Up next in 10
How To Write & Graph Inequalities
9 views
Feb 19, 2025
Section 5.3 Welcome back to our Prealgebra class, in today's session we will be discussing inequalities. We'll first learn how to write them, then we'll discuss graphing them on a number line. Chapters: 00:00 Introduction 02:01 Inequalities symbols 03:14 How to write an inequality 09:38 How To determine the truth of an inequality 14:52 How to graph an inequality on a number line
View Video Transcript
0:00
so we're going to discuss inequalities
0:03
right this is section 5.3 the last thing
0:05
that we did was 5.2 which was solving
0:08
equations with two uh variables on each
0:12
side of the equation that's where we did
0:13
the last thing before we left right then
0:15
we do like theim we did the perimeter
0:17
and all of that that's was still
0:18
equations right now we're going to talk
0:20
about
0:22
inequalities right and what is an
0:24
inequality an inequality by definition
0:27
is a mathematical sentence
0:30
that compares two quantities that are
0:33
not equal okay if you want to compare
0:35
two quantities that are not equal you
0:37
use what you call inequalities for
0:39
example my age and your age right if I
0:44
say for example nor age is less is less
0:47
than mine would I be lying if I say that
0:50
no right if I say Nora's age is less
0:57
than uh ab's age would line maybe
1:01
because I don't know right so that may
1:03
not be a true statement so why do we use
1:06
why do you use in qualities you use
1:08
inequalities to compare quantities that
1:10
are not the same all right so now
1:13
inequalities contains what we call
1:15
symbols right a symbol a symbol is a
1:17
sign something right if you see an arrow
1:19
if you driving and you see the arrow
1:21
going this way that mean you have to
1:22
turn what that right if you go to the
1:25
left is turn left so symbols are
1:27
indicators they show you some that
1:29
something's going to happen you better
1:30
write down the notes right you better
1:32
write down right now in inequalities we
1:35
also have some symbols that are
1:37
important it goes to you too don't just
1:40
look at me write down
1:42
step right so we have like four main
1:46
symbols in inequalities right we have
1:49
what you call this one some people call
1:52
it like uh I don't know they use last
1:55
year they use some funny terms right
1:57
this is called less than right less
2:01
than and this is more than this is less
2:05
than or equal and this is more than or
2:07
equal so we have four main symbols when
2:10
we come when it comes to inequalities
2:13
okay and then we're going to tr try to
2:15
use the symbols to to write uh to turn
2:19
sentences into mathematical inequalities
2:22
right
2:25
example if I say to you sure that is get
2:30
too loud for me here right for example
2:33
if I say the DVDs cost more than
2:38
$15 right how would you write that as a
2:41
mathematical uh inequality how would you
2:43
write that the DVDs cost more than $15
2:45
yes so X is greater than 15 x is greater
2:49
than 15 thank you X is greater right X
2:53
will be what DVD right that the DVDs the
2:56
cost more so X is greater so what's what
2:59
the sign for greater is it this way or
3:02
this way the other way it's going to be
3:04
more than what 15 see X is more than 15
3:08
right so the more let me put this the
3:11
right way and then 15 here so the DVD
3:14
cost more than 15 will be X is greater
3:17
than
3:18
15 does that make sense to everybody you
3:22
all got that right now if I say a dog
3:25
weighs less than 50b how would you write
3:29
that
3:33
a dog weighs less than
3:36
50 a dog weighs less than 15 his weight
3:40
is less than
3:44
that so how would you do that how would
3:48
you write that a dog weighs less than
3:50
50b yes um xal less than 15 x is what
3:56
less than 15 less than 15 x is less than
3:59
15
4:00
right so a dog weighs less than 50 x is
4:04
less than
4:05
50 okay a dog he weighs less than that I
4:08
didn't say a dog less 50 I say a dog
4:13
weighs less than 50 so that is X is less
4:17
than 50 x is less than 50 okay so again
4:22
this sign stands for less than this is
4:25
more than this is less than or equal and
4:27
this is more than or equal okay now the
4:30
next one the speed limit is less than or
4:33
equal to 65 mph now for you guys that
4:36
are not driving yet you better know this
4:38
right this the speed limit if I said the
4:41
speed limit is less than or equal to 65
4:43
milph how would you write that as a
4:46
mathematical inequality how would you
4:48
translate
4:51
that so first we need to break down the
4:54
speed limit right what letter would you
4:57
use for Speed X okay X now is less than
5:01
or equal so how you write that less a
5:04
line under it less than with a line
5:07
under it right right line under it and
5:10
then here will be 65 right that means I
5:13
can drive what I can drive 65 miles or
5:18
what over less not over less than over
5:24
not over less than right I didn't say
5:27
over so this is less than so that means
5:29
if you go 65 miles you're okay if you go
5:32
63 miles you're fine but if you're going
5:34
40 miles get out of my
5:36
way what if I'm going 70 miles
5:40
then right what's that 75 the of the
5:45
police so they're going to get
5:47
you there high mil but everybody now now
5:52
here's the thing the police officer told
5:53
me once I don't know if this is true he
5:55
says usually if the speed limit is 65
5:58
you can add 11
6:00
to it so if you go 76 you're okay but if
6:03
you go over 76 you get busted but I
6:05
don't think I want to test it so
6:08
sometimes what I do is I'll drive a
6:10
little bit like if 65 go 70 and I look
6:12
on the left and right to see there's no
6:14
cars behind me
6:16
or no you shouldn't technically that's
6:18
my sister faed her driver's test
6:23
yesterday my parents go over this time
6:26
yeah but the thing is technically they
6:28
are in every right you pretty much
6:30
violated the law right most officers
6:33
will not arrest you if you're going
6:35
under if it's like 65 most officers will
6:39
you go up to like 75 and they won't pull
6:42
they might not they might but if they do
6:44
want to do it they have every right to
6:46
yes it's just like some of them are just
6:48
like you know what I'll let you get away
6:50
with it no you can't you can also you
6:52
can't drive too slow either if you're 65
6:54
and you go 50 you may cause accidents
6:57
because this is like m out there you
7:01
know you can't do that you got to like
7:02
stay 65 like 65 60 imagine living in a
7:05
55 and over neighborhood that's where
7:07
you live yeah anyway example two right
7:11
how would you translate this you must be
7:13
at least 18 years old to
7:16
vote right you must be at least 18 years
7:19
old to vote yeah I'm actually 31 well
7:21
then how come you can't drink alcohol
7:22
until you're 21 but you can make a
7:24
decision when you're 18 why well that's
7:27
up to my hands alcohol like on a
7:33
[Music]
7:34
second you're you drink alcohol when
7:37
you're coming out of you that's very
7:40
surprising that that's is the first
7:41
thing that you think you know what I'm
7:44
saying how come you can drink
7:49
alcohol anyway since you were so bold to
7:55
give you switch my age I'm 31 so I can
7:58
drink Ando so you must be at
8:01
18 go so how you
8:04
write hey hey hey quiet yeah so that
8:08
since it's saying least I'm going to say
8:10
um like less than no actually more more
8:14
more than 18 years to vote more than or
8:17
equal right at least if I say at least
8:20
18 that means you can be 18 or what
8:23
older right so it's going what more than
8:26
with line under 18 that means if you are
8:30
18 or older you can vote in the election
8:34
right that means if you're 18 or older
8:36
this is what they determine that's why
8:38
the government determined that if you're
8:40
18 years or older you can V right all
8:43
right now now the next thing we're going
8:46
to talk about hey hey hey
8:48
that's the next thing we're going to
8:50
talk about is we're going to determine
8:53
if uh an inequality is true or false
8:58
right how would you that I'm going to
9:00
give you a statement and I'm going to
9:02
give you uh a number that you can you
9:05
use to evaluate the statement right for
9:07
example here I said 2 t + 8 is more than
9:10
7 that's the statement that I gave you
9:13
right 2T + 8 is more than
9:16
7 right 2 t + 8 is more than 7 now the
9:20
statement is this if T is equal to1 is
9:23
this still true we don't know right if T
9:26
is equal to1 will this hold
9:31
that's what this is asking we don't know
9:33
we have to we have to test it right we
9:36
have to test it that means we going to
9:37
replace T by what1 and we going to
9:40
verify this right so I'm going to go 2 *
9:44
what1 + 8 is more than S I'll put a
9:47
question mark here because I don't know
9:49
if this is true yet I need to verify
9:52
that I need to double check the
9:53
statement right so 2 * 1 is what2 -2 so
9:57
what's -2 + 8
10:00
six six now is six more than seven no no
10:04
so is this statement true no no it's a
10:06
false statement false statement right
10:11
false statement so this is how you
10:12
verify the truth of an inequality I'm
10:15
going to give you the inequality and
10:17
then I'm going to give you a number and
10:18
I say if you pluging this number in here
10:20
will this still be true right now let's
10:23
try for the next one we have P - 42 is
10:28
less than or equal to next 2 right if p
10:31
is 40 will this be true we're going to
10:33
double check that is everybody
10:36
understanding this portion of
10:37
it who's
10:40
confused all right so here what we doing
10:44
is this right let me say to you let me
10:46
give you a statement and we going to see
10:48
if this is true if I say they combine if
10:51
I combine all your ages right you all
10:55
going to be less than my age will this
10:57
be a true statement or a false statement
10:59
true false right it be a false statement
11:03
because I mean I ain't that old even
11:06
though my Korean neighbor you know he
11:08
this guy is nuts head man like I was
11:10
Shou in the snow yesterday like I said
11:12
ah he goes hey man I said how you doing
11:16
I'm I'm good he goes are you 60 years
11:18
old yes no I'm not 60 years old
11:22
something and he asked me I told him I
11:24
was like I give you my age 23 years old
11:27
and then he goes and he goes oh sorry
11:30
I'm like man how you I mean do I look 60
11:33
do I look
11:36
60 you look
11:38
55 seriously you look older than my
11:47
dad anyway I'm not talking about that
11:49
you
11:50
guys I wish you guys have something
11:53
different I mean I said you you asked
11:56
the question so you have to accept the
11:58
answer is too much I said you look
12:02
5550 that's still close to 60 I don't
12:04
want 55 no I
12:12
mean anyway so back to back to back to
12:16
this yes um if uh so I went iead and I
12:21
solv the thing yeah um if it says less
12:25
than or equal to and it's the same
12:28
number that mean it's correct it's
12:30
correct yeah so here for example right
12:33
so again back to this so I give you a
12:36
statement right I'm saying here that 2 t
12:38
+ 8 is more than 7 now verify the
12:41
statement if T is equal to1 this is what
12:44
this is asking right so basically if you
12:47
plug in negative one in here will this
12:49
be true yes this is the question that
12:51
they asking right so now what we do is
12:54
this we're going to replace T
12:55
by1 and then run it to see if this is
12:58
going to whole now when we did it I repl
13:01
T
13:01
by1 it gave me 6 is more than S this is
13:04
FAL so I'll say if T is when T is equal
13:07
to1 this is not true does that make
13:10
sense now right now here we do the same
13:13
thing I give you a statement as if P -
13:15
42 is less than or equal to -2 when p is
13:19
40 will this hold we don't know we have
13:23
to check it right we have to check it so
13:25
we're going to replace P by what 40 and
13:28
we're going to run it so so we're going
13:29
to go 40 -
13:31
42 is less than or equal to -2 and we
13:34
going to put a question mark now the
13:36
reason why I put a question mark is
13:37
because I am not sure just yet I'm
13:40
testing it right I'm testing right now
13:43
what's 40 -
13:45
42 -2 so now I have -2 is less than or
13:48
equal to -2 true or false this is true
13:51
right because -2 is less than or equal
13:54
is equal to so this is a true statement
13:58
right one is less than or equal to one
14:00
that's a true statement
14:03
right so this is true okay this is true
14:07
here all right so this is this is what
14:09
we are discussing now we're going to
14:11
learn how to graph an inequality we're
14:14
going to learn how to graph it okay so
14:16
we're going to learn how to graph it so
14:18
how do you graph an inequality that's
14:20
very simple it's actually fun okay so
14:24
let's say I have uh we're going to learn
14:28
out to graphing on the number line okay
14:31
we're going to learn how to graph this
14:32
on the number line so let's say this is
14:36
number example four where's my right
14:39
marker
14:47
number so I want you to
14:49
put a is more than six I want you to
14:53
graph this inequality on the number line
14:56
graph it
14:59
graph it on the
15:03
number line so how do you do that right
15:07
yeah is that an A or an i that's an a
15:10
let me put an X make it easy right X is
15:13
more than six and I want you to put this
15:15
on the number line so how are you going
15:17
to do it right X is more than 6 hold
15:22
up right so we're going to go
15:26
here this this is knock it so this is
15:29
our number line right we're going to put
15:31
our number line and then we're going to
15:33
put the number six all right now pay
15:36
attention here so I want X to be
15:42
what no more than I want X to be more
15:45
than 6 right if I want X to be more than
15:49
6 now pay attention here something is
15:51
important this is important right if I
15:53
want X be more than six am I going to be
15:55
on this side of six or I'm going to be
15:57
on the right side of six what side this
16:00
side the left side this side side or
16:03
this side right side the right side I'm
16:06
going to be going here right but now
16:08
there's another thing that I need to
16:09
make sure right I want X to be more than
16:12
6 can X be equal to 6 no no so because X
16:17
canot be equal to 6 I have to put a
16:19
circle here right so it's going to look
16:22
like this a circle going this way
16:25
anytime you put a open circle what you
16:28
are saying is this you're saying that X
16:30
cannot be equal to 6 right X can be
16:33
greater than 6 but X cannot be equal to
16:35
6 so you put a circle to indicate that
16:39
right so if you are to put it on the
16:40
number line you have to make sure you
16:42
put that there because there's no line
16:44
that's under the more than or equal that
16:47
means X canot be equal to 6 but X has to
16:50
be what more than 6 so therefore you put
16:53
a circle that is sh it okay let me give
16:55
you another
16:56
example so now if I say x is less than
17:02
or equal to -1 right let's try to do
17:05
that so you guys can help me with this
17:07
so if I put this here I'm going to put1
17:10
somewhere here right if x is less than
17:12
or equal to 9 I might going do one I
17:15
mean am I going to be going this way am
17:17
I going to be going to the left side
17:19
left side right now I'm going to the
17:22
left side now
17:24
is1 can 1 be included in there yes or no
17:28
yes because X what less than or equal so
17:30
now I'm no longer going to put a open
17:32
circle but what am I going to do now is
17:34
going to be what a Clos Circle closed
17:37
Circle the closed Circle shows you that1
17:41
is also included in this does that make
17:43
sense right so anytime you have an open
17:46
circle that means the number is not
17:48
included and anytime you have a closed
17:51
Circle that means the number is included
17:52
now what I want to show you I want you
17:54
guys to do this I'm going to write an
17:57
inequality I'm going to graph it and you
17:59
you guys are going to try to interpret
18:00
it by telling me what the statement is
18:02
right let's try to do that let's play a
18:03
little game here so what if I have this
18:06
line right and I have
18:10
zero and open circle going this way so
18:13
how would you write this
18:14
inequality this is zero now
18:18
yes X is more than zero excellent X is
18:22
more than zero going to get that yes or
18:27
no so is here I'm going to the right and
18:30
I put a
18:31
circle that means can X be equal to zero
18:35
no but I'm going to the right side of
18:37
zero can X be less than zero no it has
18:40
to be what more right more than zero yes
18:44
wait what's the FI circle mean fi Circle
18:47
means the number is included included
18:50
included that means let's say for
18:52
example I have the same thing right and
18:54
this is fi all I would change is put a
18:57
line under that
18:59
X is less than or
19:01
equal when you hear the word or equal
19:04
that means the number is included feel
19:07
Circle that make sense we do some other
19:09
example let let me do let me show you
19:11
another
19:13
one so if I
19:16
have uh let me draw a number the number
19:19
line and I have this here -10 right and
19:22
I go here so how would you do
19:26
this um X mhm
19:35
look where am I going that X is less
19:39
okay
19:41
what 10 but because this is a
19:44
filed Circle you have to put a line
19:47
under the line that's under shows that s
19:50
is less than or equal
19:52
to1 when there's no line under that
19:55
means the -1 is not in and or and or
20:00
yeah n or equal right so let give you
20:02
another example what time is this class
20:05
n doesn't matter if the Bell
20:08
Rings it does so let's put 17 here right
20:14
and here so how would you write
20:17
this by leaving the last what what way
20:22
am I going don't say just randomly so
20:25
that means what if I'm going here
20:27
greater greater because if I'm going am
20:29
I going is the next number going to be
20:31
17 or 18 could be 18 19 20 right they
20:35
all going to be greater so greater than
20:38
what x is greater
20:40
than is am I
20:43
done am I that is this
20:47
it this is a fi Circle that means
20:51
what it has to be greater than or equal
20:54
to
20:55
17 what are you sming about what sm all
21:00
right so anytime you have that so let's
21:02
let me show you another one here right
21:04
you guys can try to see if you can
21:05
figure this out X is less or equal to
21:11
uh5 so go ahead and try to see if you
21:13
can write this on the number l x is less
21:15
than oral
21:17
to you can use a red
21:20
one so X is less than or equal
21:24
to5 you guys locked out cuz today is a
21:27
Wednesday I can't give you no
21:33
homework was very
21:37
excited i' rather that than do homework
21:39
right now
21:42
I rather I rather jump that's is less or
21:46
equal to em I don't know do so
21:51
here what way am I going left or right
21:55
left is fing Circle open circle gentl
21:59
fill in so it's going to be this it's
22:01
going to be going this way right now
22:03
what I want you to do is this I want you
22:05
to tell me how you write this
22:10
right 23 ther or whatever it's
22:15
called what is it called again I want
22:17
you to I want you to write the stat this
22:20
I also want you to do
22:23
this
22:25
no I'm about to eat this
22:29
all right so you going do
22:31
this we have like minutes
22:35
so so 23 going this way so how would you
22:39
write that statement
22:43
it's X is what 23 x is greater than 23
22:48
what about
22:57
here right
22:59
to what 90 all right so tomorrow we're
23:02
going to have a work day that I'm going
23:04
to give you