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How To Solve Multi Step Equations & Inequalities part 2 ( Null Set & Identities)
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Feb 19, 2025
welcome back to our class, in part 2 of our session on solving multi step equations and inequalities, we'll be discussing solving more complex equations and inequalities. They'll include instances where the equations lead to a null set or an identity. Chapters: 00:00 Introduction 00:21 Solving multi step inequalities 09:58 The Null Set and Identities
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0:00
all right
0:01
so so we're going to continue part two
0:05
on our solving multistep equations and
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inequalities I Spar you yesterday
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because we did not do this kind okay so
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now we have we are going to learn how to
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solve a multi-step inequality and I'm
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expecting everybody to take notes okay
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because that's really important that you
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take notes all right because this
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chapter requires a lot of brain power to
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understand okay now we have 5x - 8 is
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greater than or equal to 4 and then x -
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3 we want to solve this okay we want to
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solve this we want to solve this
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inequality so our first step is always
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to Use the distributive law to blow out
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those parentheses right we have to blow
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out the parentheses we have to open them
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up so I'm going to go 5x - 8 is greater
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than or equal to 4 * X right which is 4
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xus I keep my sign in the middle - 4 * 3
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right we have this here so it's 4 * X
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and you keep your sign minus
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4
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* 3 all right so that gives us 5x - 8 is
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greater than or equal to 4x - 12 all
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right this is what we have right now now
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what do I do from
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here our goal is to always keep the
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variables we trying to keep them on the
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left side so what would you do here I
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would subtract 4X you're going to
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subtract 4X that's perfect right 4X and
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then what we get
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now
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X right minus what 8 is great or equal
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to -12 and then
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what then you want to get the variable
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by itself so you would add 18 you that's
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she as you talk through a problem is
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easier to solve yes we subtract eight we
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don't subtract it
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we add it why why do we add instead of
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subtracting it's negative negative
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that's right right so as you talk to a
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problem it's easier to solve it all
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right so that gives
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us hey what are you doing taking not
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you're supposed to take notes I am
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taking no that's no no that you're
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drawing Dash lines that's that's not
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what I asked for see how it's not so X
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is -2 + 8 is X is greater than or equal
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to4 right so that's what we have here
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all right so what I want you to do is I
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want you to try 2 y + 5 is greater than
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or equal to 3 y - 6 and then we also
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going to try this one all right we have
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a negative I made it complicated because
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the negatives are what tend to be a
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little bit problematic right so we going
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to try this one you guys going to try
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that and tell me what you got you got
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about 2 minutes to get this done try
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both or just one of them try the first
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one and then we're going to do the
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second one as well okay so do the first
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one why do you put us
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through thank you what if I don't like
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his job well guess what what's the word
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do it he's going to do it that's
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right sing her little
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song Nobody
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Like that's
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[Music]
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okay for
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that can't happen you're smart can't do
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this
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done all right Nora can you walk us
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through
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please why me I can do
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it cuz I chose
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you why not you I want you do the
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distributive
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property go by what do you get uh 2 y +
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5 is greater than or = to 3 y - 18 mhm
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um then
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you I don't even know at 18 18
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good so now what do we have now we have
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2 y + 23 is greater than or equal to 3 y
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all right and um you subtract three two
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y all
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right you get 23 is greater than or
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equal to y That's it there you go right
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23 is greater equal to y or if you want
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to say you want you can say Y is less
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than or equal to 23 either way it's fine
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right 23 is greater than or equal to y
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or Y you can also say or Y is less than
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or equal to 23 is the same thing you're
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saying the same thing right okay good
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all right I did differently I think I
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did a different order where I had to
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divide by at the end yeah and then you
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get so yeah you can do that too cuz you
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you you you kept your variable on the
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left so if you did that then you're
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going to get a negative right yeah good
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all right so now the next
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one next one you have -2 and then x +
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one right is greater than -16 + 5x let's
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do
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it why do we have to have math in the
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morning because your brain is at best
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capacity in the morning like Optimal
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that is not true I'm still asleep it's
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the
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opposite in the morning that's when you
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are freshest all right you're the
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freshest in the
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morning that's
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awesome I got fres right
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now French Prince of
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belir what will fres in the morning
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right after my shower then it all goes
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downhill
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from you take a shower in the morning
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yes got my ears pierced because I have
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to wash them every day shower so I'm
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taking it all right let's work we talk
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about other things extra Cal activities
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later on so right now we doing the
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work all right the candidate now is
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Kenzie you going to walk me through it
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right see I'm as lost as no you're not
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lost why am I all right so tell me what
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you've done so far tell me what you've
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done
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uhhuh so um
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I wrote it down
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uhuh step by step yeah I
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wrot yeah2 * X
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okay uh you do that also one so and what
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did you
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get2 -2 good you see that good all right
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and
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[Music]
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then I didn't I haven't done anything
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else yet all right so why would you what
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would you what would be your next
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logical
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step um mhm cancel out something cancel
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out right so pick what you want to
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cancel out right now plus 16 plus 16
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that's good plus 16 all
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right and then this goes away this goes
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away right and then what's -2 +
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16 18 or 18 14 right 14 so yeah -2 2x +
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14 is greater than 5x then what would
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you do
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here you do it again you do what again
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canulation yeah what what you want to
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combine like terms right if you do 14
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you want to you want to keep all the
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variables on one side right so the
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number is here you want to move these
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guys so how would you move it flip them
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why would you flip it how do you I have
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-2X so what do I
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do I can't see anything over there right
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so what is it it's
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2x my God so you add it right you're GNA
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add it because you want to keep this on
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the side right you're going to add it so
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you got that
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and quiet quiet
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quiet so we have what
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14 right 14 is greater than 7 x and you
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divide seven by seven you divide and now
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you have two
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or if you
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want you also done it the other way you
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could have done it the other way
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yeah div by negative only by negative
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right okay so now now we have this now
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we're going to talk about we're going to
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finishes talk about the N the null set
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and identities that's important right
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so I hey
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stop we're going to talk about the N set
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and identities here all right so the the
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null set this is called the null set and
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this are identity null why are you
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making me do this okay so the first
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thing you want to do is we want to try
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to solve this problem and then we see
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what happen here so let's do it together
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right so if I have this here 3 y - 5 +
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25 I'm going to go 3 * Y is 3 y right
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keep my negative in the middle minus 3 *
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* 5 which is 15 + 25 = 3 y + 10 right
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now watch what happens I can combine
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these two right -5 + 25 what is that
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what is that
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equal -5 + 25 10 so I have 3 y +
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10 = 3 y + 10 right now what do you see
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that's interesting here both 3 y + 10
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how often is that going to be how often
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is 3 y + 10 = 3 y + 10 not very always
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right always this is always going to be
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true 3 y + 10 is always going to be
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equal to 3 y + 10 is it going to change
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at any given time never no matter what
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number you put here this is always going
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to be the same right if y takes any
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value 3 y + 10 is always going to be
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equal to 3 y + 10 does that make sense
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because you have the same expression on
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either side of the equation right so
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this is called an ident
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this is color identity so the solution
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is what infinite
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numbers CU this is always going to be
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true
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always
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going to be
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true no matter what right if you put in
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any number for y this is always going to
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hold is he ever going to change never if
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you replace y by 1 3 + 10 is is going to
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be equal to 3 + 10 if you replace y by 2
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3 * 2 is 6 16 is going to be equal to 16
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if you repl y by 3 19 is going to be
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equal to 19 all so this is a NE number
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of solutions yeah would be the same if
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you I was going to ask repl it with
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netive yeah still going to be the same
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because 3 y y is y 3 y + is always going
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to Beal 3 y+ so therefore all is going
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to be true the solution is an infinite
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number of solution infinite right now
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we're going to see the second type of
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solution which is different this is
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called a uh null set null set all right
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we're going to see why we call it a null
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set it's going to be
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beautiful math is not beautiful math is
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beautiful you are breaking my heart
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unless I understand it it's
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ugly y are breaking my heart right now
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it's okay everybody goes through heart
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this is breaking my brain so how am I
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love it all right now watch here right
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so we have -5x - 14 = 2 * 2x + 3 - 9 is
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let's see what happened here right so
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I'm going to go I'm going to keep this
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here because that doesn't change right
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I'm going to distribute this 2 * 2x is
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4X 4 plus 2 * 3 that is 6 right - 9x now
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before I solve it I'm going to combine
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the like hey
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that's enough I'm say again all right
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start so 4x - 9x right they're on the
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same side so I don't I'm just going to
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do them together right so this stay the
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same so now
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4X right - 9x that is equal to what -5x
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right so -5x +
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6 now watch what happened I want to keep
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my numbers together and my variables on
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the same side right so let me start with
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the numbers so I'm going to do I'm going
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to add what 14 right I'm going to add 14
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here so I get
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-5x is = to
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-5x + 20 right that's what I have right
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now and then I'm going to add
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what I'm going to add 5x not 5x this 5
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I'm going to add 5x right I want to
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cancel
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now watch what is 5x +
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5x zero right it's zero so 0 is equal to
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20 what what do you think the solution
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is 20 0 is equal to
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20 I see positive faces subtract 20 0 is
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equal to 20 listen to my statement 0 is
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equal to 20 true not true that's
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impossible can 0 be equal to 20 not true
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right so therefore what what do we say
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no
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what no no solution no solution this is
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impossible imposs this is impossible 0
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cannot be equal to 20 right that is
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impossible so this is called a n that's
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n set and this this is a
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sign all right sorry so the non set
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right the non set the solution for the
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non set is what this
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if you this is the set this is equal to
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the nor whenever you see this if you see
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this that means there's no
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solution right it's almost like a assign
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language in mathematic if you see that
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zero with a slash in it that's no
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solution that means there's no solution
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Nonet all right Nonet it's no solution
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anytime you get you say oh no solution
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you see this there no
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solution okay so that is a no solution
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set okay so so now we we've we've pretty
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much uh wrapped up this chapter now the
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rest is what practice right now we've
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learned the material now we have to
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learn to practice what we've learned all
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right so we're going to go
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