WEBVTT
Kind: captions
Language: en

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welcome to the Cypress College math
review on linear inequalities

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in this video you are going to learn to
solve linear inequalities and express

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the solution in various forms in
objective one you will learn to write

00:00:22.470 --> 00:00:26.210
interval and set builder notation

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given the inequality X is greater than
two there are different ways to writing

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a solution to this inequality first note
that X is greater than two means that X

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can be three four five six ten a
thousand a million a zillion

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furthermore X can also be two point one
because two point one is greater than

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two 2 point 2, 2 point 3, 2 point three
three, 2 point three, 2 four so

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basically there are many numbers that
are greater than two so we can represent

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the solutions of this inequality by
graphing it on a real number line

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so here is a real number line where we
mark the negative values to the left of

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zero and the positive values to the
right of zero since X is greater than 2

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then X can be any numbers to the right
of 2 and we graph it by shading the

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region to the right of 2
another representation of this

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inequality is interval notation note
that on the real number line the numbers

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go infinitely to the right where we
represented by positive infinity and on

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the left the numbers go infinitely to
the left and that is represented by

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negative infinity
we look at the shaded region on the

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graph and see that it starts from 2 and
it goes infinitely to the right to

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positive infinity therefore we write 2
comma infinity the number on the left is

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where the shaded region on the graph
starts from and the number on the right

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is where it ends
the third representation of this

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inequality is the set builder notation
this is read

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the set of all X such that X is greater
than 2 this bar here means such that

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note that what goes after the bar is
exactly the given inequality

00:02:47.150 --> 00:02:51.319
here is the quick chart to help you
distinguish between the three types of

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writing the solutions to the inequality

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suppose X is the variable and a is some
arbitrary number if we have X is greater

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than a then we graph by first marking a
on the number line and shade to the

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right of a because greater implies
shading to the right so the graph goes

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infinitely to the right or positive
infinity then based on this graph we get

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the interval notation as I mentioned
earlier we look at the graph from left

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to right it starts from a and it goes to
infinity therefore in the interval

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notation the left endpoint is a and the
right endpoint is infinity and for the

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set builder notation we have the set of
all X such that X is greater than a

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recall that this bar means such that and
what goes after the bar is exactly the

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given inequality here we have X is
greater than or equal to a so we still

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mark a on the line and shade to the
right of a because X is greater than so

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the graph goes infinitely to the right
or to positive infinity to indicate that

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X can equal to a we put a bracket at a
so the bracket means the endpoint is

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included note in the previous one the
inequality does not include equal two

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which means X cannot be a so we use a
parenthesis around a to indicate that a

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is not included
now we get the interval notation of this

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by looking at the graph the shaded
region starts from a on the left and

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ends at infinity on the right so we have
a comma infinity note that we use a

00:04:57.229 --> 00:05:03.680
bracket around a to indicate that a is
included we always use a parenthesis

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around infinity or negative infinity
because infinity is not a number but

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rather a notation that indicates a
large quantity

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for set-builder notation once again note
that what goes after the bar is exactly

00:05:20.100 --> 00:05:23.330
the given inequality

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here we have X is less than a less than
means shading to the left so we mark a

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on the number line and shade to the left
of a so the graph goes infinitely to the

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left or to negative infinity since X is
strictly less than a we use a

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parenthesis on a then based on this
graph we get the interval notation on

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the left side of the shaded region we
have negative infinity and the shaded

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region ends up a and of course we use
parentheses around a because X does not

00:06:05.710 --> 00:06:08.699
equal to a

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for the set-builder notation once again
what goes after the bar is exactly the

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given inequality

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here we have X is less than or equal to
a so this means a is included since X is

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less than a then we shade to the left of
a and it goes infinitely to negative

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infinity also since X is equal to a we
use a bracket on a then for the interval

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notation we once again based on the
graph on the Left we have negative

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infinity and it ends at a for the set
builder notation once again what goes

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after the bar is exactly the given
inequality

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so there are four things you need to pay
attention to if the variable is greater

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than a number then we shade to the right
of that number if the variable is less

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than or less than or equal to a number
then we shade the left side of the

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number if the inequality does not
include equal to then we use a

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parenthesis around the number but if the
inequality sign includes the equal to

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then we use bracket and always remember
to use parentheses around infinity and

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negative infinity

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let's look at this example we want to
graph the inequality and then write the

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solutions in interval notation and in
set-builder notation for x is less than or

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equal to negative 4 we mark negative 4
on the number line since x is less than

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negative 4 that means X is negative 5
negative 6 negative 7 so on and so forth

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which is the region left of negative 4

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since the inequality sign has equal 2
then we use a bracket at negative 4

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then we look at the graph to get the
interval notation the graph goes

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infinitely to negative infinity so on
the left of the shaded region we have

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negative infinity and on the right side
we have negative for around negative 4

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we use a bracket around negative
infinity we always use a parenthesis now

00:08:54.570 --> 00:09:03.660
for set builder notation we have the set
of all X such that that's the bar X less

00:09:03.660 --> 00:09:08.000
than or equal to negative 4

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in this example we have X is greater
than five we want to graph the

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inequality then write the solutions in
interval notation and in set-builder

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notation we begin by marking five on the
number line

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since x is greater than five means x is
six seven eight nine ten and even the

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decimal values between those points that
means we shade to the right of five and

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since the inequality symbol does not
have the equal sign then we use a

00:09:51.110 --> 00:09:57.740
parenthesis around five so this graph
goes infinitely to the right or positive

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infinity you don't need to put infinity
here like I did but I just want to write

00:10:03.889 --> 00:10:08.899
it there to help you see when we write
the interval notation so based on the

00:10:08.899 --> 00:10:16.519
graph we have the left value is five and
the right is infinity since we use

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parenthesis around five in the graph we
also use parenthesis in the interval

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notation and as always we use
parentheses around infinity so remember

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that when you write the interval
notation that does not include the

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endpoint use parentheses for set-builder
notation we have the set of all X such

00:10:38.720 --> 00:10:43.029
that X is greater than five

00:10:47.170 --> 00:10:52.439
pause the video and try these problems

00:11:01.700 --> 00:11:09.750
objective two solving linear inequalities
with variables on one side when finding

00:11:09.750 --> 00:11:15.240
the solutions to an inequality solve for
the variable like you would as if it

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were an equation just remember when you
divide or multiply by a negative number

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you need to flip the inequality
let's look at a few examples for

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instance if you have negative 5 X less
than 15 then to solve for X we would

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divide by negative 5

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negative 5 divided by negative 5 is 1 so
we have X 15 divided by negative 5 is

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negative 3 and since we divided by a
negative number then this inequality

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sign is flipped to greater than

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another example negative 2 X is greater
than or equal to negative 10 so we would

00:12:08.200 --> 00:12:15.820
divide by negative 2 to get X by itself
we get X on the Left negative 10 divided

00:12:15.820 --> 00:12:22.030
by negative 2 is positive 5 and since
we're dividing by a negative number then

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we switch the inequality sign and this
becomes less than or equal to

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another example 4x is less than negative
12 so we divide by 4 we get X on the

00:12:38.949 --> 00:12:46.480
left negative 12 divided by 4 is
negative 3 however we are not dividing

00:12:46.480 --> 00:12:51.699
by a negative number
so the inequality remains as is so you

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see that it does not matter what the end
value is it could be positive or

00:12:56.139 --> 00:13:00.160
negative
the only time we flip the inequality is

00:13:00.160 --> 00:13:05.160
when we divide or multiply by a negative
number

00:13:08.160 --> 00:13:13.830
let's solve for x in this inequality
then express the solution in the three

00:13:13.830 --> 00:13:19.970
forms that we just learned so we would
solve for this as if it were an equation

00:13:19.970 --> 00:13:26.450
we want to isolate X so we first need to
add 8 to both sides

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negative eight plus eight is zero so we
are left with 2x negative four plus

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eight is four we bring down the
inequality now we divide by two

00:13:46.580 --> 00:13:55.430
2 divided by 2 is 1 so we have X on the
left 4 divided by 2 is positive 2 since

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we are not dividing by a negative number
so the inequality remains the same x is

00:14:01.550 --> 00:14:06.250
greater than 2
to graph this we marked you on the

00:14:06.250 --> 00:14:11.860
number line and just for the purpose of
convenience we do not need to label more

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numbers than two
X is greater than 2 so which side do you

00:14:17.940 --> 00:14:20.959
think we need to shade

00:14:21.180 --> 00:14:24.890
yes the right side

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now do we use a bracket or parentheses

00:14:32.330 --> 00:14:38.150
parentheses yes that's right because
there isn't an equal in the inequality

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please try and write the interval
notation yourself

00:14:43.390 --> 00:14:48.600
it should be two comma infinity

00:14:50.950 --> 00:14:59.550
and the set builder notation is the set
of all X such that X is greater than 2

00:14:59.550 --> 00:15:07.050
once again what goes after the bar is
the inequality there

00:15:10.100 --> 00:15:15.940
to solve for P in this inequality we
first need to combine the like terms

00:15:15.940 --> 00:15:23.260
negative 3p minus 4p is negative 7p

00:15:25.500 --> 00:15:30.710
then we divide both sides by negative
seven

00:15:32.050 --> 00:15:38.870
negative seven divided by negative seven
is just one so we have P on the left

00:15:38.870 --> 00:15:45.080
hand side 28 divided by negative 7 is
negative 4

00:15:45.080 --> 00:15:49.779
since we're dividing by a negative
number then we must flip the inequality

00:15:49.779 --> 00:15:57.980
so it becomes less than or equal to now
we graph this by marking negative 4 on

00:15:57.980 --> 00:16:02.390
the number line and which size should we
shade

00:16:02.390 --> 00:16:11.670
yes the left side because P is less than
less than or less than or equal to means

00:16:11.670 --> 00:16:18.930
shading to the left do we use a bracket
or parentheses here yes we use a bracket

00:16:18.930 --> 00:16:23.540
because there is an equal in the
inequality

00:16:23.540 --> 00:16:29.620
please pass the video and write in the
interval and set builder notation

00:16:29.930 --> 00:16:37.430
you should get negative infinity comma
negative for bracket at negative 4

00:16:37.430 --> 00:16:43.670
because we use a bracket in the graph
this also means that negative 4 is

00:16:43.670 --> 00:16:51.050
included for the set builder notation we
have the set of all P since the variable

00:16:51.050 --> 00:16:59.210
used in the problem is P such that we
copy this P is less than or equal to

00:16:59.210 --> 00:17:02.200
negative 4

00:17:07.300 --> 00:17:17.160
to self okay we need to combine 5k with
4k first that gives 9k

00:17:20.830 --> 00:17:26.340
now we divide both sides by 9 to get K
by itself

00:17:26.340 --> 00:17:35.460
negative 18 divided by 9 is negative 2 9
K divided by 9 is K

00:17:35.460 --> 00:17:42.970
do we flip the inequality no because
we're not dividing by a negative number

00:17:42.970 --> 00:17:48.340
now we should always repay first
although it is written on the right hand

00:17:48.340 --> 00:17:55.510
side this is actually K it's greater
than negative 2 or to make it simple we

00:17:55.510 --> 00:18:03.270
write K on the left so we have K is
greater than negative 2

00:18:03.590 --> 00:18:10.499
notice the inequality opens towards K so
when you change K to the left the

00:18:10.499 --> 00:18:16.109
inequality should still be opening
towards K the reason that I recommend

00:18:16.109 --> 00:18:21.149
that you write k on the left is because
it makes it easy to graph and to write

00:18:21.149 --> 00:18:24.049
the set builder notation

00:18:24.250 --> 00:18:30.730
we mark negative two on the number line
since K is greater than we shade to the

00:18:30.730 --> 00:18:36.010
right of negative two
since there is an equal in the

00:18:36.010 --> 00:18:42.640
inequality then we use parentheses
then for the interval notation we have

00:18:42.640 --> 00:18:49.540
the shaded region begins from negative
two and it goes infinitely to positive

00:18:49.540 --> 00:18:55.080
infinity we use parentheses around
negative two

00:18:55.080 --> 00:19:01.919
and for the set-builder notation we have
the set of all K since the variable used

00:19:01.919 --> 00:19:09.980
in the problem is K such that K is
greater than negative 2

00:19:14.290 --> 00:19:19.590
pass the video and try these problems

00:19:23.900 --> 00:19:25.960
you

00:19:29.000 --> 00:19:34.230
objective three solving linear
inequalities with variables on both

00:19:34.230 --> 00:19:40.250
sides
example soft graph the solution set and

00:19:40.250 --> 00:19:45.880
write the solution in interval and set
builder notation for this inequality

00:19:45.880 --> 00:19:51.800
notice we have the variable written on
both sides of the inequality so to solve

00:19:51.800 --> 00:19:57.800
for X we first need to gather all the X
terms to one side so we're going to

00:19:57.800 --> 00:20:01.300
subtract 2x from both sides

00:20:03.980 --> 00:20:10.160
negative 5x minus 2x is negative 7x

00:20:11.179 --> 00:20:19.960
two X minus two x is 0 so we are left
with 1 then we add 3 to both sides

00:20:21.730 --> 00:20:31.250
negative 3 plus 3 is 0 1 plus 3 is 4 and
now we divide by negative 7 to get X by

00:20:31.250 --> 00:20:38.780
itself we have X on the right 4 divided
by negative 7 is negative 4 over 7 and

00:20:38.780 --> 00:20:44.150
it is okay to get a fraction since we
are dividing by a negative number we

00:20:44.150 --> 00:20:50.300
need to flip this inequality so it
becomes less than we have X is less than

00:20:50.300 --> 00:20:56.249
negative 4 over 7
another way we can solve for x in this

00:20:56.249 --> 00:21:02.369
inequality is to bring X to the
right-hand side by adding 5x to both

00:21:02.369 --> 00:21:11.909
sides negative 5x plus 5 X is zero so we
have negative 3 is greater than 2x plus

00:21:11.909 --> 00:21:14.840
5x is 7x

00:21:15.060 --> 00:21:21.860
then subtract one from both sides so
that the X term is by itself on one side

00:21:21.860 --> 00:21:27.369
negative three minus one is negative
four

00:21:28.260 --> 00:21:34.510
one minus one is zero and now we divide
by seven

00:21:34.510 --> 00:21:41.490
negative 4 divided by 7 is negative 4/7
and then we get X on the right hand side

00:21:41.490 --> 00:21:47.910
since we are not dividing by a negative
number the inequality remains the same

00:21:47.910 --> 00:21:53.860
comparing the two inequalities we have
less than here and greater than over

00:21:53.860 --> 00:21:59.710
here however as I have mentioned in
objective 2 we should always read the

00:21:59.710 --> 00:22:06.460
variable first so this is read X is less
than negative 4 over 7 which is

00:22:06.460 --> 00:22:12.700
essentially the same as what we got on
the Left we can rewrite this to have X

00:22:12.700 --> 00:22:19.720
on the left so it should be X is less
than negative 4 over 7 so my

00:22:19.720 --> 00:22:26.080
recommendation is to always move X to
the left hand side so that it is easy to

00:22:26.080 --> 00:22:30.970
graph and to write the set builder
notation later

00:22:30.970 --> 00:22:38.040
now to graph this we mark negative 4/7
on the number line

00:22:38.920 --> 00:22:45.300
which side do we shade
yes the left-hand side since X is less

00:22:45.300 --> 00:22:47.600
than

00:22:49.389 --> 00:22:56.020
and do we use a bracket or parentheses
at negative 4/7

00:22:56.020 --> 00:23:03.730
yes parentheses because X is strictly
less than there isn't equal to in the

00:23:03.730 --> 00:23:09.100
inequality please pause the video and
write the interval and set builder

00:23:09.100 --> 00:23:12.179
notation yourself

00:23:12.670 --> 00:23:20.950
the interval notation should be negative
infinity comma negative 4/7 once again

00:23:20.950 --> 00:23:25.480
we look at the shaded part since the
graph goes infinitely to negative

00:23:25.480 --> 00:23:31.570
infinity then we have negative infinity
on the left and it ends at negative 4

00:23:31.570 --> 00:23:38.650
servants around negative infinity we use
parentheses and since negative 4/7 is

00:23:38.650 --> 00:23:44.830
not included we use parentheses which is
also what we used in the graph for the

00:23:44.830 --> 00:23:51.400
set builder notation we get the set of
all X such that X is less than negative

00:23:51.400 --> 00:23:55.050
4 servants

00:23:58.740 --> 00:24:04.559
notice we have the variable written on
both sides of the inequality so to solve

00:24:04.559 --> 00:24:11.390
this we need to gather them to one side
as I had mentioned before it is more

00:24:11.390 --> 00:24:16.040
convenient to move the variable to the
left hand side so we are going to

00:24:16.040 --> 00:24:21.550
subtract 8 Y from both sides

00:24:23.290 --> 00:24:33.220
why - 8 Y is negative 7 y bring down
negative 6 and the inequality 8y minus

00:24:33.220 --> 00:24:41.340
8y is 0 so we have 15 on the right
please pass the video and try to solve

00:24:41.340 --> 00:24:44.030
for y yourself

00:24:44.160 --> 00:24:52.710
let's go over this we add six to both
sides negative six plus six is zero

00:24:52.710 --> 00:25:04.170
so we have negative 7y 15 plus 6 is 21
then we divide by negative seven

00:25:04.170 --> 00:25:13.060
negative 7y divided by negative 7 is y
21 divided by negative 7 is negative 3

00:25:13.060 --> 00:25:19.060
and since we are dividing by a negative
number we flip the inequality so we get

00:25:19.060 --> 00:25:27.310
Y is greater than or equal to negative 3
let's graph this we mark negative 3 on

00:25:27.310 --> 00:25:32.500
the number line and we shade to the
right of negative 3 since Y is greater

00:25:32.500 --> 00:25:41.450
than an at negative 3 we use a bracket
because Y is also equal to negative 3

00:25:41.450 --> 00:25:47.330
based on the graph we get the interval
notation to be negative three comma

00:25:47.330 --> 00:25:54.260
infinity and we put a bracket around
negative three for the set builder

00:25:54.260 --> 00:26:00.820
notation we get the set of all Y since
the variable used in the problem is Y

00:26:00.820 --> 00:26:07.420
such that Y is greater than or equal to
negative three

00:26:07.420 --> 00:26:11.790
and this completes the problem

00:26:12.830 --> 00:26:18.100
pause the video and try these problems

00:26:22.700 --> 00:26:24.760
you

00:26:28.220 --> 00:26:33.530
objective for
solving linear inequalities by using the

00:26:33.530 --> 00:26:36.250
distributive property

00:26:37.130 --> 00:26:42.860
we would solve for this as if it were an
equation so to solve for x in this

00:26:42.860 --> 00:26:49.160
inequality we first need to distribute
five to remove the parentheses so we

00:26:49.160 --> 00:26:58.750
take five times seven which gives 35
five times 2x is 10x

00:26:58.750 --> 00:27:05.500
then bring down the terms on the right
side now we gather the terms involving X

00:27:05.500 --> 00:27:11.880
to the left-hand side
so we add X to both sides

00:27:14.200 --> 00:27:19.499
ten X plus one X is 11 X

00:27:20.080 --> 00:27:26.430
negative x plus x is zero so we get 8 on
the right hand side

00:27:26.430 --> 00:27:32.280
please pause the video and try to finish
the problem yourself restart the video

00:27:32.360 --> 00:27:39.800
when you are ready to check your answer
let's go over this so we want to

00:27:39.800 --> 00:27:45.910
subtract 35 from both sides
so that we have the variable on one side

00:27:45.910 --> 00:27:56.080
and constant on the other 35 - 35 is
zero so we get 11 X on the left 8 minus

00:27:56.080 --> 00:28:04.450
35 is negative 27 then we divide both
sides by 11 to get X by itself

00:28:04.450 --> 00:28:11.549
we get X is greater than or equal to
negative 27 over 11

00:28:11.549 --> 00:28:16.869
notice that the inequality stays the
same because we're not dividing by a

00:28:16.869 --> 00:28:24.489
negative number now to graph this we
mark negative 27 over 11 on the number

00:28:24.489 --> 00:28:29.259
line and it's okay to get a fraction
whatever the number we get which is

00:28:29.259 --> 00:28:36.159
market on the number line we shade to
the right side of negative 27 over 11

00:28:36.159 --> 00:28:42.129
because X is greater than we put a
bracket at this endpoint since X does

00:28:42.129 --> 00:28:48.010
equal to negative 27 over 11
then we look at the graph to get the

00:28:48.010 --> 00:28:54.490
interval notation it starts from
negative 27 over 11 and it goes

00:28:54.490 --> 00:29:02.590
infinitely to positive infinity
for the set-builder notation we get the

00:29:02.590 --> 00:29:08.890
set of all X such that X is greater than
or equal to negative 27

00:29:08.890 --> 00:29:11.910
over 11

00:29:12.550 --> 00:29:16.740
and this completes the problem

00:29:19.429 --> 00:29:24.589
the first thing we need to do in this
inequality is to distribute to remove

00:29:24.589 --> 00:29:29.779
the parentheses there is an invisible
negative one in front of this

00:29:29.779 --> 00:29:37.969
parenthesis so we have negative 1 times
4x is negative 4x negative 1 times

00:29:37.969 --> 00:29:42.309
negative 2 is plus 2

00:29:42.509 --> 00:29:48.469
on the right hand side take negative 5
times both terms inside the parentheses

00:29:48.469 --> 00:29:56.429
negative 5 times X is negative 5 X
negative 5 times 1 is negative 5 then

00:29:56.429 --> 00:30:02.610
bring this negative 5 down what do you
think we need to do next

00:30:02.610 --> 00:30:10.759
yes we need to combine the like terms
8 plus 2 is 10

00:30:12.020 --> 00:30:16.250
over here we combine negative five with
negative five since there are both

00:30:16.250 --> 00:30:21.070
constant and they give negative ten

00:30:21.720 --> 00:30:26.220
now we gather the terms involving X to
the left-hand side

00:30:26.220 --> 00:30:31.490
so we add 5x to both sides

00:30:32.480 --> 00:30:37.779
negative 4x plus 5 X is 1x

00:30:38.539 --> 00:30:44.979
negative five x plus five x is zero so
we get negative ten on the right side

00:30:44.979 --> 00:30:53.380
now we subtract 10 from both sides
ten minus ten is zero so we get X is

00:30:53.380 --> 00:30:59.580
less than negative ten minus 10 is
negative 20

00:30:59.580 --> 00:31:05.820
we graph this by marking negative 20 on
the number line and we shade to the left

00:31:05.820 --> 00:31:13.200
of negative 20 since X is less than and
we use a parenthesis at negative 20

00:31:13.200 --> 00:31:20.820
since X is strictly less than negative
20 the graph goes infinitely to negative

00:31:20.820 --> 00:31:27.139
infinity on the left and it ends at
negative 20

00:31:27.549 --> 00:31:34.539
and the set builder notation is the set
of all X such that X is less than

00:31:34.539 --> 00:31:37.830
negative 20

00:31:42.720 --> 00:31:47.930
for this last example please pause the
video and try the problem yourself

00:31:47.930 --> 00:31:53.650
restart the video when you are ready to
check your answer

00:31:53.650 --> 00:32:00.309
let's go over this so you see here we
need to first distribute on both sides

00:32:00.309 --> 00:32:11.080
so we take 3 times X to get 3x 3 times
negative 5 to get negative 15 2 times 2x

00:32:11.080 --> 00:32:18.820
is 4x 2 times negative 1 is negative 2
and then we subtract 4x from both sides

00:32:18.820 --> 00:32:23.280
so that now we have the X term on the
Left

00:32:25.559 --> 00:32:36.010
3x minus 4x is negative X bring down the
negative 15 4x minus 4x is 0 so we get

00:32:36.010 --> 00:32:43.170
just negative 2 on the right now add 15
to both sides

00:32:43.500 --> 00:32:52.000
negative 15 plus 15 is zero so we get
negative X on the Left negative 2 plus

00:32:52.000 --> 00:32:58.660
15 is 13
there is an invisible negative one in

00:32:58.660 --> 00:33:04.020
front of X so we divide by negative one

00:33:04.220 --> 00:33:11.970
negative X divided by negative 1 is
positive X 13 divided by negative 1 is

00:33:11.970 --> 00:33:18.470
negative 13 and since we are dividing by
a negative number we flip the inequality

00:33:18.470 --> 00:33:25.230
so it becomes X is greater than or equal
to negative 13

00:33:25.230 --> 00:33:32.610
to graph this we mark negative 13 on the
number line and we shade to the right

00:33:32.610 --> 00:33:41.380
side because X is greater than and we
use a bracket at negative 13

00:33:41.380 --> 00:33:47.440
we look at the graph to get the interval
notation it starts from negative 13 and

00:33:47.440 --> 00:33:53.610
it goes infinitely to positive infinity
on the right

00:33:53.799 --> 00:34:01.779
and finally the set builder notation is
the set of all X such that X is greater

00:34:01.779 --> 00:34:06.570
than or equal to negative 13

00:34:10.660 --> 00:34:16.180
pause the video and try these problems