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How To Solve Inequalities By Adding, Subtracting, Multiplying Or Dividing
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Feb 19, 2025
In our last session, we learned how to write and graph inequalities. In this chapter, we'll learn how to solve them. Chapters: 00:00 Introduction 01:14 Solving inequalities by adding or subtracting 03:40 Solving inequalities involving fractions 09:24 Solving inequalities by adding or multiplying 13:46 Solving inequalities involving negatives
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0:00
all right so now we went from like
0:03
defining inequalities right to now
0:06
solving inequalities right we're going
0:09
to learn how to solve
0:12
them really was ni so I need to
0:17
like
0:19
okay so we're going to learn how to
0:21
solve the inequalities now right so I
0:23
want to ready to take notes and I want
0:25
youbody to pay attention because it's
0:28
important right you better take notes so
0:31
because when it comes like solving
0:33
inequalities when we have multiplication
0:35
and division especially when you have
0:37
negative numbers it becomes a little bit
0:39
complicated so I need you to really take
0:44
notes all right so now how do you solve
0:48
an inequality right to solve an
0:50
inequality you solve it just like you
0:52
were solving a regular equation the only
0:56
difference is in this case we're going
0:59
to graph the solution and when it comes
1:01
to negative the sign or the symbol is
1:04
going to flip when you divide or
1:06
multiply by negative you're going to get
1:07
to that in a second right but first
1:10
thing comes first we need to learn how
1:11
to solve an inequality by adding or
1:13
subtracting which we've done with
1:15
equations themselves right if I have x +
1:20
7 is greater than 12 how do I solve for
1:24
x yes subtract seven subtract seven
1:28
right
1:30
it's just like a regular equation now
1:32
this is going to go away and now you
1:34
have X is greater than what what's 12 -
1:39
7 it should be five five right X is
1:42
greater than five now in this section we
1:46
not only going to just give the solution
1:47
but we also going to graph it right so
1:49
nor how would I graph this solution X is
1:51
greater than 5 how would you graph that
1:54
dra number line draw a number line put
1:57
five on the number line five on the
1:59
number line have an empty circle with a
2:02
arrow going to the right to the right
2:04
that's correct right so that's the
2:06
solution so the solution is five not
2:09
included to the right yeah how do you
2:12
graph and whatever it's
2:16
called that's just what we just did
2:19
that's what we just did X is greater
2:20
than five so you go here right now we
2:22
have solved y + 10 is less than or equal
2:26
to -6 and I'm trying to solve for y
2:31
Vicki what do I do
2:33
here I have y + 10 is less than or equal
2:37
to -6 I am trying to solve for y so what
2:41
am I going to
2:44
do all right so who knows yeah um
2:47
subract 10 subtract 10 why are we
2:50
subtracting
2:51
10 why
2:53
it's this is positive 10 right I want to
2:56
get y by itself right so I'm going to
2:58
take away the 10
3:01
right so I'm going to subtract the 10
3:04
and now what do I have I have y is less
3:07
than or equal to
3:08
what -6 right and if I was to write this
3:13
on my number
3:15
line what 16 you do a circle colored in
3:21
colored in and I go left or right left
3:24
right again if you're struggling with
3:26
this and you find someone that knows how
3:28
to do it then you should talk a and then
3:30
work with that person this is how you
3:32
get better right or watch the lesson now
3:37
now we have another one here right so
3:40
now we both in both of these cases to
3:42
solve it we did what we subtracted right
3:46
now here I have 3 is less than or equal
3:49
to b - 1 and 1/3 right now what do I do
3:54
here first before I even solve it what
3:55
do I need to
3:57
do yes
4:00
how you get what you had your hand up no
4:02
I did well I assume you did so now you
4:05
can have me what do I do
4:08
[Music]
4:11
first
4:12
add3 add one one3 right but we first
4:15
need to also change this into What
4:18
improper fraction and what would that be
4:20
4 43 right 43 so now we're going to add
4:24
4/3 on both side right add four over
4:27
three so now we I'm going to bring bring
4:29
this back here right so we have
4:31
3 + 4 3 is less than or equal to B right
4:36
now how do I deal with
4:39
this excuse me um I I I was I was Sol
4:44
like them myself seeing if I could do it
4:48
and I just went I um did it with the
4:50
mixed fraction and cuz when I add three
4:53
to one and 13 is just 4 and 13 4 and 1/3
4:57
yeah that's right that's correct 4 and
4:58
1/3 yeah so I I turn it into a mix yeah
5:02
you can do that 3 and 1/3 which is 4 and
5:04
1/3 which gives you what 13 over 4 right
5:07
you can do it this well you can do it
5:09
her way that's fine you can also do that
5:11
right CU some people like to deal with
5:13
like mix they like mixed fraction I
5:14
personally don't like them I just like
5:16
to turn them into improper fraction but
5:17
if you can do that that's fine right
5:19
once we get here we can turn this into
5:21
what we have to find the lowest common
5:24
denominator which is
5:26
what I have 3 + 4/3 right I need to put
5:29
this over what one right and then I need
5:32
to find my lowest common denominator
5:34
what is it three three right so I need
5:37
to multiply this by three and this by
5:39
three so I get 9 right over 3 + 4 3 is
5:44
less than or equal to B which gives me
5:47
13 / 3 is less than or equal to B right
5:52
does anybody have a problem with this no
5:55
all right
5:57
huh somebody say so long WR I heard
6:01
something I don't know what I heard
6:02
anyway so now how would I put this on
6:05
the number
6:07
line uh
6:10
yes all right this 13 over3 is here
6:14
where am I going am I going right or
6:16
left think think very carefully
6:20
H am I going right or left right right
6:24
that's correct fill in right I'm going
6:27
right because 13 over 3 is left less
6:29
than or equal to B which means B is
6:32
greater than or equal to
6:35
133 right so you going to the right all
6:38
right does that make sense all right now
6:41
we have
6:43
x + 5 over 7 is greater than 2 x + 5
6:50
7 is greater than 2 how do I solve this
6:53
here
6:55
VI x + 5 7 is greater than 2 so how do I
6:59
solve this
7:04
I still don't understand how to do it
7:07
what are you trying to solve for x x so
7:09
what needs to go
7:12
away 5 over 7 so how do you get rid of
7:15
it what's the operation because this is
7:17
adding so I'm going to do what sub
7:20
subtract so I'm going to subtract the 5
7:21
over s
7:23
right so I'm going to subtract the 5
7:25
over 7 and now we're going to have
7:27
another fraction here so now we have we
7:29
have X is greater than 2 - 5 /
7:35
7
7:38
right and how do I solve this from here
7:40
I still need to turn this over what 2
7:43
over
7:45
K is 2 - 5 7 what do I do here to solve
7:49
this
7:50
fraction I need to put the two over
7:54
what one first right and I need to find
7:57
our most Comm so what's the lowest
8:00
common denominator s seven so this is be
8:03
multip by 7 7 right so I get X is
8:06
greater than
8:08
14 - 5 / 7 which means X is greater than
8:15
9/ 7 right does that make sense you got
8:20
it and then I can put this on a number
8:24
line right so Lena what am I going to do
8:28
here on the number line what was
8:31
here 97 and
8:35
then ccle
8:38
circle to the right right so the
8:41
solution will be 9 over 7 Circle move to
8:45
the right because X has to be greater
8:48
than 9 7 it's not less than or equal or
8:51
more than or equal so it's 9/ 7 is going
8:53
to the right it gets a little bit
8:55
complicated when you have fractions this
8:57
is why we need to do a lot of practice
8:59
right right now now we're going to learn
9:01
how to solve inequalities by multiplying
9:04
or dividing the first thing that we did
9:06
was adding or subtracting okay now we're
9:11
going to learn how to do by
9:12
multiplication or division so I give you
9:15
a word problem and then we're going to
9:17
use that word problem because you know a
9:19
lot of times the question is when am I
9:21
going to use this right we can to use it
9:23
here now Macy she's making bracelets
9:27
okay Macy is making bracelets for each
9:29
of us seven friends I find it
9:31
interesting that she has seven friends
9:32
right seven what what's seven in the
9:35
Bible completion number of completion
9:38
like number of God completion
9:40
completeness so she has seven friends so
9:42
she has seven friends I mean she's just
9:46
like perfect yeah right and she does not
9:49
want to spend more than $49 for her
9:51
seven friends right so find the maximum
9:55
cost for each bracelet so how would you
9:57
set this cuz the key word here is this
10:00
she does not want to spend right she
10:03
does not want to spend more than $49
10:05
hour right that's a that's her cut off
10:09
she does not want to spend more than7
10:12
$49 so how would you write an equ an
10:15
inequality
10:17
specific
10:20
Kaleb are we running up a tax it has to
10:24
be so how many friends does she have
10:27
seven friends right she has seven
10:28
friends right
10:30
do we know how much she going to
10:32
be we don't know how many uh how much
10:34
the bra are going to cost right
10:36
technically we can find it but we don't
10:37
know so she has seven friends and each
10:40
one of those friends is going to have a
10:41
bracelet right how much money does she
10:43
have to
10:44
spend under 49 49 right yeah under 49 so
10:49
that means it has to be like this and
10:51
now we don't know how many bracelet she
10:53
can make but we know that this is going
10:54
to be what x because the cost is unknown
10:59
right
11:00
so 7 x is less than or equal to 49 does
11:03
that make sense she has seven friends
11:06
and each one of those is going to get
11:08
the same type of bracet it's going to
11:09
cost her about 7 * X less than or equal
11:13
to 49 because that's how much money she
11:15
has in total and she wants to make a
11:18
bracet for each one of them so each one
11:19
of them is going to cost them a same
11:21
amount of money right so 7even X less
11:24
than equal to 49 she has $49 to spend
11:27
correct $ so $7 so that means she has
11:31
the maximum she can spend for each
11:33
friend is what seven how do you do it
11:35
you're going to divide by seven remember
11:38
that right every time we multiply to get
11:40
rid of the multiplication we use what
11:42
division okay so therefore X is going to
11:46
be what less than or equal to 7 so that
11:48
means for each friend she's going to
11:50
spend either $7
11:52
or less right so $7 or less that means
11:58
this is how what you want to you she
11:59
want to spend $7 or less for each friend
12:01
so to represent that it's going to be
12:03
seven here right full circle going here
12:07
okay I think Miss is a little cheap $7
12:10
that's that's kind of cheap to make a
12:12
bracelet
12:14
right you could get a I could get a
12:16
rainbow you get a nice um bracelet kit
12:20
for $49 and make individual bracelet out
12:23
of that you probably more than seven
12:25
bracelets yeah no just buy a rainbow Lo
12:27
all right
12:29
so now new New Concept here we're going
12:33
to learn how to multiply multiply and
12:36
divice store my
12:39
bracelet go and buy like a little rubber
12:41
band yeah just like you put them on your
12:43
fingers and like so the next thing is
12:46
and I take them apart
12:48
multiplication I love doing
12:51
that and division properties right IND
12:54
division properties word problems no
12:57
it's not a word problem it's just
13:00
what happen when you divide by a
13:01
negative number right negative number
13:04
what if I already forgot all the past
13:05
lessons we've already
13:07
done well then guess what then you go
13:10
back and review that's that's the whole
13:12
reason why you do that I won't need
13:14
those guess what I forgot your name
13:16
somewh I forgot Kenzie name yesterday
13:19
and I forgot uh Alina's name I had to
13:21
like remember okay and oh yeah this is
13:24
Kenzie Salena and K Kinsley so he
13:27
happens I forgot iig name he was only 2
13:29
weeks were like we were all for like 16
13:31
days I don't even know what the last
13:33
unit was what I don't even know what the
13:36
last unit even was area um it was the
13:40
areas and of the triangles and
13:42
rectangles
13:44
all now I want you all to pay attention
13:46
here right I want you all to be very
13:49
careful here and pay
13:51
attention right so before we even solve
13:54
this I want to show you something right
13:57
[Music]
13:59
two is less than three right yeah we all
14:02
agree is anybody think this is not true
14:05
is two less than three yes or no right
14:08
it's it's not it's not I disagree if you
14:11
disagree then you have to prove it why
14:12
you disagree right if you can't prove it
14:14
then you shouldn't disagree right now
14:17
let me do this here right and I want I
14:20
want to see what you guys think if I
14:21
multiply each of these things by let's
14:25
say4 right then two would be greater
14:28
than three now let's see what do I
14:30
what's 4 *
14:32
2
14:33
right now what's uh 3 *
14:39
-42 now what if I keep it this way is
14:42
this a true statement no why because 8
14:45
is closer to zero uhhuh right so that
14:49
makes it so what can you can you make
14:51
can you make up a rule based on this
14:54
here yeah what do you mean anytime you
14:57
multiply right an inequality by a
15:00
negative number what happen to the
15:03
symbol it Chang it changes Direction it
15:06
flips right does that make sense it
15:09
flips because we just realized it right
15:12
something happened here right now let's
15:14
look here now I have
15:16
-5x is less than or equal to
15:20
45 now this is where this is going to be
15:22
important remember in in in when we were
15:24
working on like equations a lot of you
15:26
guys were forgetting to divide by
15:27
negative and every time I I will take
15:30
out like 25 like come on come on come on
15:33
here you can't do that because it's
15:35
going to completely give you a different
15:39
answer so we're not dividing by we are
15:42
dividing by neg five but most of you
15:43
guys a lot of time in our previous
15:45
sections people forget to put negative
15:48
right here you can't do that yeah so
15:51
less than or less than or equal to now
15:54
here this is less right so now I want to
15:56
solve this so what do I do here yeah
16:00
divide by divide by five right and
16:03
divide by five but remember in the last
16:06
thing any time you divide by negative
16:08
what has to happen to this
16:11
thing you must flip it you must right
16:16
yeah know how you going to remember that
16:18
you
16:20
must flip
16:23
it every time every time you divide
16:32
or
16:35
multiply by a negative
16:38
number right a negative number that
16:42
is a rule that must never be
16:45
forgotten every time you divide or
16:47
multiply by negative number on both side
16:49
of the equation you have to flip the
16:53
symbol is that understood right because
16:57
if you don't and you turn something in
16:59
like I'm going to mark it wrong I'm not
17:01
going to give you half of the credit be
17:02
like well you try no cuz you're getting
17:05
the wrong answer
17:06
right does that make sense so now if you
17:11
divide this by5 so we have X and then
17:13
automatically this is going to be now
17:15
what greater than 455 which
17:22
is9 right every time we divide by
17:25
negative or multiply by negative we have
17:27
to flip the sign because it
17:30
changes the direction of the equation
17:33
the inequality it changes this the the
17:35
meaning of it right okay that's what
17:38
happen he changes the meaning so
17:39
therefore now you have a negative you
17:42
have a negative number so he changes the
17:43
meaning of it so now let me give you
17:45
another example can I erase this
17:48
here
17:49
this the the that FP it you must flip it
17:54
right every time you divide you must
17:56
flip you must flip the symbol I mean the
17:59
symol
18:03
right every time you divide or multiply
18:05
by a negative number on either side
18:08
ofquality more
18:17
okay and uh let me uh get something here
18:24
so can I raise this now ra everything
18:27
mhm all right thank you
18:29
all right now let's assume that we have
18:31
this
18:35
now X over
18:39
-3 is greater than or less or more than
18:43
no greater than or equal to uh four
18:48
right and I'm trying to solve for x now
18:51
look what's the difference here now I
18:53
have a fraction right is x -3 is greater
18:58
than or = to 4 so how do I solve for x
19:01
here
19:02
yeah time3 * -3 right I'm going to
19:07
multiply by -3 and I'm going to multiply
19:10
this by -3 so now when I do this right
19:13
what need to come into my mind I need to
19:15
do what first change change this right
19:18
now it's going to be instead of more
19:20
than or equal it's going to be what less
19:22
than or equal right and then I know that
19:24
this cancels this
19:25
out right so I get X is 4 * -3
19:32
is2 would
19:34
say7 my head Char do division division
19:38
right so it's -12 here all right and I
19:41
flip the sign I flip the symbol right so
19:45
we're going to do some more exercise
19:47
like that and then to get the hang of it
19:50
so now I assume I have this here
19:53
2x is less than or equal to 10 or -1
19:58
right
19:59
how do I solve this 2x is less than oral
20:02
to10 yeah um so the first thing I would
20:06
do is divide by two right so do I flip
20:11
it do I flip it no no
20:16
why did I divide by negative no so I
20:19
don't flip it you see how I tricked you
20:20
cuz I put1 you on purpose right I don't
20:22
CLI it here because I divide by what a
20:26
positive number does not change meaning
20:29
to be negative number it has to be a
20:31
negative every time you divide by a
20:32
negative that's when you flip it every
20:35
time you multiply or divide by negative
20:36
you flip it here it's a positive number
20:39
so it does not change it right it's
20:41
almost like saying this is X now is less
20:44
or equal to5 now let me do this this way
20:47
now what if I have -2X is less than or
20:50
equal to -10 now
20:53
what now you flipping now you flipping
20:56
right you divided by -2 now you flipping
20:59
now this
21:00
flips now you get five here right now
21:04
you
21:04
flip no all right now you flip
21:08
it uh let's try this here
21:39
all right so how would you do this one
21:41
here -2 3x is less than 5 over
21:46
three there's a lot of things going on
21:48
here right I want you all to think a
21:50
little
21:55
bit so we have -2 over 3 x is less than
21:59
5 over
22:10
3 you have any grave soul I want to try
22:14
that want to all right so how would you
22:16
do it okay so I would first multiply -2
22:22
over three by its reciprocal ah 3 two ah
22:27
did you hear that right reciprocal right
22:30
so let me go here -2 3x is less than 53
22:33
so I'll multiply this by
22:36
what reciprocal which is
22:39
now because we have a negative so
22:42
what3 over2 right the symbol would flip
22:46
your symbol will flip that's correct do
22:48
you see that right we multiply this by
22:50
the reciprocal right it's -3 over2 and
22:54
the symbol automatically flip
22:59
are we all on the same page yes all
23:01
right so now this cancel this out right
23:03
so we have X is now what greater right
23:08
and you have five the thre can cancel
23:11
out right so now you have what five over
23:15
what
23:17
two you see how this works it becomes
23:21
fun
23:23
right we just for I think it's fine it's
23:27
fine when you you don't want
23:29
it's fun but it's not the most terrible
23:30
thing ever say what it's not fun but
23:33
it's not terrible yeah it's not fun may
23:35
not be fun but it's not ter once you get
23:36
the G like it's like a walk in the park
23:40
now right you just enjoy it right so now
23:43
let's do some let's do a couple world
23:44
problem we can end the day
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