
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do is this we're going to have two two
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separate cases and then we going to go
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to the multi-step absolute value so the
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first case is when we use less than
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right less than so less than so that
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means I have here absolute value of x is
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less than or equal to
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5 so how do I solve this problem so we
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talked about how absolute value is
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actually the distance between the number
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and the given center right in this case
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the center is five so basically I'm
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saying here that I want the distance
1:01
between X and five to be less than what
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less than or equal to five right but
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because it's absolute value you have to
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go to the left and to the right because
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we have two cases does that make sense
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so far no yes who says no why does it
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have
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to because it's absolute value right is
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actual value of x I don't know the value
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of x x can be either what5 or or
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negative so that can go either way so as
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long as the distance is less than five
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right so now I want you all to think
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about what we talked about two days ago
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right so if x is in between here what am
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I referring to is that imply or or does
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it imply
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end end right because X is what in
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between -5 and what five so therefore to
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solve this problem I want my X to be
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what
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between and 5 that means I want the
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distance between zero between um X and
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this number to be five or less and the
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only way that's going to happen if x is
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in this spacing here does that make
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sense right so this is how you deal with
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this when this is less than we're going
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to come up with a general case where
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every time you have a less than you're
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always going to end up with these
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cases all right does that make sense now
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we're going to move on to the more
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right so more so in this case I want the
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distance to be more than or equal to
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what five so that means X has to be
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where either
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here or here does that make sense right
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so in this case what is the key word is
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it n or or or or so that's the only two
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cases we have so this is why we needed
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to do the other thing on on on Tuesday
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so that we can build off of that right
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so when you have that to solve it you're
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going to have your two
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cases does that make sense yes all right
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so now let's take some example assume I
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have um I won't do this yet cuz this is
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a little tough right so assume I have
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absolute value of x is less than 7 how
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am I going to solve
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this x is
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and X is L that's right you can either
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write this way or you can say x is more
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than7 and X is less than 7 either one of
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those will be fine you can just do it
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this way and then just save yourself a
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lot of trouble does that Mak sense are
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we good now assume I change
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it now I have absolute value of y is
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greater than 20 how do I solve this
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okay well how would you set this up now
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abute value of y is more than 20 you'd
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have mhm um Y is less
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than y is less than uhuh less than
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20 oh five 20 oh negative 20 or or Y is
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less or greater than 20 20 does that
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make sense that's all you need to worry
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about right so once you have this just
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know you always going to have the same
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cases every time I know you guys love so
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is this confusing or was I hyping it a
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little too
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much it's not confusing right no all
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right good cuz last year when I did it
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do you know I'm like whatever so so I
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was thinking okay maybe you know maybe
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you learn you go and you learn cuz I
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guess maybe this is different maybe your
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brains are a little bit more stand
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there all now that we know that right so
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how would you set this up it's the same
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way how would you set this
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up uh Jack knowing that how would you
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set this up here it's the same
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principle uh 6x - 5 is less than 13 and
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then 6x - 5
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is-3 that's it you see that's it and now
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we can solve this right does that make
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sense
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um Gavin so what if I had 6x - 5 greater
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than or equal to 13 how would you set
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that up before you solve it
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um you just do the same thing but the
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sides so how would you do
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it 6x - 5 oh if you do that's this year
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remember when you have a less than it
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implies
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what a right we have a more than it
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implies what or so we going to have what
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6x - 5 is less than what -3 or 6x - 5 is
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greater than 13 does that make sense so
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if you can retain these if you can
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retain this information and that you
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will be fine right so let's just go
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let's say if absolute value of x is less
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than a in general X has to be between a
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and a and if absolute value of x is more
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than a then in general X has to be less
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than a or X has to be more than a and
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then you can go ahead and solve it yes
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sir I'm just pointing out you put less
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than instead of less than or equal to oh
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yeah yeah thank you for cing
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that he get oh on a second that's not
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that's not an error that's just an
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Omission that's an omission
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no
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G5 that's an Omission that's an
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Omission that's Omission on accident you
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omitted it by mistake yeah accident it
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happens to every one of
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us I never claim I was perfect he that
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so you guys know what to do here right
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so now this is what we're going to work
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on so with that say I want to go to the
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book and then work on some problems so
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if you open your book and if you go on
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page 43 right if you go on page 43
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there's a table there that gives you how
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to set it up if you if you forgot what I
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said here you can go back to your book
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on page 43 and there's a way to remember
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that right now if I'm if I'm generous on
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the test I might give you that right but
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I don't know how I'm going to feel that
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day page 43 if you look on page 43 at
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the bottom you're going to see a table
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that that tells you exactly how to set
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them up
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please at least LEL give us the forance
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but the honor she wouldn't
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give so at least all level
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get
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so I want you all to solve
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this uh I'll
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oh
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no go ahead and solve these two all
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right go ahead and solve these two