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How To Solve Multi Step Equations & Inequalities part 2 ( Null Set & Identities)
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
0:08
inequalities I Spar you yesterday
0:10
because we did not do this kind okay so
0:14
now we have we are going to learn how to
0:16
solve a multi-step inequality and I'm
0:19
expecting everybody to take notes okay
0:22
because that's really important that you
0:24
take notes all right because this
0:27
chapter requires a lot of brain power to
0:31
understand okay now we have 5x - 8 is
0:36
greater than or equal to 4 and then x -
0:39
3 we want to solve this okay we want to
0:42
solve this we want to solve this
0:43
inequality so our first step is always
0:46
to Use the distributive law to blow out
0:49
those parentheses right we have to blow
0:51
out the parentheses we have to open them
0:53
up so I'm going to go 5x - 8 is greater
0:58
than or equal to 4 * X right which is 4
1:03
xus I keep my sign in the middle - 4 * 3
1:09
right we have this here so it's 4 * X
1:13
and you keep your sign minus
1:16
4
1:18
* 3 all right so that gives us 5x - 8 is
1:24
greater than or equal to 4x - 12 all
1:27
right this is what we have right now now
1:29
what do I do from
1:31
here our goal is to always keep the
1:34
variables we trying to keep them on the
1:35
left side so what would you do here I
1:37
would subtract 4X you're going to
1:39
subtract 4X that's perfect right 4X and
1:43
then what we get
1:46
now
1:48
X right minus what 8 is great or equal
1:53
to -12 and then
1:55
what then you want to get the variable
1:57
by itself so you would add 18 you that's
2:00
she as you talk through a problem is
2:02
easier to solve yes we subtract eight we
2:05
don't subtract it
2:07
we add it why why do we add instead of
2:10
subtracting it's negative negative
2:12
that's right right so as you talk to a
2:15
problem it's easier to solve it all
2:17
right so that gives
2:18
us hey what are you doing taking not
2:22
you're supposed to take notes I am
2:24
taking no that's no no that you're
2:25
drawing Dash lines that's that's not
2:27
what I asked for see how it's not so X
2:30
is -2 + 8 is X is greater than or equal
2:33
to4 right so that's what we have here
2:36
all right so what I want you to do is I
2:38
want you to try 2 y + 5 is greater than
2:41
or equal to 3 y - 6 and then we also
2:44
going to try this one all right we have
2:46
a negative I made it complicated because
2:49
the negatives are what tend to be a
2:51
little bit problematic right so we going
2:54
to try this one you guys going to try
2:55
that and tell me what you got you got
2:57
about 2 minutes to get this done try
2:58
both or just one of them try the first
3:01
one and then we're going to do the
3:02
second one as well okay so do the first
3:05
one why do you put us
3:09
through thank you what if I don't like
3:11
his job well guess what what's the word
3:15
do it he's going to do it that's
3:18
right sing her little
3:21
song Nobody
3:24
Like that's
3:27
[Music]
3:28
okay for
4:16
that can't happen you're smart can't do
4:25
this
4:27
done all right Nora can you walk us
4:29
through
4:30
please why me I can do
4:34
it cuz I chose
4:37
you why not you I want you do the
4:40
distributive
4:42
property go by what do you get uh 2 y +
4:48
5 is greater than or = to 3 y - 18 mhm
4:54
um then
4:56
you I don't even know at 18 18
5:03
good so now what do we have now we have
5:06
2 y + 23 is greater than or equal to 3 y
5:09
all right and um you subtract three two
5:15
y all
5:18
right you get 23 is greater than or
5:21
equal to y That's it there you go right
5:25
23 is greater equal to y or if you want
5:28
to say you want you can say Y is less
5:30
than or equal to 23 either way it's fine
5:33
right 23 is greater than or equal to y
5:36
or Y you can also say or Y is less than
5:39
or equal to 23 is the same thing you're
5:41
saying the same thing right okay good
5:44
all right I did differently I think I
5:48
did a different order where I had to
5:50
divide by at the end yeah and then you
5:52
get so yeah you can do that too cuz you
5:55
you you you kept your variable on the
5:57
left so if you did that then you're
5:58
going to get a negative right yeah good
6:01
all right so now the next
6:03
one next one you have -2 and then x +
6:07
one right is greater than -16 + 5x let's
6:11
do
6:15
it why do we have to have math in the
6:17
morning because your brain is at best
6:20
capacity in the morning like Optimal
6:23
that is not true I'm still asleep it's
6:26
the
6:27
opposite in the morning that's when you
6:29
are freshest all right you're the
6:31
freshest in the
6:33
morning that's
6:37
awesome I got fres right
6:40
now French Prince of
6:43
belir what will fres in the morning
6:46
right after my shower then it all goes
6:48
downhill
6:48
from you take a shower in the morning
6:52
yes got my ears pierced because I have
6:55
to wash them every day shower so I'm
6:58
taking it all right let's work we talk
7:00
about other things extra Cal activities
7:03
later on so right now we doing the
7:10
work all right the candidate now is
7:13
Kenzie you going to walk me through it
7:14
right see I'm as lost as no you're not
7:20
lost why am I all right so tell me what
7:23
you've done so far tell me what you've
7:25
done
7:26
uhhuh so um
7:30
I wrote it down
7:32
uhuh step by step yeah I
7:36
wrot yeah2 * X
7:41
okay uh you do that also one so and what
7:46
did you
7:47
get2 -2 good you see that good all right
7:51
and
7:53
[Music]
7:55
then I didn't I haven't done anything
7:57
else yet all right so why would you what
7:59
would you what would be your next
8:01
logical
8:03
step um mhm cancel out something cancel
8:07
out right so pick what you want to
8:08
cancel out right now plus 16 plus 16
8:12
that's good plus 16 all
8:15
right and then this goes away this goes
8:18
away right and then what's -2 +
8:20
16 18 or 18 14 right 14 so yeah -2 2x +
8:26
14 is greater than 5x then what would
8:29
you do
8:34
here you do it again you do what again
8:37
canulation yeah what what you want to
8:40
combine like terms right if you do 14
8:43
you want to you want to keep all the
8:45
variables on one side right so the
8:48
number is here you want to move these
8:49
guys so how would you move it flip them
8:52
why would you flip it how do you I have
8:54
-2X so what do I
8:56
do I can't see anything over there right
8:59
so what is it it's
9:01
2x my God so you add it right you're GNA
9:04
add it because you want to keep this on
9:06
the side right you're going to add it so
9:10
you got that
9:13
and quiet quiet
9:16
quiet so we have what
9:19
14 right 14 is greater than 7 x and you
9:24
divide seven by seven you divide and now
9:26
you have two
9:30
or if you
9:33
want you also done it the other way you
9:36
could have done it the other way
9:40
yeah div by negative only by negative
9:46
right okay so now now we have this now
9:51
we're going to talk about we're going to
9:53
finishes talk about the N the null set
9:56
and identities that's important right
9:59
so I hey
10:03
stop we're going to talk about the N set
10:06
and identities here all right so the the
10:08
null set this is called the null set and
10:11
this are identity null why are you
10:13
making me do this okay so the first
10:15
thing you want to do is we want to try
10:16
to solve this problem and then we see
10:19
what happen here so let's do it together
10:20
right so if I have this here 3 y - 5 +
10:23
25 I'm going to go 3 * Y is 3 y right
10:27
keep my negative in the middle minus 3 *
10:29
* 5 which is 15 + 25 = 3 y + 10 right
10:35
now watch what happens I can combine
10:38
these two right -5 + 25 what is that
10:41
what is that
10:42
equal -5 + 25 10 so I have 3 y +
10:47
10 = 3 y + 10 right now what do you see
10:51
that's interesting here both 3 y + 10
10:56
how often is that going to be how often
10:58
is 3 y + 10 = 3 y + 10 not very always
11:04
right always this is always going to be
11:06
true 3 y + 10 is always going to be
11:08
equal to 3 y + 10 is it going to change
11:10
at any given time never no matter what
11:13
number you put here this is always going
11:16
to be the same right if y takes any
11:19
value 3 y + 10 is always going to be
11:21
equal to 3 y + 10 does that make sense
11:24
because you have the same expression on
11:26
either side of the equation right so
11:28
this is called an ident
11:30
this is color identity so the solution
11:32
is what infinite
11:34
numbers CU this is always going to be
11:37
true
11:39
always
11:42
going to be
11:45
true no matter what right if you put in
11:48
any number for y this is always going to
11:51
hold is he ever going to change never if
11:54
you replace y by 1 3 + 10 is is going to
11:58
be equal to 3 + 10 if you replace y by 2
12:01
3 * 2 is 6 16 is going to be equal to 16
12:04
if you repl y by 3 19 is going to be
12:06
equal to 19 all so this is a NE number
12:09
of solutions yeah would be the same if
12:13
you I was going to ask repl it with
12:16
netive yeah still going to be the same
12:19
because 3 y y is y 3 y + is always going
12:22
to Beal 3 y+ so therefore all is going
12:24
to be true the solution is an infinite
12:27
number of solution infinite right now
12:29
we're going to see the second type of
12:32
solution which is different this is
12:34
called a uh null set null set all right
12:40
we're going to see why we call it a null
12:42
set it's going to be
12:45
beautiful math is not beautiful math is
12:48
beautiful you are breaking my heart
12:49
unless I understand it it's
12:52
ugly y are breaking my heart right now
12:55
it's okay everybody goes through heart
12:57
this is breaking my brain so how am I
12:59
love it all right now watch here right
13:01
so we have -5x - 14 = 2 * 2x + 3 - 9 is
13:08
let's see what happened here right so
13:10
I'm going to go I'm going to keep this
13:13
here because that doesn't change right
13:15
I'm going to distribute this 2 * 2x is
13:18
4X 4 plus 2 * 3 that is 6 right - 9x now
13:25
before I solve it I'm going to combine
13:28
the like hey
13:30
that's enough I'm say again all right
13:33
start so 4x - 9x right they're on the
13:38
same side so I don't I'm just going to
13:40
do them together right so this stay the
13:42
same so now
13:44
4X right - 9x that is equal to what -5x
13:50
right so -5x +
13:52
6 now watch what happened I want to keep
13:55
my numbers together and my variables on
13:58
the same side right so let me start with
14:00
the numbers so I'm going to do I'm going
14:02
to add what 14 right I'm going to add 14
14:08
here so I get
14:11
-5x is = to
14:13
-5x + 20 right that's what I have right
14:18
now and then I'm going to add
14:22
what I'm going to add 5x not 5x this 5
14:26
I'm going to add 5x right I want to
14:28
cancel
14:31
now watch what is 5x +
14:34
5x zero right it's zero so 0 is equal to
14:38
20 what what do you think the solution
14:40
is 20 0 is equal to
14:44
20 I see positive faces subtract 20 0 is
14:49
equal to 20 listen to my statement 0 is
14:53
equal to 20 true not true that's
14:57
impossible can 0 be equal to 20 not true
15:00
right so therefore what what do we say
15:03
no
15:04
what no no solution no solution this is
15:09
impossible imposs this is impossible 0
15:12
cannot be equal to 20 right that is
15:16
impossible so this is called a n that's
15:18
n set and this this is a
15:21
sign all right sorry so the non set
15:25
right the non set the solution for the
15:27
non set is what this
15:30
if you this is the set this is equal to
15:32
the nor whenever you see this if you see
15:35
this that means there's no
15:37
solution right it's almost like a assign
15:40
language in mathematic if you see that
15:42
zero with a slash in it that's no
15:44
solution that means there's no solution
15:47
Nonet all right Nonet it's no solution
15:50
anytime you get you say oh no solution
15:52
you see this there no
15:54
solution okay so that is a no solution
15:57
set okay so so now we we've we've pretty
16:01
much uh wrapped up this chapter now the
16:04
rest is what practice right now we've
16:08
learned the material now we have to
16:10
learn to practice what we've learned all
16:12
right so we're going to go
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