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In this chapter, we'll talk about how to solve inequalities and represent them on the number line.
Chapters
00:00 Introduction
00:00 How To Solve one step equalities
01:28 How To graph the solution on a number line
03:24 Multiplication property of Inequalities
06:01 Division property of Inequalities
06:55 Solving Inequalities by Multiplication or division
08:51 Solving multi step inequalities
11:06 Working on a real life problem involving inequalities
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0:00
so we're going to talk about solving
0:02
inequalities all right again this is
0:05
chapter is still we still on the easy
0:08
side of things so solving inequalities
0:10
if you've done it in algebra one even Al
0:13
pre-algebra this should be just like a
0:15
walking apart it's not that hard okay
0:18
walking apart so the first thing I want
0:21
us to discuss is we're going to first
0:24
solve one step inequalities right one
0:27
step means one step you don't have to do
0:29
too much
0:30
okay we're going to learn how to solve
0:32
it and then we also going to learn how
0:33
to graph the solution on a number line
0:36
all right so how do you solve this
0:38
equation here we have y - 6 is less than
0:43
3 if I want to solve it I'm going to
0:46
solve this just like I was solving the
0:47
regular equation right tell me what do I
0:49
do
0:52
here I'm going to add six to the either
0:55
side right so I get Y is less than 9 now
0:58
the difference between this kind of
1:00
inequality equation is y is less than 9
1:04
here so it's not just a number you want
1:08
the number line a lot of things can
1:10
satisfy this inequality right give me an
1:12
example a number that's less than
1:16
n negative
1:19
what2 yes sir um s seven eight right 8.9
1:23
99 a bunch of numbers so to graph the
1:26
solution we have to draw a number line
1:28
right this is our number line line and I
1:31
have to locate nine nine is here so I
1:33
want everything to be less than nine am
1:36
I going to go left or
1:39
right left right but the thing is this
1:43
I'm going left but there's also
1:44
something else I need to make sure I do
1:46
right you guys know about the closed
1:49
Circle and the open circle right because
1:52
nine is not included is he have does he
1:54
have to be a closed or open
1:56
circle open open is they we're
1:58
comfortable with this open Circle right
2:00
so it's going to be from
2:02
9 open circle to Infinity right this is
2:07
called this is on the number line now if
2:08
you want to represent this as a interval
2:10
it's going to be Infinity to 9 we are
2:13
not there yet so I'm just giving it to
2:15
you as a of knowing okay but this is
2:18
what we have negative Infinity to 9 and
2:20
9 is not included so I put the
2:22
parenthesis not a bracket okay so what
2:25
if I change this problem and I had less
2:28
than or equal why what would be
2:30
different in this case
2:34
yes you'll be a close Circle right
2:37
because now 9 is included if I change
2:40
this to less than or equal 9 will be
2:42
included and I'm still going to be going
2:43
to Infinity negatively but nine is
2:46
included and now instead of parenthesis
2:48
what do I
2:49
have I have a bracket this is important
2:53
right A lot of times people get hung up
2:55
on this they forget to put the bracket
2:56
they put parenthesis that's false
2:59
because if you put par is you're saying
3:00
that n is not included so you don't have
3:02
the same set you have to put a bracket
3:05
right so the one thing that you need to
3:07
remember is this closed Circle closed um
3:11
bracket this
3:14
open that implies a pareses that implies
3:17
a Clos bracket all right so now we going
3:19
to learn about the multiplic
3:22
multiplication property of inequalities
3:25
again very simple step now put a
3:27
statement there 3 is less than 5 we
3:29
agree yes or no unless you live in a
3:32
different Universe three is always going
3:34
to be less than five right now I've
3:37
heard of someone that say that he he
3:39
knew someone that conceived of a squared
3:42
Circle A Squared Circle squared
3:46
Circle yeah can you think of a square
3:52
circle happen right so three three is
3:57
less than five do we agree right now
4:00
what if I times this by -2 and -2 right
4:04
and I write this here
4:08
-6 is less
4:12
than - 10 is this true
4:18
why there a negative right so what do I
4:21
need to do in order to make this
4:22
inequality true yes sir with the sign is
4:26
flip it right anytime you multiply by
4:30
negative we have to make sure we flip
4:33
the inequality
4:36
position right anytime you times by a
4:39
negative you have to flip it it has to
4:40
change because in negativ -6 is greater
4:44
than -10 okay that's just a must now 12
4:49
is more than -7 same thing if I times it
4:53
by-1
4:55
right what am I going to do I'm going to
4:58
get -12
5:00
is what less than postive 7 you have to
5:03
flip it okay and the rule applies also
5:08
for
5:11
what why did you
5:13
use I just used them I just chose this I
5:17
just wanted to does it have to be any
5:20
speci number no I I mean I'm just I'm
5:23
just saying I just this is an example
5:25
right yeah what be7 and not
5:31
where what you
5:37
talk seven because it's 7 * 1 I'm
5:41
multiplying is a
5:43
multiplication this
5:46
one 12 -2 is less than 8 yeah that's a
5:50
completely different problem yeah that's
5:52
true oh I'm saying this is a different
5:55
problem all right this is this is
5:58
totally different okay now this is the
6:00
division property of inequalities right
6:02
I have -12 is less than 8 right if I
6:05
divide this
6:06
by1 what do I get what's -12
6:12
or1 yes 12 12 do I leave this like this
6:18
obviously not I have to change it right
6:21
that becomes eight so again with the
6:23
division property of inequalities
6:25
anytime you divide by a negative you
6:27
have to flip the sign that's just a must
6:30
right so this is why I'm introducing
6:31
this here so if you divide by negative
6:34
or you multiply by negative and
6:35
inequalities are involved the the next
6:38
thing to do is to make sure you flip the
6:40
sign I've seen this too many of too
6:42
often people leave signs like this and
6:45
oh I just forgot so remember that this
6:47
is why I'm recalling this rule okay same
6:50
thing here now we're going to solve
6:53
using multiplication or division here
6:54
right I have -4x this is an example
6:57
solving using multiplication or division
7:00
-4x is less than or equal to 24 how do I
7:03
solve this for X um Co what do I do
7:07
here4 less or equal to 24 divide div4
7:12
and then what switch the sign switch the
7:15
sign so we get
7:17
X is greater than what or equal to -6
7:22
right if I put this on a number line if
7:25
-6 is here where do I go go uh
7:30
right or left right right and then what
7:33
else do I
7:34
do close or open close right because
7:38
it's more than or equal okay now I have
7:40
-1
7:43
/x -1 /
7:46
2x is more than5 how do I solve for x
7:50
here
7:53
kay I don't know what you're asking to
7:55
it right
7:57
here yeah but okay don't know that I'm
8:00
solving for x so what do I do I want to
8:03
get X yes sir
8:06
um oh what else can I do instead of
8:09
dividing you multiply by
8:11
what -2 right the reciprocal right NE -2
8:16
and then this cancels out so now I get
8:19
one or X is what less than 10 again do
8:25
not forget to make sure because you're
8:28
dividing by a negative you multiply by
8:30
negative you have to flip the sign and
8:31
as you put this on the number
8:33
line negative positive 10 is here open
8:37
circle we're going here okay this is
8:40
negative Infinity to 10 open circle now
8:44
on the last one the very last one we are
8:47
solving multistep
8:50
inequalities is Mary missing today Mary
8:53
and Stephanie are both gone
8:56
okay and GG when do the leaves change
9:00
oh I don't know fall season right is
9:03
fall I don't know probably at the end of
9:04
the month I don't
9:05
know so now I'm trying to solve for x
9:08
here so what's my first step this is a
9:10
multi step right what do I need to do
9:15
first times by
9:17
six right on both Sid do you like to
9:20
flip the
9:21
sign no right so I got
9:25
-4x is less than or equal to 5x + 58
9:30
what's the next
9:34
step before we divide we have an X here
9:36
so what I do subract subtract the 5x
9:41
right so I get -
9:44
29x is less than or equal to 58 then
9:48
what's the last step
9:51
now you have a -29 / -29 and do not
9:55
forget
9:56
to flip the sign all right you have to
9:59
flip it so what's 29 into 58 two two so
10:05
this going to be -2 number line going
10:08
from
10:09
-2 to positive Infinity all right so
10:13
this is what this section is all about
10:15
so now what I'm going to do is I'm going
10:16
to give you guys some work to do and
10:19
then we're going to call it a day okay
10:22
so let's do that mhm so why when you
10:26
multip them by six why didn't you put
10:28
the sign there
10:30
it's not netive because it's not
10:32
negative it's positive
10:35
6 very negative right only when you
10:38
multiply by negative or divide by
10:40
negative do you still want your video on
10:42
this or something uh Yes actually let's
10:46
do one word problem and then we going to
10:49
after that we we're going to start
10:53
um I got one more problem after that
10:56
I'll give you all some work to do
11:07
so
11:11
Enrique
11:14
pays uh
11:17
Josh to
11:22
advertise
11:24
on his website
11:32
now Josh's
11:37
website
11:41
earns $15 a
11:46
month
11:49
plus
11:52
0.05 every
11:56
time a visitor
12:02
clicks on the ad right the question is
12:07
what is the
12:13
least number of
12:19
clicks per
12:23
month that
12:26
Josh needs
12:29
in
12:31
order to
12:34
earn
12:37
$50
12:41
month all right so how would you
12:44
translate this problem that's an
12:46
inequality you guys I don't know if you
12:48
know a lot about websites cuz we people
12:51
always ask me what I do I say I'm a
12:53
writer I write on my website I have four
12:56
no five websites that I run right one of
12:58
them is for shoes and stuff like that
13:01
and then people say how do you make
13:02
money on the website you make money from
13:04
ad revenues okay people add on your
13:06
website you make money from there this
13:08
is called passive income you write an
13:10
article once and then people are
13:12
clicking reading the article ODS are
13:14
showing if they click on that you get
13:15
paid so some website makes a gang lot of
13:18
money YouTube is sitting in billions I
13:21
mean probably a lot of money right
13:23
because this is how to make money now
13:25
this guy Enrique he pays Josh to
13:28
advertise on his website maybe Enrique
13:30
is selling whatever maybe earrings may
13:33
maybe he's selling fidget Spinners I
13:35
don't know what he's selling but he
13:36
wants to get more what more exposure
13:40
right so he's paying this guy to
13:43
advertise on his website and joshes
13:45
earns about $15 a month plus
13:48
$0.05 every time somebody clicks on the
13:51
app right so a th000 people click on the
13:53
ad he makes about probably not that much
13:56
$50 right so the question is what is the
13:59
least number of clicks per month that
14:01
Josh needs in order to earn
14:04
$50 so how would you set that up Jack
14:08
let's find an inequality in order to
14:10
earn the least is the key word here
14:13
right least least implies
14:16
what what I say least minimum minimum so
14:19
it's something or more right so how much
14:22
money does you earn a
14:24
month 15 15 we start with that right $15
14:29
plus
14:30
what 0 0.05 since we don't know how much
14:36
uh how many people are clicking we're
14:38
going to put X right and how much are we
14:41
trying to earn at least what 50 so at
14:43
least how am I going to put write that
14:46
uh greater than or equal greater than or
14:48
equal to 50 50 and now we need to solve
14:53
for x right so we need to know how many
14:55
clicks does he need to earn that 50 so
14:58
we have to subract the 15
15:01
first so we get 0.05 X is more than or
15:06
equal to 35 and then we divide this by
15:09
05
15:11
05 so do that in your calculator tell me
15:14
what you
15:20
got
15:21
yeah how much so and that crazy so he
15:26
needs 700 people to click on this in
15:30
order to make at least 50
15:33
bucks guess what if you do that what
15:36
happens if you yourself click on those
15:38
ads this is called this is Shady stuff
15:41
so Google going to find out and you
15:43
going to get banned so people have done
15:45
that people tell other people click on
15:46
those ads they know cuz they have ways
15:48
to figure it out right so this has to be
15:52
people that you don't tell to do that
15:53
it's just people that are coming you
15:55
have to have genuine visitors but the
15:57
fee that he charges is just way too
15:59
little he need to charge a little more
16:02
CU 5 Cent is nothing right cuz you you
16:04
have to get 700 clicks to only make $50
16:07
mean for 500 you need at least 7,000
16:09
clicks right and for for 5,000 you need
16:12
70,000 clicks on that good luck that
16:14
ain't happen so you need to charge more
16:16
money okay all right so that concludes
16:19
that lecture so to speak but now we're
16:21
going to work on some individual
16:23
problems that
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