Direct inverse and joint variation

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.700
Determine whether the data in the table is

00:00:02.700 --> 00:00:06.970
an example of direct inverse or joint
variation.

00:00:06.970 --> 00:00:10.720
Then identify the equation that represents
the relationship.

00:00:10.720 --> 00:00:14.730
So let's just think about what direct
inverse or joint variation even means.

00:00:14.730 --> 00:00:17.360
So if you have direct variation.

00:00:17.360 --> 00:00:18.870
Direct variation.

00:00:18.870 --> 00:00:23.840
So if y varied directly with x it
literally means that y is equal to some

00:00:23.840 --> 00:00:29.290
constant multiple of x, or if you divide
both sides of this by x it means that

00:00:29.290 --> 00:00:35.020
y over x is equal to k so the ration
between y and x is a constant.

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And you could go the other way around.

00:00:36.660 --> 00:00:39.000
You could also say that x is equal to some

00:00:39.000 --> 00:00:42.710
constant not, not going to be the same
constant times y.

00:00:42.710 --> 00:00:47.340
Or that x over y is going to be equal to
some other constant.

00:00:47.340 --> 00:00:49.350
So these aren't necessarily the same k.

00:00:49.350 --> 00:00:52.240
All I'm just saying is that it's a
constant relationship.

00:00:52.240 --> 00:00:55.330
These are all examples of direct
variation.

00:00:55.330 --> 00:00:58.350
In dir, or I should say inverse variation
is to

00:00:58.350 --> 00:01:02.390
some degree the opposite depending on how
you view the opposite.

00:01:02.390 --> 00:01:03.730
And before I even talk about that, let's

00:01:03.730 --> 00:01:06.660
think about the telltale signs of direct
variation.

00:01:06.660 --> 00:01:10.140
If x increases, y should increase.

00:01:10.140 --> 00:01:11.480
So if x increases.

00:01:11.480 --> 00:01:13.710
Let me do that in the same yellow.

00:01:13.710 --> 00:01:16.764
So the telltale signs of direct variation,
if

00:01:16.764 --> 00:01:19.988
x increases then y will increase and vice
versa.

00:01:19.988 --> 00:01:21.714
The other telltale sign is.

00:01:21.714 --> 00:01:25.600
Is if you increase x by some, by some
factor.

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So, if you have x going to 3x then y
should also increase by that same factor.

00:01:33.880 --> 00:01:35.500
And we could see that with some examples.

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So, I mean, you could pick a K, let's say
that, let's say that K was one.

00:01:39.290 --> 00:01:42.868
So if y is equal to x, if you take, if x
goes from

00:01:42.868 --> 00:01:45.340
one to three, then y is also going to go
from one to three.

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So that's all we're talking about here.

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Let me actually, y should actually to

00:01:49.365 --> 00:01:52.000
three times y, that's what I'm talking
about.

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If you triple x, you're also gonna end up
tripling y.

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Inverse variation.

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You have y being equal to some constant
times one over x.

00:02:02.790 --> 00:02:06.650
So instead of an x here you have a one
over x or if you

00:02:06.650 --> 00:02:13.100
multiply both sides by x you get x times y
is equal to some constant.

00:02:13.100 --> 00:02:17.790
And you could switch the x's and the y's
around as well for inverse variation.

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Now what are the tale tale signs?

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Well if you increase x, if x goes up, then
what happens to y?

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If x goes up then this becomes a smaller
value

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cuz it's one over x so then y will go
down.

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Then y will go down.

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And if you take X and if you're to say
increase it

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by a factor of three then what's going to
happen to Y?

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Well if you increase this by a factor of

00:02:43.685 --> 00:02:47.570
three, you're actually going to decrease
this whole value by

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a factor of one-third, so Y is going to

00:02:50.140 --> 00:02:53.820
go, so then you're going to have one-third
of y.

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So that's, these are the tell-tale signs
for inverse variation.

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Now finally they talk about something
called joint variation, and

00:03:00.840 --> 00:03:03.780
this one you won't necessarily see in
introductory algebra course.

00:03:03.780 --> 00:03:06.520
But joint variation deals with more than
one variable.

00:03:06.520 --> 00:03:12.680
So if I told you, if I told you that area
of a rectangle is equal to the width of

00:03:12.680 --> 00:03:17.950
a rectangle times the length of rectangle,
this is an example of joint variation.

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Area is proportionally to two.

00:03:20.750 --> 00:03:23.540
Is the proportional to two different
quantities?

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So the main tell tale sign here for joint
variation

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frankly is you're gonna be dealing with
more than two variables.

00:03:28.786 --> 00:03:34.000
Joint, Joint, Joint variation.

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So when you look at this example, there
only gi, giving us two variables.

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So you can rule out joint variation just
right from the get go.

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Now let's look at the tell tale signs.

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So as x is increasing, as x goes from one
to two, what is happening to y?

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Y went from 12 to six.

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So as x is going up by a factor of two,

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y is going, is, is going by a factor of
one half.

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Or y is being multiplied by one half.

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So as x goes from one to three, it's being
multiplied by three.

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Y is being multiplied or I guess could say
is, is multiplied by one-third.

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So it's definitely not direct variation.

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As x increases, y is decreasing.

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So it's definitely not direct variation.

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And then really you can just rule out,
since we rule out the

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other two, you can probably guess this is
going to be inverse variation.

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But we can validate it.

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When X increases, Y is decreasing.

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When X increases by a certain factor, Y is

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increasing by one over that factor, which
is actually decreasing.

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So, if you go from one to three, if X is
being

00:04:34.040 --> 00:04:40.030
multiplied by three, then Y essentially
becomes one third of its original value.

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When X is one Y is 12 when X is three Y is
four so we have inverse variation in play.

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Now they ask us identify the equation that
represents the relationship.

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Well we know with inverse variation the
product of

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x and y need to be equal to some constant.

00:04:59.650 --> 00:05:02.280
So that if we take x times y over here so
lets just

00:05:02.280 --> 00:05:06.715
multiply lets make another column here
call this the x times y column.

00:05:06.715 --> 00:05:08.000
One times 12 is 12.

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Two times six is 12.

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Three times four is 12.

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Four times three is 12.

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So, clearly in every situation, x times y
is, is a constant and it is 12.

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So, the equation that represents the
relationship, it is, X,

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Y is equal to 12 and that is clearly an
inverse


Another Take on the Rate Problem

WEBVTT
Kind: captions
Language: en

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I want to do the same problem
that I did in the last video.

00:00:02.930 --> 00:00:04.410
But I want to show you
that there's multiple ways

00:00:04.410 --> 00:00:06.679
to solve these problems that
should hopefully give you

00:00:06.679 --> 00:00:08.720
the right answer if you
do them in the right way.

00:00:08.720 --> 00:00:10.920
So once again, one
pipe can fill a pool

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1 and 1/2 times faster
than a second pipe.

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If both pipes are open, the
pool can be filled in 6 hours.

00:00:17.850 --> 00:00:20.130
If only the slower
pipe is open, how long

00:00:20.130 --> 00:00:22.080
would it take to fill the pool?

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So as I said in the last video,
whenever you do these problems,

00:00:24.920 --> 00:00:27.400
it's very important to
think in terms of rate.

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The rate at which
the slower pipe

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can fill a pool and the
rate at which the faster

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pipe can fill a pool.

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Right over here,
I'm going to define

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that the slower pipe
can fill a pool in r.

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So its rate is r pools per hour.

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So if its rate is r pools
per hour-- and r for rate--

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what is the faster pool's rate?

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The faster pool's rate is
going to be 1.5 times r.

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It's 1 and 1/2 times faster.

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It has 1 and 1/2 times the rate.

00:01:00.020 --> 00:01:05.950
So it's going to be able
do 1.5r pools per hour.

00:01:05.950 --> 00:01:07.640
And what's their combined rate?

00:01:07.640 --> 00:01:09.080
Well, you can add the rates.

00:01:09.080 --> 00:01:12.980
r plus 1.5r-- let me
write this in orange--

00:01:12.980 --> 00:01:20.101
so if you say their combined
rate is r plus 1.5--

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what am I doing-- plus
1.5r pools per hour,

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which is equal to
2.5r pools per hour.

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And they're also telling us
that their combined rate,

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if both pipes are open, the
pool can be filled in 6 hours.

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So they're telling
us that the combined

00:01:47.310 --> 00:01:51.380
rate is 1 pool in every 6 hours.

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So they're saying that
the combined rate is also

00:01:56.030 --> 00:02:01.690
equal to 1 pool in every 6
hours or 1/6 of a pool per hour.

00:02:01.690 --> 00:02:07.740
1 pool in every 6 hours
is the same thing--

00:02:07.740 --> 00:02:11.120
so that's this line right over
here-- 1 pool in every 6 hours

00:02:11.120 --> 00:02:13.870
is the same thing as
1/6 of a pool per hour.

00:02:16.860 --> 00:02:20.460
And then, once again, we
have 2.5r is equal to 1/6.

00:02:20.460 --> 00:02:22.760
We can multiply both sides.

00:02:22.760 --> 00:02:24.810
So let me do it this way.

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We can multiply both sides by 6.

00:02:27.360 --> 00:02:29.600
Let me just rewrite it so
I have some real estate.

00:02:29.600 --> 00:02:31.830
2.5r.

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The units are the
same on both sides,

00:02:33.610 --> 00:02:35.630
so really this needs
to be equal to this.

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So 2.5r needs to
be equal to 1/6.

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You can multiply
both sides by 6.

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This just becomes 1.

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This right over here,
6 times 2.5 is 15.

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So you get 15r is equal to
1 or r is equal to 1/15.

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And let's see how
that might make sense.

00:02:55.280 --> 00:02:59.070
So we know that the slower
pool is r pools per hour.

00:02:59.070 --> 00:03:02.510
Or we could say it
is 1/15 of a pool.

00:03:06.080 --> 00:03:09.670
1/15 of a pool per hour.

00:03:09.670 --> 00:03:15.210
Or you could view it as to do
one pool, it takes 15 hours.

00:03:15.210 --> 00:03:18.160
So once again, to answer our
question, if only the slower

00:03:18.160 --> 00:03:21.730
pipe is open, how
long would it take?

00:03:21.730 --> 00:03:23.240
15 hours.

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So once again, the
exact same problem we

00:03:25.016 --> 00:03:27.390
did in the last video,
just a different way.

00:03:27.390 --> 00:03:28.974
And frankly, a
little bit simpler.

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Although, I went into a
little bit more detail

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in the last video
so I don't know

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if I can make that comparison.


Direct variation application

WEBVTT
Kind: captions
Language: en

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So, in this problem, they're telling us in
outer space, the distance

00:00:03.260 --> 00:00:08.160
an object travels varies directly with the
amount of time that it travels.

00:00:08.160 --> 00:00:10.340
And, that's of course assuming that it's
not accelerating, and

00:00:10.340 --> 00:00:12.010
there's no net force, and all of that on
it.

00:00:12.010 --> 00:00:14.030
So, I guess they're talking about a
specific object.

00:00:14.030 --> 00:00:18.560
So, some specific object, the amount, the
distance that it travels is directly,

00:00:18.560 --> 00:00:23.810
it varies directly, it varies directly
with the amount of time that it travels.

00:00:23.810 --> 00:00:26.178
So, if we think of in, in, in terms of

00:00:26.178 --> 00:00:30.991
constants of proportionality, and direct
variation, we could say that the

00:00:30.991 --> 00:00:34.249
distance, we could say that the distance
is equal to

00:00:34.249 --> 00:00:38.670
some constant, times the time, times the
time that it travels.

00:00:38.670 --> 00:00:42.228
The distance varies directly with the
amount

00:00:42.228 --> 00:00:45.500
of time that travels for this particular
object.

00:00:45.500 --> 00:00:49.960
If an asteroid travels 3000 miles, in, if
the asteroid travels

00:00:49.960 --> 00:00:55.060
3000 miles in 6 hours, what is the
constant of variation?

00:00:55.060 --> 00:01:01.646
So the distance is 3000, so we have d is
equal to 3000 miles.

00:01:01.646 --> 00:01:05.730
We have 3,000 miles is the distance, and
that's

00:01:05.730 --> 00:01:08.540
going to be equal to the constant of
variation.

00:01:08.540 --> 00:01:12.900
The constant of variation times the time,
times 6 hours.

00:01:12.900 --> 00:01:14.500
So, if we wanna solve for the constant of

00:01:14.500 --> 00:01:17.720
variation, we can just divide both sides
by 6 hours.

00:01:17.720 --> 00:01:23.978
6 hours, and we divide the right-hand side
by 6 hours.

00:01:23.978 --> 00:01:31.070
And, so, 3,000 divided by 6 is 500, and 6
divided by 6 is 1.

00:01:31.070 --> 00:01:34.050
The hours also cancel out if you care
about the units.

00:01:34.050 --> 00:01:39.270
And, so, the concept of proportionality,
the left-hand side is just

00:01:39.270 --> 00:01:44.440
500, 500, and then we have miles per hour,
miles per hour.

00:01:44.440 --> 00:01:48.760
Fired 500 miles per hour, and that is
equal to k.

00:01:48.760 --> 00:01:51.830
So, the constant proportionality is 500
miles per hour, or you

00:01:51.830 --> 00:01:55.890
could say 500 if you're not too worried
about the units.

00:01:55.890 --> 00:01:57.650
Or, we should say, the constant of
variation, to

00:01:57.650 --> 00:02:00.110
use the terminology that they actually use
in the question.


Proportionality constant for direct variation

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.130
y is directly proportional to x.

00:00:04.130 --> 00:00:06.700
If y equals 30 when
x is equal to 6,

00:00:06.700 --> 00:00:10.130
find the value of
x when y is 45.

00:00:10.130 --> 00:00:12.480
So let's just take this
each statement at a time.

00:00:12.480 --> 00:00:15.080
y is directly proportional to x.

00:00:15.080 --> 00:00:16.960
That's literally
just saying that y

00:00:16.960 --> 00:00:20.480
is equal to some
constant times x.

00:00:20.480 --> 00:00:22.360
This statement can
literally be translated

00:00:22.360 --> 00:00:25.315
to y is equal to some
constant times x.

00:00:25.315 --> 00:00:27.770
y is directly proportional to x.

00:00:27.770 --> 00:00:36.060
Now, they tell us, if
y is 30 when x is 6--

00:00:36.060 --> 00:00:38.420
and we have this constant
of proportionality--

00:00:38.420 --> 00:00:40.390
this second statement
right over here

00:00:40.390 --> 00:00:42.000
allows us to solve
for this constant.

00:00:42.000 --> 00:00:44.730
When x is 6, they
tell us y is 30

00:00:44.730 --> 00:00:46.740
so we can figure out
what this constant is.

00:00:46.740 --> 00:00:52.530
We can divide both
sides by 6 and we

00:00:52.530 --> 00:00:56.210
get this left-hand side is
5-- 30 divided by 6 is 5.

00:00:56.210 --> 00:00:58.830
5 is equal to k or
k is equal to 5.

00:00:58.830 --> 00:01:04.030
So the second sentence tells us,
this gives us the information

00:01:04.030 --> 00:01:06.930
that y is not just k
times x, it tells us

00:01:06.930 --> 00:01:12.330
that y is equal to 5 times
x. y is 30 when x is 6.

00:01:12.330 --> 00:01:17.810
And then finally, they say, find
the value of x when y is 45.

00:01:17.810 --> 00:01:23.010
So when y is 45 is
equal to-- so we're just

00:01:23.010 --> 00:01:27.580
putting in 45 for y--
45 is equal to 5x.

00:01:27.580 --> 00:01:30.690
Divide both sides
by 5 to solve for x.

00:01:30.690 --> 00:01:36.830
We get 45 over 5 is 9, and
5x divided by 5 is just x.

00:01:36.830 --> 00:01:42.804
So x is equal to 9 when y is 45.


Adding and subtracting rational expressions

WEBVTT
Kind: captions
Language: en

00:00:00.574 --> 00:00:03.041
- [Voiceover] Perform
the indicated operations,

00:00:03.041 --> 00:00:04.695
then simplify.

00:00:04.695 --> 00:00:07.078
So we have three rational
expressions here,

00:00:07.078 --> 00:00:08.286
and we're gonna subtract this one,

00:00:08.286 --> 00:00:10.004
and we're going to add this one.

00:00:10.004 --> 00:00:11.584
So just like if we were adding

00:00:11.584 --> 00:00:13.001
and subtracting a bunch of fractions,

00:00:13.001 --> 00:00:15.823
we really have to find a
common denominator here.

00:00:15.823 --> 00:00:17.227
And to figure out a common denominator,

00:00:17.227 --> 00:00:18.679
we have to figure out
the least common multiple

00:00:18.679 --> 00:00:21.304
of x plus five, of 2x minus 10,

00:00:21.304 --> 00:00:23.907
and x-squared minus 25.

00:00:23.907 --> 00:00:25.716
And to find the least common multiple,

00:00:25.716 --> 00:00:27.161
we really want to factor these out

00:00:27.161 --> 00:00:29.125
and make sure that that
least common multiple,

00:00:29.125 --> 00:00:30.355
that that common denominator,

00:00:30.355 --> 00:00:33.223
has all of the constituent factors in it.

00:00:33.223 --> 00:00:34.872
So this can't be factored anymore.

00:00:34.872 --> 00:00:37.465
X plus five, that's about
as simple as you can get it.

00:00:37.465 --> 00:00:38.869
2x minus 10.

00:00:38.869 --> 00:00:40.674
We can factor out a two, and it becomes

00:00:40.674 --> 00:00:44.070
two times x minus 5.

00:00:44.070 --> 00:00:45.515
So we factor that out.

00:00:45.515 --> 00:00:47.539
And then x-squared minus 25

00:00:47.539 --> 00:00:50.082
you might recognize as
a difference of squares,

00:00:50.082 --> 00:00:51.963
so that's the same thing as

00:00:51.963 --> 00:00:56.237
x plus five times x minus five.

00:00:56.237 --> 00:00:58.282
Just factored that out.

00:00:58.282 --> 00:01:01.603
And so if we want to have
a least common multiple,

00:01:01.603 --> 00:01:03.300
so let me rewrite each of these

00:01:03.300 --> 00:01:04.912
with that common denominator here,

00:01:04.912 --> 00:01:06.700
that the least common multiple,

00:01:06.700 --> 00:01:07.993
the least common multiple

00:01:09.472 --> 00:01:12.142
is going to have to be something
that has x plus five in it,

00:01:12.142 --> 00:01:13.477
it's gonna have to have a two in it,

00:01:13.477 --> 00:01:15.802
it's gonna have an x minus five in it,

00:01:15.802 --> 00:01:17.083
and we've already said it has to have an

00:01:17.083 --> 00:01:19.288
x plus five and an x minus five in it.

00:01:19.288 --> 00:01:20.786
So it's gotta have a two.

00:01:20.786 --> 00:01:22.144
It's gotta have a two.

00:01:22.144 --> 00:01:23.723
It's gotta have an x plus five.

00:01:25.743 --> 00:01:27.565
It's gotta have an x minus five in it.

00:01:30.078 --> 00:01:32.574
And it's gotta have an x plus
five times an x minus five.

00:01:32.574 --> 00:01:33.815
Well, we've already got that there,

00:01:33.815 --> 00:01:35.429
so no need to write it again.

00:01:35.429 --> 00:01:38.422
So that right there is
our common denominator,

00:01:38.422 --> 00:01:39.653
so let's write it out.

00:01:39.653 --> 00:01:41.424
So let's rewrite each of these expressions

00:01:41.424 --> 00:01:42.596
with that common denominator.

00:01:42.596 --> 00:01:45.197
So two times x plus five

00:01:45.197 --> 00:01:47.440
times x minus five.

00:01:47.440 --> 00:01:50.075
And then right over here
I'll write it again.

00:01:50.075 --> 00:01:52.339
Two times x plus five

00:01:52.339 --> 00:01:54.370
times x minus five.

00:01:54.370 --> 00:01:55.113
Once you get used to this,

00:01:55.113 --> 00:01:57.762
you might be able to just write
out the common denominator

00:01:57.762 --> 00:02:00.202
once and then figure out the appropriate

00:02:00.202 --> 00:02:02.042
things to multiply the numerators by,

00:02:02.042 --> 00:02:04.006
but I want to make sure
we go through every step

00:02:04.006 --> 00:02:05.259
for these videos just so we make sure

00:02:05.259 --> 00:02:07.338
we understand what's going on.

00:02:07.338 --> 00:02:10.675
So to go from x plus five
to two times x plus five

00:02:10.675 --> 00:02:11.881
times x minus five.

00:02:11.881 --> 00:02:14.656
What do we have to
multiply x plus five by?

00:02:14.656 --> 00:02:16.726
Well, we have to multiply
it in order to get at this.

00:02:16.726 --> 00:02:19.547
We're multiplying it by a two.

00:02:19.547 --> 00:02:21.486
We're multiplying the
x plus five by a two,

00:02:21.486 --> 00:02:23.645
and we're also multiplying it by

00:02:23.645 --> 00:02:25.526
an x minus five.

00:02:25.526 --> 00:02:27.753
So if we're multiplying
the denominator by a two

00:02:27.753 --> 00:02:29.031
and an x minus five,

00:02:29.031 --> 00:02:30.895
we have multiply the numerator by a two

00:02:31.988 --> 00:02:33.466
and an x minus five.

00:02:33.466 --> 00:02:36.973
So if we multiply two times
x minus five times one,

00:02:36.973 --> 00:02:40.014
we'll get two times x minus five,

00:02:40.014 --> 00:02:42.034
and we don't even have to write this one.

00:02:42.034 --> 00:02:44.344
Now we're subtracting over here.

00:02:44.344 --> 00:02:48.211
And to get from two times x
minus five to this over here,

00:02:48.211 --> 00:02:49.964
what to we have to multiply it by?

00:02:49.964 --> 00:02:53.658
Well, we have the two times x
minus five right over there.

00:02:53.658 --> 00:02:55.954
We have to multiply it by an x plus five.

00:02:55.954 --> 00:02:57.788
This is the part that is new.

00:02:57.788 --> 00:03:00.830
We have to multiply it by an x plus five.

00:03:00.830 --> 00:03:02.409
We have to multiply this
denominator right here

00:03:02.409 --> 00:03:04.509
by x plus five to get this denominator.

00:03:04.509 --> 00:03:07.191
So if you multiply the
denominator by x plus five,

00:03:07.191 --> 00:03:09.675
you want to multiply the
numerator by x plus five as well.

00:03:09.675 --> 00:03:12.915
So the numerator's going to
be two times x plus five.

00:03:12.915 --> 00:03:15.613
Just to put two, multiplied
it by x plus five.

00:03:15.613 --> 00:03:16.940
Exactly what we did to the denominator.

00:03:16.940 --> 00:03:19.274
We multiplied it by x plus five.

00:03:19.274 --> 00:03:23.361
Then we have this last
expression right over here.

00:03:25.491 --> 00:03:27.282
I'm going to do this in a new color.

00:03:27.282 --> 00:03:29.870
We went from x plus
five times x minus five

00:03:29.870 --> 00:03:32.099
to x plus five times x minus five

00:03:32.099 --> 00:03:34.074
times two.

00:03:34.074 --> 00:03:35.944
So to go from this denominator
to this denominator

00:03:35.944 --> 00:03:37.834
we just have to multiply it by a two.

00:03:37.834 --> 00:03:39.728
If you multiplied the denominator by two,

00:03:39.728 --> 00:03:41.951
you should multiply the numerator by two.

00:03:41.951 --> 00:03:44.634
So multiply two times 2x.

00:03:44.634 --> 00:03:47.746
You're going to get 4x.

00:03:47.746 --> 00:03:51.784
So all I did here is I
multiplied it by two.

00:03:51.784 --> 00:03:53.037
Let me make it clear.

00:03:53.037 --> 00:03:54.883
To get from this to this,

00:03:56.313 --> 00:03:59.450
I multiplied it by x plus five.

00:03:59.450 --> 00:04:00.529
X plus five.

00:04:02.709 --> 00:04:04.916
And then to go from there to there,

00:04:04.916 --> 00:04:09.176
I multiplied it by two times x minus five.

00:04:09.176 --> 00:04:11.022
And over here I should use,

00:04:11.022 --> 00:04:12.537
instead of writing multiplied
by two right there,

00:04:12.537 --> 00:04:13.763
I should multiply it by two.

00:04:13.763 --> 00:04:15.643
Don't want to get the
multiplication symbol confused

00:04:15.643 --> 00:04:17.547
with the x variable.

00:04:17.547 --> 00:04:20.182
And of course, there
was a positive out here.

00:04:20.182 --> 00:04:23.725
So we're now ready to do
our adding and subtracting.

00:04:23.725 --> 00:04:26.151
And we could view this
just for simplicity,

00:04:26.151 --> 00:04:27.150
just make it clear.

00:04:27.150 --> 00:04:29.170
Subtracting this thing is the same thing

00:04:29.170 --> 00:04:31.516
as adding the negative of it.

00:04:31.516 --> 00:04:33.048
And that makes it a little bit simpler.

00:04:33.048 --> 00:04:35.672
So let's put our common denominator down.

00:04:35.672 --> 00:04:37.750
Let's put our common denominator down.

00:04:37.750 --> 00:04:39.422
So all of this business is going to be,

00:04:39.422 --> 00:04:42.720
and our denominator's going
to be two times x plus five

00:04:42.720 --> 00:04:44.984
times x minus five.

00:04:44.984 --> 00:04:47.365
And our numerator is now going to be,

00:04:47.365 --> 00:04:49.176
and I'm going to multiply this out,

00:04:49.176 --> 00:04:51.325
so that we can properly add terms.

00:04:51.325 --> 00:04:55.002
So two times x minus five is 2x minus 10.

00:04:55.972 --> 00:05:00.058
And then we have a negative
two times x plus five,

00:05:00.058 --> 00:05:02.942
so that's going to be negative 2x,

00:05:02.942 --> 00:05:06.251
and then negative two
times five is negative 10.

00:05:06.251 --> 00:05:09.017
And then we're going to have a plus 4x.

00:05:09.017 --> 00:05:11.572
Plus 4x over here.

00:05:11.572 --> 00:05:15.046
Plus 4x, and then we
can simplify this thing.

00:05:15.046 --> 00:05:16.923
And so you have a 2x.

00:05:16.923 --> 00:05:18.386
You're subtracting a 2x.

00:05:18.386 --> 00:05:19.988
Those guys cancel out.

00:05:22.268 --> 00:05:24.427
And then you're left with just a 4x.

00:05:24.427 --> 00:05:26.586
So you're left with just a 4x,

00:05:26.586 --> 00:05:28.560
and then minus 10 minus 10.

00:05:28.560 --> 00:05:30.594
So 4x minus 20

00:05:32.135 --> 00:05:35.510
over two times x plus five

00:05:35.510 --> 00:05:37.794
times x minus five.

00:05:39.555 --> 00:05:42.555
Let's see, we can factor
out a four up here.

00:05:42.555 --> 00:05:44.088
So if we factor out a four,

00:05:44.088 --> 00:05:48.790
this becomes four times x minus five.

00:05:50.770 --> 00:05:53.048
If we assume that we're not dealing

00:05:53.048 --> 00:05:54.813
with x is equal to positive five,

00:05:54.813 --> 00:05:57.840
'cause that would make
the denominator undefined,

00:05:57.840 --> 00:06:00.878
we can divide the numerator
and the denominator

00:06:00.878 --> 00:06:02.642
by x minus five.

00:06:02.642 --> 00:06:04.720
And so if we divide the
numerator and the denominator

00:06:04.720 --> 00:06:07.298
by x minus five, those cancel out.

00:06:07.298 --> 00:06:09.294
And we can divide the numerator
and the denominator by two,

00:06:09.294 --> 00:06:11.237
since they're both divisible by two.

00:06:11.237 --> 00:06:12.812
That four divided by two is two.

00:06:12.812 --> 00:06:14.832
Two divided by two is one.

00:06:14.832 --> 00:06:17.766
So the numerator, we're
just left with a two,

00:06:17.766 --> 00:06:21.435
and in the denominator, we're
just left with an x plus five.

00:06:21.435 --> 00:06:23.548
We're just left with an x plus five.

00:06:23.548 --> 00:06:26.951
Now, I want to be very careful here,

00:06:26.951 --> 00:06:29.552
because this expression up here,

00:06:29.552 --> 00:06:33.174
if we really want this
final answer that we got

00:06:33.174 --> 00:06:36.808
to be truly identical to
this expression over here,

00:06:36.808 --> 00:06:38.747
just a simplified version,

00:06:38.747 --> 00:06:41.290
we have to make sure that
it has the same domain.

00:06:41.290 --> 00:06:43.344
If say this was part of
the function definition,

00:06:43.344 --> 00:06:44.931
you want to make sure that it's valid

00:06:44.931 --> 00:06:46.604
for the same x's and not any more,

00:06:46.604 --> 00:06:48.544
if you want them to be truly identical.

00:06:48.544 --> 00:06:51.081
And this up here is not valid.

00:06:51.081 --> 00:06:54.599
So the domain if you viewed
this as one entire expression,

00:06:54.599 --> 00:06:56.422
or one entire function definition,

00:06:57.742 --> 00:06:59.088
the domain would be

00:07:00.486 --> 00:07:02.679
all real x's,

00:07:04.726 --> 00:07:05.568
except

00:07:07.465 --> 00:07:10.078
x cannot be equal to negative five,

00:07:10.078 --> 00:07:13.848
or x cannot be equal to positive five,

00:07:13.848 --> 00:07:16.332
'cause that would make this
denominator equal zero,

00:07:16.332 --> 00:07:18.004
or x is equal to five.

00:07:18.004 --> 00:07:21.069
And so if you want these
two things to be identical,

00:07:21.069 --> 00:07:23.044
they have to have the same domain.

00:07:24.324 --> 00:07:25.727
Most algebra classes,

00:07:25.727 --> 00:07:28.062
people will not be particular about this.

00:07:28.062 --> 00:07:30.446
And maybe you don't want
them to be the same function.

00:07:32.161 --> 00:07:33.539
Maybe you want it to be defined

00:07:33.539 --> 00:07:35.780
when x is equal to a positive five.

00:07:35.780 --> 00:07:38.114
But if you want them to be
truly the same expression,

00:07:38.114 --> 00:07:40.102
you have to have the same domain.

00:07:40.102 --> 00:07:44.148
So your domain you would want to say

00:07:44.148 --> 00:07:46.962
it is all real

00:07:47.931 --> 00:07:50.436
x's except...

00:07:50.440 --> 00:07:51.976
If you were just to do
the domain for this thing,

00:07:51.976 --> 00:07:53.753
if you just solved this
expression right here and say

00:07:53.753 --> 00:07:55.871
okay, x can't be equal to negative five.

00:07:57.591 --> 00:08:00.526
So let me write except x
is equal to negative five.

00:08:03.867 --> 00:08:06.652
And so this right here is
just a domain for this thing,

00:08:06.652 --> 00:08:07.990
but if we wanted to have the same domain

00:08:07.990 --> 00:08:09.314
as the original expression,

00:08:09.314 --> 00:08:13.729
we'd also throw in and x is equal to five.


Solving for a variable

WEBVTT
Kind: captions
Language: en

00:00:00.950 --> 00:00:06.137
Solve P equals 2l plus 2w for l.

00:00:06.137 --> 00:00:07.970
So this right here,
this is just the formula

00:00:07.970 --> 00:00:10.290
for the perimeter
of a rectangle.

00:00:10.290 --> 00:00:12.480
Perimeter is equal
to 2 times the length

00:00:12.480 --> 00:00:14.060
plus 2 times the width.

00:00:14.060 --> 00:00:16.940
But they just want us to solve
this equation right here,

00:00:16.940 --> 00:00:17.750
solve for l.

00:00:17.750 --> 00:00:18.670
So let's do that.

00:00:18.670 --> 00:00:24.900
So we have P is equal to
2 times l plus 2 times w.

00:00:24.900 --> 00:00:26.650
So we need to solve for l.

00:00:26.650 --> 00:00:29.106
So let's isolate all of
the l terms on one side.

00:00:29.106 --> 00:00:30.730
And the best way, we
could just do that

00:00:30.730 --> 00:00:32.670
by leaving it here
on the right and then

00:00:32.670 --> 00:00:33.637
getting rid of this 2w.

00:00:33.637 --> 00:00:35.220
And the best way to
get rid of this 2w

00:00:35.220 --> 00:00:37.217
is to subtract it
from the right.

00:00:37.217 --> 00:00:39.300
But if you're going to
subtract it from the right,

00:00:39.300 --> 00:00:41.050
you also have to
subtract it from the left

00:00:41.050 --> 00:00:43.100
if you want this
equality to hold.

00:00:43.100 --> 00:00:45.500
So you have to also
subtract it from the left.

00:00:45.500 --> 00:00:48.127
And so the left-hand
side becomes P minus 2w.

00:00:50.970 --> 00:00:53.830
And the right-hand
side, you get--

00:00:53.830 --> 00:00:56.290
this 2w minus 2w cancels out.

00:00:56.290 --> 00:00:58.620
You just have a 2l.

00:00:58.620 --> 00:01:00.120
And then if you
want to solve for l,

00:01:00.120 --> 00:01:04.080
you just have to divide both
sides of this equation by 2.

00:01:04.080 --> 00:01:06.850
You just divide both sides
of this equation by 2.

00:01:06.850 --> 00:01:08.800
And we have isolated our l.

00:01:08.800 --> 00:01:12.100
We get l is equal to
P minus 2w over 2.

00:01:12.100 --> 00:01:14.160
Or if we wanted to
write it the other way,

00:01:14.160 --> 00:01:20.350
you could write l is equal
to P minus 2w over 2.

00:01:20.350 --> 00:01:22.742
And we are done.


Extraneous solutions to rational equations

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.050
Solve and eliminate any
extraneous solutions.

00:00:04.050 --> 00:00:06.390
And what they mean by
extraneous solutions are,

00:00:06.390 --> 00:00:10.210
in the course of solving this
rational equation right here,

00:00:10.210 --> 00:00:12.510
we might get some solutions
that if we actually put it

00:00:12.510 --> 00:00:17.830
into the original problem would
give us undefined expressions.

00:00:17.830 --> 00:00:19.830
And so those solutions
are extraneous solutions.

00:00:19.830 --> 00:00:21.170
They actually don't apply.

00:00:21.170 --> 00:00:22.830
You actually want
to throw them out.

00:00:22.830 --> 00:00:24.288
And so let's look
at this equation.

00:00:24.288 --> 00:00:27.910
We have x squared over x plus
2 is equal to 4 over x plus 2.

00:00:27.910 --> 00:00:29.950
So right from the
get-go, we don't

00:00:29.950 --> 00:00:31.450
know if this is
going to necessarily

00:00:31.450 --> 00:00:33.380
be a solution to this equation.

00:00:33.380 --> 00:00:34.810
But we know, just
looking at this,

00:00:34.810 --> 00:00:36.670
that if x is equal
to negative 2,

00:00:36.670 --> 00:00:39.489
then this denominator and this
denominator are going to be 0.

00:00:39.489 --> 00:00:40.530
And you're dividing by 0.

00:00:40.530 --> 00:00:41.886
It would be undefined.

00:00:41.886 --> 00:00:43.260
So we can, right
from the get-go,

00:00:43.260 --> 00:00:45.440
exclude x is equal
to negative 2.

00:00:45.440 --> 00:00:47.430
So x cannot be
equal to negative 2.

00:00:47.430 --> 00:00:49.730
That would make either
of these expressions

00:00:49.730 --> 00:00:52.410
undefined, on either
side of the equation.

00:00:52.410 --> 00:00:54.920
So with that out of the
way, let's try to solve it.

00:00:54.920 --> 00:00:57.270
So as a first step, we
want to get the x plus 2

00:00:57.270 --> 00:00:58.570
out of the denominator.

00:00:58.570 --> 00:01:03.520
So let's multiply both
sides by x plus 2.

00:01:03.520 --> 00:01:06.300
X plus 2 divided by
x plus 2 is just 1.

00:01:06.300 --> 00:01:09.180
And we can assume
that x plus 2 isn't 0.

00:01:09.180 --> 00:01:10.580
So it's going to be defined.

00:01:10.580 --> 00:01:13.790
x plus 2 divided by
x plus 2 is just 1.

00:01:13.790 --> 00:01:19.830
And so our equation has
simplified to x squared

00:01:19.830 --> 00:01:21.387
is equal to 4.

00:01:21.387 --> 00:01:23.220
And you could probably
do this in your head,

00:01:23.220 --> 00:01:24.810
but I want to do it properly.

00:01:24.810 --> 00:01:25.850
So you can write this.

00:01:25.850 --> 00:01:27.790
You could subtract
4 from both sides.

00:01:27.790 --> 00:01:30.830
Do it in kind of the proper
quadratic equation form.

00:01:30.830 --> 00:01:34.040
So x squared minus
4 is equal to 0.

00:01:34.040 --> 00:01:37.152
I just subtracted 4 from
both sides over here.

00:01:37.152 --> 00:01:38.360
And so you could factor this.

00:01:38.360 --> 00:01:39.760
This is a difference of squares.

00:01:39.760 --> 00:01:45.970
You get x plus 2 times
x minus 2 is equal to 0.

00:01:45.970 --> 00:01:48.970
And then if this is equal to
0, if the product of two things

00:01:48.970 --> 00:01:51.390
are equal to 0, that means
either one or both of them

00:01:51.390 --> 00:01:52.190
are equal to 0.

00:01:52.190 --> 00:01:59.340
So this tells us that x plus
2 is equal to 0 or x minus 2

00:01:59.340 --> 00:02:00.770
is equal to 0.

00:02:00.770 --> 00:02:04.190
If you subtract 2 from both
sides of this equation right

00:02:04.190 --> 00:02:06.910
here, you get x is
equal to negative 2.

00:02:06.910 --> 00:02:10.509
If you add 2 to both sides of
this equation right over here,

00:02:10.509 --> 00:02:13.310
you get x is equal to 2.

00:02:13.310 --> 00:02:15.310
And we're saying
that either of these

00:02:15.310 --> 00:02:17.500
would make this
last expression 0.

00:02:17.500 --> 00:02:20.080
Now, we know that we need
to exclude one of them.

00:02:20.080 --> 00:02:24.230
We know that x cannot
be equal to negative 2.

00:02:24.230 --> 00:02:27.490
So x equals negative 2 is
an extraneous solution.

00:02:30.150 --> 00:02:33.670
It's not really a solution
for-- it is a solution for this,

00:02:33.670 --> 00:02:36.110
once we got rid of the
rational expressions.

00:02:36.110 --> 00:02:38.320
But it's not a solution
for this original problem

00:02:38.320 --> 00:02:41.020
up here, because it would make
the expressions undefined.

00:02:41.020 --> 00:02:43.290
It would cause you
to divide by 0.

00:02:43.290 --> 00:02:47.810
So the only solution
here is x is equal to 2.

00:02:47.810 --> 00:02:49.400
And you can check it yourself.

00:02:49.400 --> 00:02:53.760
If you do 2 squared, you
get 4, over 2 plus 2 is 4.

00:02:53.760 --> 00:02:57.310
And that should be
equal to 4 over 2

00:02:57.310 --> 00:03:00.000
plus 2, over 4, which
it definitely does.

00:03:00.000 --> 00:03:02.263
1 is equal to 1.


Rational equations

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.960
Solve the equation and
find excluded values.

00:00:04.960 --> 00:00:08.100
And what they're talking about,
about finding excluded values,

00:00:08.100 --> 00:00:12.000
is we need to think about what
values would make these either

00:00:12.000 --> 00:00:14.119
side of this equation undefined.

00:00:14.119 --> 00:00:15.660
And the reason why
we want to do that

00:00:15.660 --> 00:00:17.200
is because as we
manipulate this,

00:00:17.200 --> 00:00:19.824
we might lose things
in the denominator.

00:00:19.824 --> 00:00:21.240
And then we might
get some answer.

00:00:21.240 --> 00:00:22.739
But if it's one of
those things that

00:00:22.739 --> 00:00:26.890
made the original, the original
expressions, or either side

00:00:26.890 --> 00:00:29.220
of the original
equation undefined

00:00:29.220 --> 00:00:31.110
then that wouldn't be
a legitimate solution.

00:00:31.110 --> 00:00:33.530
So that's what they're talking
about, the excluded values.

00:00:33.530 --> 00:00:35.488
So what values do we have
to exclude here right

00:00:35.488 --> 00:00:36.460
from the get go?

00:00:36.460 --> 00:00:39.790
Well 4 over p minus
1 won't be defined

00:00:39.790 --> 00:00:43.050
if p is 1 because if p is
1, then this, then you're

00:00:43.050 --> 00:00:44.330
going to be dividing by 0.

00:00:44.330 --> 00:00:45.440
And that's undefined.

00:00:45.440 --> 00:00:48.930
So we know that p
cannot be equal to 1.

00:00:48.930 --> 00:00:51.170
And over here, if p
was at negative 3,

00:00:51.170 --> 00:00:54.080
then this denominator would be
0 and it would be undefined.

00:00:54.080 --> 00:00:57.410
And so p cannot be equal
to 1 or negative 3.

00:00:57.410 --> 00:01:00.460
So these right here are
our excluded values.

00:01:00.460 --> 00:01:02.240
So now let's try to solve.

00:01:02.240 --> 00:01:04.027
Let's try to solve
this equation.

00:01:04.027 --> 00:01:05.610
And I'm going to
rewrite it over here.

00:01:05.610 --> 00:01:11.170
So we have 4 over p minus 1
is equal to 5 over p plus 3.

00:01:11.170 --> 00:01:12.920
So the first thing we
could do, especially

00:01:12.920 --> 00:01:15.070
because we can assume now that
neither of these expressions

00:01:15.070 --> 00:01:15.520
are 0.

00:01:15.520 --> 00:01:17.353
And this is going to
be defined, since we've

00:01:17.353 --> 00:01:21.082
excluded these values of
p, is to get the p minus 1

00:01:21.082 --> 00:01:22.040
out of the denominator.

00:01:22.040 --> 00:01:25.080
We can multiply the left
hand side by p minus 1.

00:01:25.080 --> 00:01:26.650
But remember, this
is an equation.

00:01:26.650 --> 00:01:28.590
If you want them to
continue to be equal,

00:01:28.590 --> 00:01:30.690
anything you do
left hand side, you

00:01:30.690 --> 00:01:32.710
have to do to the
right hand side.

00:01:32.710 --> 00:01:35.230
So I'm multiplying by p minus 1.

00:01:35.230 --> 00:01:38.020
Now I also want to get this p
plus 3 out of the denominator

00:01:38.020 --> 00:01:39.650
here on the right hand side.

00:01:39.650 --> 00:01:42.550
So the best way to do that is
multiply the right hand side

00:01:42.550 --> 00:01:43.850
by p plus 3.

00:01:43.850 --> 00:01:45.530
But if I do that to
the right hand side,

00:01:45.530 --> 00:01:47.950
I also have to do that
to the left hand side.

00:01:47.950 --> 00:01:50.780
p plus 3.

00:01:50.780 --> 00:01:51.850
And so what happens?

00:01:51.850 --> 00:01:53.930
We have a p minus 1 in
the numerator, p minus 1

00:01:53.930 --> 00:01:55.760
at the denominator.

00:01:55.760 --> 00:01:57.099
These cancel out.

00:01:57.099 --> 00:01:58.765
So you have just a 1
of the denominator,

00:01:58.765 --> 00:02:00.580
or you have no
denominator anymore.

00:02:00.580 --> 00:02:02.370
And the left hand
side simplifies

00:02:02.370 --> 00:02:06.200
to 4 times p plus
3, or if you were

00:02:06.200 --> 00:02:10.060
to distribute the
4, 4 times p plus 3.

00:02:10.060 --> 00:02:14.410
So that is 4p plus 12.

00:02:14.410 --> 00:02:16.590
And then the right
hand side, you

00:02:16.590 --> 00:02:19.040
have plus 3 canceling
with a p plus 3.

00:02:19.040 --> 00:02:21.870
This is p plus 3
divided by p plus 3.

00:02:21.870 --> 00:02:24.530
And all you're left with
is 5 times p minus 1.

00:02:24.530 --> 00:02:29.500
If you distribute the
5 you get 5p minus 5.

00:02:29.500 --> 00:02:31.930
And now this is a pretty
straightforward linear equation

00:02:31.930 --> 00:02:32.700
to solve.

00:02:32.700 --> 00:02:34.750
We just want to isolate
the p's on one side

00:02:34.750 --> 00:02:36.410
and the constants on the other.

00:02:36.410 --> 00:02:39.730
So let's subtract
5p from both sides.

00:02:39.730 --> 00:02:41.030
I'll switch colors.

00:02:41.030 --> 00:02:44.690
So let's subtract
5p from both sides.

00:02:44.690 --> 00:02:48.340
And we get on the
left hand side,

00:02:48.340 --> 00:02:52.050
4p minus 5p is
negative p plus 12.

00:02:52.050 --> 00:02:55.650
Is equal to, these cancel
out, is equal to negative 5.

00:02:55.650 --> 00:02:57.785
And then we could subtract
12 from both sides.

00:03:01.400 --> 00:03:05.740
And we get, these cancel out,
we get negative p is equal to,

00:03:05.740 --> 00:03:08.970
negative 5 minus
12 is negative 17.

00:03:08.970 --> 00:03:09.910
And we're almost done.

00:03:09.910 --> 00:03:11.720
We can multiply both
sides by negative 1

00:03:11.720 --> 00:03:13.450
or divide both
sides by negative 1

00:03:13.450 --> 00:03:16.160
depending on how
you want to view it.

00:03:16.160 --> 00:03:19.900
And we get, negative
one times negative p is.

00:03:19.900 --> 00:03:21.700
So let me just scroll
down a little bit

00:03:21.700 --> 00:03:23.610
so I have a little
bit of real estate.

00:03:23.610 --> 00:03:28.260
That's positive
p is equal to 17.

00:03:28.260 --> 00:03:31.160
p is equal to 17.

00:03:31.160 --> 00:03:33.590
And let's verify that
this really works.

00:03:33.590 --> 00:03:36.090
Well, it wasn't one of our
excluded values, but just

00:03:36.090 --> 00:03:38.480
in case, let's verify
that it really works.

00:03:38.480 --> 00:03:41.630
If we go, if we have p is 17.

00:03:41.630 --> 00:03:49.780
We get 4 over 17 minus 1,
needs to be equal to 5 over 17

00:03:49.780 --> 00:03:50.550
plus 3.

00:03:50.550 --> 00:03:54.330
I'm just putting 17 in for p,
because that's our solution.

00:03:54.330 --> 00:03:58.270
So this is the same
thing as 4 over 16,

00:03:58.270 --> 00:04:02.370
needs to be the same
thing as 5 over 20.

00:04:02.370 --> 00:04:04.900
Or 4/16 is the
same thing as 1/4,

00:04:04.900 --> 00:04:06.990
and that needs to be
the same thing as 5/20,

00:04:06.990 --> 00:04:09.330
which is the same thing as 1/4.

00:04:09.330 --> 00:04:10.990
So it all checks out.

00:04:10.990 --> 00:04:12.230
So these are excluded values.

00:04:12.230 --> 00:04:15.590
And lucky for us, this
wasn't one of them.


Simplifying first for subtracting rational expressions

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.139
This is the exact same problem
that I did in the last video,

00:00:03.139 --> 00:00:04.930
but I realized after
looking at the problem

00:00:04.930 --> 00:00:06.910
that I just mindlessly
did the problem.

00:00:06.910 --> 00:00:09.280
But there's actually a much
simpler way of doing it.

00:00:09.280 --> 00:00:11.720
So once again, they're telling
us to subtract and state

00:00:11.720 --> 00:00:13.070
the difference in simplest form.

00:00:13.070 --> 00:00:14.570
And what I did in
the last video is,

00:00:14.570 --> 00:00:16.750
I just said, OK, let's
find a common denominator.

00:00:16.750 --> 00:00:19.110
And then I found it, and
then I did the subtraction.

00:00:19.110 --> 00:00:20.610
But what I could
have done, and that

00:00:20.610 --> 00:00:22.193
would have simplified
all of the math,

00:00:22.193 --> 00:00:25.250
is I could have simplified each
of these expressions first.

00:00:25.250 --> 00:00:26.550
So let's do that.

00:00:26.550 --> 00:00:29.140
So this first expression right
over here, both the numerator

00:00:29.140 --> 00:00:31.760
and the denominator
is divisible by m.

00:00:31.760 --> 00:00:33.370
So let's divide them both by m.

00:00:33.370 --> 00:00:35.510
If you divide the
numerator by m, you get 1.

00:00:35.510 --> 00:00:38.430
You divide the denominator
by m, m squared divided by m

00:00:38.430 --> 00:00:40.380
is just m.

00:00:40.380 --> 00:00:43.400
The second term right
over here, both of them,

00:00:43.400 --> 00:00:46.470
both the numerator and the
denominator, are divisible by n

00:00:46.470 --> 00:00:48.180
to the third power.

00:00:48.180 --> 00:00:50.830
So let's divide the numerator
by n to the third power.

00:00:50.830 --> 00:00:51.920
You get a 1.

00:00:51.920 --> 00:00:54.340
You divide the denominator,
you divide n to the fourth

00:00:54.340 --> 00:00:57.540
by n to the third power,
you just get an n.

00:00:57.540 --> 00:01:01.700
And so we can rewrite
this as 6-- 6 times 1,

00:01:01.700 --> 00:01:08.440
so that's just 6-- 6
over 7mn, minus 5--

00:01:08.440 --> 00:01:12.130
because that's just
5 times 1-- over 3mn.

00:01:12.130 --> 00:01:14.840
And this really
simplifies the problem.

00:01:14.840 --> 00:01:17.690
So let's think about a
common denominator over here.

00:01:17.690 --> 00:01:20.420
So it's going to be
something minus something.

00:01:20.420 --> 00:01:24.070
So a common denominator has to
be divisible by both 7 and 3.

00:01:24.070 --> 00:01:28.150
So the least common
multiple of 7 and 3 is 21.

00:01:28.150 --> 00:01:30.530
And it has to be
divisible by m, and it

00:01:30.530 --> 00:01:32.180
has to be divisible by n.

00:01:32.180 --> 00:01:35.300
21mn, and that also
takes care of this.

00:01:35.300 --> 00:01:38.050
21mn is clearly
divisible by 7mn,

00:01:38.050 --> 00:01:40.170
and it's clearly
divisible by 3mn.

00:01:40.170 --> 00:01:41.170
It's got the mn there.

00:01:41.170 --> 00:01:42.910
It's divisible by 3,
it's divisible by 7.

00:01:42.910 --> 00:01:46.250
So that's our least
common multiple,

00:01:46.250 --> 00:01:48.800
or you could say it's
our common denominator.

00:01:48.800 --> 00:01:52.772
And to get from 7mn to 21mn,
you've got to multiply by 3.

00:01:52.772 --> 00:01:54.730
And if you're multiplying
the denominator by 3,

00:01:54.730 --> 00:01:56.870
you've got to multiply
the numerator by 3.

00:01:56.870 --> 00:02:01.110
And you get 3 times
6 is equal to 18.

00:02:01.110 --> 00:02:05.570
And to go from 3mn to 21mn,
you've got to multiply by 7.

00:02:05.570 --> 00:02:07.870
I'll do it over here, you've
got to multiply it by 7.

00:02:07.870 --> 00:02:10.250
And if you multiply
the denominator by 7,

00:02:10.250 --> 00:02:12.290
you've got to multiply
the numerator by 7

00:02:12.290 --> 00:02:14.830
if you don't want to change
the value of the expression.

00:02:14.830 --> 00:02:19.710
So 5 times 7 is equal to 35.

00:02:19.710 --> 00:02:23.820
And then so this is going
to be equal to 18 minus 35,

00:02:23.820 --> 00:02:25.720
since we have-- let
me write it down--

00:02:25.720 --> 00:02:29.362
we have a common
denominator here of 21mn.

00:02:29.362 --> 00:02:31.790
And we're going to have
to calculate 18 minus 35

00:02:31.790 --> 00:02:32.550
in the numerator.

00:02:32.550 --> 00:02:37.070
I'll just write it over here,
18 minus 35 in the numerator.

00:02:37.070 --> 00:02:41.430
And 18 minus 35 is negative 17.

00:02:41.430 --> 00:02:46.870
So we get negative 17 in
the numerator, over 21mn,

00:02:46.870 --> 00:02:48.116
and we are done.

00:02:48.116 --> 00:02:49.740
And I think that was
a good bit simpler

00:02:49.740 --> 00:02:52.560
than the way we did
it in the last video.


Subtracting rational expressions

WEBVTT
Kind: captions
Language: en

00:00:01.057 --> 00:00:03.730
We're asked to subtract and state the difference

00:00:03.730 --> 00:00:05.319
in simplest form.

00:00:05.319 --> 00:00:08.863
So we're subtracting one rational expression from another.

00:00:08.863 --> 00:00:12.046
And just like the case when you are subtracting fractions,

00:00:12.046 --> 00:00:14.729
or adding fractions for that matter,

00:00:14.729 --> 00:00:17.462
you have to make sure that you have a common denomintator.

00:00:17.462 --> 00:00:21.147
And over here it looks like we have kind of the same components,

00:00:21.147 --> 00:00:23.063
but they're not quite the same thing.

00:00:23.063 --> 00:00:29.617
So we need to find the least common multiple of 7m^2n and 3mn^4,

00:00:29.617 --> 00:00:32.495
and then that we can use as a common denominator.

00:00:32.495 --> 00:00:34.296
Let's see if we can do that.

00:00:34.296 --> 00:00:36.813
So, let me find a common denominator:

00:00:36.813 --> 00:00:40.147
we have a negative sign out here and a common denominator over here.

00:00:40.147 --> 00:00:42.481
So a common denominator is going to have ...

00:00:42.481 --> 00:00:44.258
let's just look at each of the numbers.

00:00:44.258 --> 00:00:47.147
We have a 7 here and a 3 over here.

00:00:47.147 --> 00:00:51.045
So the common denominator has to be divisible by both 7 and 3.

00:00:51.045 --> 00:00:52.680
Or another way to think about it,

00:00:52.680 --> 00:00:55.112
the least common multiple of 7 and 3 is 21.

00:00:55.112 --> 00:00:59.462
So we're going to have to have 21 in our common denominator.

00:00:59.462 --> 00:01:02.529
And I'm getting that by encorporating the 7 and the 3.

00:01:02.529 --> 00:01:04.897
That's what gives me the 21.

00:01:04.897 --> 00:01:07.110
Now let's look at the m's.

00:01:07.110 --> 00:01:09.745
This denominator has 2 "m"s if you were to factor it.

00:01:09.745 --> 00:01:11.529
It's m times m.

00:01:11.529 --> 00:01:13.129
This only has one m.

00:01:13.129 --> 00:01:15.446
So as long as we're divisible by m^2

00:01:15.446 --> 00:01:17.481
then we're also going to be divisible by m

00:01:17.481 --> 00:01:19.713
So we just have to put an m^2 over here

00:01:19.713 --> 00:01:22.746
and an m^2 over here.

00:01:22.746 --> 00:01:25.813
Notice this is ... right now this is both divisible by

00:01:25.813 --> 00:01:29.080
7, 3, m and m^2.

00:01:29.080 --> 00:01:31.530
So we're cool through the "m"s.

00:01:31.530 --> 00:01:33.463
Now let's look at the "n" terms.

00:01:33.463 --> 00:01:35.112
It has to be divisible by n,

00:01:35.112 --> 00:01:37.283
and it also has to be divisible by n^4.

00:01:37.283 --> 00:01:39.318
Well, if we're divisible by n^4,

00:01:39.318 --> 00:01:41.544
we're also divisible by n, since this is one of the factors of n^4

00:01:41.544 --> 00:01:45.880
This to the fourth power is n to the fourth.

00:01:45.880 --> 00:01:48.612
Anyway, we have to be divisible by n^4

00:01:48.612 --> 00:01:51.862
If we're divisible by n^4 we are divisible by n

00:01:51.862 --> 00:01:53.540
So that's our common denominator.

00:01:53.540 --> 00:01:57.000
We're divisible by 7m^2n, definitely divisible by 3mn^4.

00:01:57.000 --> 00:02:02.195
So that's our common denominator.

00:02:02.195 --> 00:02:06.148
So to go from 7m^2n, to this right over here,

00:02:06.148 --> 00:02:09.597
we have to multiply it by 3 times

00:02:09.597 --> 00:02:12.663
let's see, the m^2, we didn't have to multiply it by anything

00:02:12.663 --> 00:02:16.062
to get from n to n^4, we have to multiply it by n^3

00:02:16.062 --> 00:02:19.086
So we have to multiply this by 3n^3

00:02:19.086 --> 00:02:22.731
So we multiply the denominator by 3n^3 to get this over here

00:02:22.731 --> 00:02:26.317
we also have to multiply the numerator by 3n^3

00:02:26.317 --> 00:02:29.547
3n^3 times 6m is 18m,

00:02:29.547 --> 00:02:36.596
one right over there

00:02:36.596 --> 00:02:38.753
and then we have an n^3

00:02:38.753 --> 00:02:43.696
now let's do it with the second term right over here

00:02:43.696 --> 00:02:46.797
or the thing we are subtracting from this first rational expression

00:02:46.797 --> 00:02:49.112
so to get from 3m,

00:02:49.112 --> 00:02:51.129
so let's see, to get from that to that

00:02:51.129 --> 00:02:52.993
what did we have to do?

00:02:52.993 --> 00:02:54.212
to get from a 3 to a 21

00:02:54.212 --> 00:02:56.230
you have to multiply it by a 7

00:02:56.230 --> 00:02:58.179
you have to multiply it by 7

00:02:58.179 --> 00:03:03.148
and to get from an m to an m^2 you have to multiply it by m

00:03:03.148 --> 00:03:07.229
And to get from n^4 to n^4, you didn't have to multiply by anything

00:03:07.229 --> 00:03:09.528
So to get from this expression in the denominator

00:03:09.528 --> 00:03:11.045
to this expression in the denominator

00:03:11.045 --> 00:03:13.148
you have to multiply it by 7m.

00:03:13.148 --> 00:03:15.393
So if you multiply the denominator by 7m

00:03:15.393 --> 00:03:18.346
you also have to ultiply the numerator by 7m

00:03:18.346 --> 00:03:21.547
And so you have 5 time 7 is 35

00:03:21.547 --> 00:03:25.600
and you have 1m, right over here,

00:03:25.600 --> 00:03:27.647
and then we have n^3

00:03:27.647 --> 00:03:29.648
and now we are ready to subtract

00:03:29.648 --> 00:03:31.480
since we have the same denominator

00:03:31.480 --> 00:03:33.730
Let me write the denominator here

00:03:33.730 --> 00:03:38.713
So the denominator is 21m^2n^4

00:03:38.713 --> 00:03:40.962
And in our numerator we have

00:03:40.962 --> 00:03:52.596
18mn^3 minus 35mn^3

00:03:52.596 --> 00:03:54.196
Now this is interesting,

00:03:54.196 --> 00:03:56.730
because we have 18 of something

00:03:56.730 --> 00:03:58.379
of mn^3

00:03:58.379 --> 00:04:00.745
and we're going to subtract 35 of that same something

00:04:00.745 --> 00:04:02.063
mn^3

00:04:02.063 --> 00:04:03.564
we have the same power on m,

00:04:03.564 --> 00:04:05.529
same power on n

00:04:05.529 --> 00:04:08.481
So this is 18 of something -35 of that something

00:04:08.481 --> 00:04:10.346
which is going to be equal to what?

00:04:10.346 --> 00:04:12.480
18 minus 35,

00:04:12.480 --> 00:04:14.805
this is just practicing subtracting negative numbers

00:04:14.805 --> 00:04:17.111
is -17

00:04:17.111 --> 00:04:19.614
Let me do this in a new color

00:04:19.614 --> 00:04:23.613
So this is going to be -17

00:04:23.613 --> 00:04:29.180
-17mn^3

00:04:29.180 --> 00:04:34.211
right, 18-35: 35-18 would be 17

00:04:34.211 --> 00:04:37.711
So if you swap them around you get -17

00:04:37.711 --> 00:04:39.120
yup that works

00:04:39.120 --> 00:04:49.481
and then all of that over 21m^2n^4.

00:04:49.481 --> 00:04:51.861
Now let's see if we can simplify this further

00:04:51.861 --> 00:04:53.345
both the numerator ...

00:04:53.345 --> 00:04:56.314
17 is prime, so you can't simplify that with 21

00:04:56.314 --> 00:05:00.079
both the numerator and denominator are divisible by m

00:05:00.079 --> 00:05:02.261
So let's divide them both by m

00:05:02.261 --> 00:05:06.013
this becomes a 1, this becomes just an m

00:05:06.013 --> 00:05:09.085
both the numerator and denominator are divisible by n^3

00:05:09.085 --> 00:05:13.412
so you divide the numerator by n^3, that becomes 1

00:05:13.412 --> 00:05:16.563
you divide the denominator by n^3, this becomes an n

00:05:16.563 --> 00:05:21.344
We are left with, in simplest form,

00:05:21.354 --> 00:05:31.129
-17 over, I'll do it in the same colors, 21mn.

00:05:31.129 --> 00:05:33.302
And we are done.


Multiplying and simplifying rational expressions

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:04.399
We're asked to multiply and express as a simplified rational.

00:00:04.399 --> 00:00:08.297
So when you multiply any rational expressions, it's just like multiplying any fractions.

00:00:08.297 --> 00:00:11.796
You multiply the numerators times each other, and the denominators times each other.

00:00:11.796 --> 00:00:14.779
So if we multiply the numerators here, and when you multiply--

00:00:14.779 --> 00:00:18.378
When you're just multiplying a bunch of numbers, it doesn't matter what order you do them in.

00:00:18.378 --> 00:00:20.479
So we want to multiply in an order that makes it a little bit clearer.

00:00:20.479 --> 00:00:22.713
We can multiply the coefficients first.

00:00:22.713 --> 00:00:25.249
We can multiply 3 times 14.

00:00:25.249 --> 00:00:27.916
3 times 14 gives us

00:00:27.916 --> 00:00:29.832
-3 times 10 is thirty-

00:00:29.832 --> 00:00:31.781
-3 times 4 is 12-

00:00:31.781 --> 00:00:33.965
It gives us 42.

00:00:33.965 --> 00:00:36.765
So that's the 3 and the 14.

00:00:36.765 --> 00:00:41.748
Then we have an a-squared, 42a-squared.

00:00:41.748 --> 00:00:45.632
Then we have a b.

00:00:45.632 --> 00:00:48.116
I'll write these in the order that the letters in the alphabet-- alphabetical order.

00:00:48.116 --> 00:00:51.799
And then we have an x-squared -and then we have an x-squared.

00:00:51.799 --> 00:00:55.348
So I just changed the order, but I just multiplied all of these things in both numerators.

00:00:55.348 --> 00:00:58.081
That's the numerators multiplied.

00:00:58.081 --> 00:00:59.899
Let's do the same thing for the denominators.

00:00:59.899 --> 00:01:03.490
So I'll first do the constant -or I should say the coefficients-

00:01:03.490 --> 00:01:06.116
I'll multiply 2 times 18, which is 36.

00:01:06.116 --> 00:01:12.167
And then I have an a over here - just one a- 36a

00:01:12.167 --> 00:01:16.984
Then I have a just one b over here -times b-

00:01:16.984 --> 00:01:21.250
Then I just have one x over here -times x-

00:01:22.173 --> 00:01:25.682
So we've multiplied the two, and now we just need to simplify them.

00:01:25.682 --> 00:01:28.237
Let's just think about this a little bit.

00:01:28.237 --> 00:01:30.770
Both 42 and 36 are divisible by 6.

00:01:30.770 --> 00:01:32.487
That's their greatest common divisor.

00:01:32.487 --> 00:01:33.986
Let's divide both of them by 6.

00:01:33.986 --> 00:01:36.738
42 divided by 6 is 7.

00:01:36.738 --> 00:01:39.422
36 divided by 6 is 6.

00:01:40.591 --> 00:01:43.844
Both a-squared and a are divisible by a, so let's divide them both by a.

00:01:43.844 --> 00:01:46.396
a-squared divided by a is a.

00:01:46.396 --> 00:01:48.895
a divided by a is one.

00:01:48.895 --> 00:01:50.664
And just remember, I'm just dividing in both

00:01:50.664 --> 00:01:54.812
--in any of these steps I'm just dividing the numerator and the denominator by the same number.

00:01:54.812 --> 00:01:57.862
And that's not changing the overall value of the expression.

00:01:59.216 --> 00:02:01.443
Then I have a b divided by a b

00:02:01.443 --> 00:02:04.526
-well I can divide the numerator and the denominator by b-

00:02:05.711 --> 00:02:09.551
And so, those two characters will cancel out.

00:02:09.551 --> 00:02:11.801
To just 1 and 1.

00:02:11.801 --> 00:02:13.751
You don't even have to write it when you multiply by 1.

00:02:13.751 --> 00:02:15.018
It doesn't change the value.

00:02:15.018 --> 00:02:16.968
And then I have an x-squared divided by x.

00:02:16.968 --> 00:02:18.468
They're both divisible by x.

00:02:18.468 --> 00:02:20.551
x-squared divided by x is x.

00:02:20.551 --> 00:02:22.951
x divided by x is 1.

00:02:22.951 --> 00:02:29.236
So, in the numerator we are left with 7ax.

00:02:29.236 --> 00:02:32.002
We can ignore the b, because it's just multiplying by 1 now.

00:02:32.002 --> 00:02:36.301
We have 7 -let me do that same shade of green-

00:02:37.947 --> 00:02:44.324
We have 7 times a times x, in the numerator.

00:02:44.324 --> 00:02:48.306
And then in the denominator, we 6 times 1 times 1 times 1.

00:02:48.306 --> 00:02:53.306
We have all sixes.

00:02:53.306 --> 00:02:57.741
Now I want to be a little bit careful here, because if we want this expression.

00:02:57.741 --> 00:03:02.306
So we have done what they've asked us to do, but this begs a little bit of a question here.

00:03:02.306 --> 00:03:09.724
If we want this expression to truly be the same expression as the product of these two things,

00:03:09.724 --> 00:03:12.207
we have to constrain its domain.

00:03:12.207 --> 00:03:15.156
We have to constrain the variable values here.

00:03:15.156 --> 00:03:17.391
Because this expression right over here.

00:03:17.391 --> 00:03:19.474
This is defined for any a's or x's.

00:03:19.474 --> 00:03:22.706
You could put any a or x here, and you will get a value here.

00:03:22.706 --> 00:03:26.958
But this product over here, this was not defined for any a's or x's.

00:03:26.958 --> 00:03:30.508
The b's cancel out, but it was not defined for any a's or x's.

00:03:30.508 --> 00:03:38.842
In this situation, if either a or x equal to 0, this expression right over here would be undefined.

00:03:38.842 --> 00:03:42.453
In fact if b was equal to 0, this expression would also be undefined.

00:03:42.453 --> 00:03:46.404
So if you really this to be the exact same expression.

00:03:46.404 --> 00:03:48.088
Say this was a function definition.

00:03:48.088 --> 00:03:54.537
If you want it to be the exact same function definition as this over here, you would have to have the same domain.

00:03:54.537 --> 00:03:56.203
So I'll just write down the domain here.

00:03:56.203 --> 00:03:57.590
They have to have the same domain.

00:03:57.590 --> 00:03:59.195
If you want them to be the exact same function.

00:03:59.195 --> 00:04:02.362
If you don't care about that, then you could just say, ok let's just go with this.

00:04:02.362 --> 00:04:07.203
But you should constrain it if you want it to be an equivalent expression.

00:04:07.203 --> 00:04:16.686
And so in either case, you should constrain the domain to be all real numbers.

00:04:16.686 --> 00:04:17.757
And this is a bit different, because

00:04:17.757 --> 00:04:20.659
-- this is essentially like a two variable function.

00:04:20.659 --> 00:04:23.492
If you view a as a possible variable here.

00:04:23.492 --> 00:04:41.126
So, all real numbers for a, b and x, except (when) a, b or x is equal to 0.

00:04:41.126 --> 00:04:44.425
So we're just saying that a, b and c can be anything other than 0.

00:04:44.425 --> 00:04:47.625
We're getting it from our original expression, after we multiplied.

00:04:47.625 --> 00:04:50.808
We're saying it has to be the same, it has to have the same domain.

00:04:50.808 --> 00:04:54.258
Most algebra classes, they wont make you be this particular about it.

00:04:54.258 --> 00:04:56.636
They'll just say, simplify the expression.

00:04:56.636 --> 00:04:58.270
And you'll just simplify it, and you'll give this answer.

00:04:58.270 --> 00:05:03.172
But if you really want it to be the exact same expression, you would really have to have the same domain.


Simplifying rational expressions example 2

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.877
Simplify the rational
expression and state the domain.

00:00:04.877 --> 00:00:06.460
So we have a rational
expression here.

00:00:06.460 --> 00:00:10.290
One expression being
divided by another one.

00:00:10.290 --> 00:00:12.280
And so to simplify
it, I am guessing

00:00:12.280 --> 00:00:15.360
that we will have to
factor this top expression

00:00:15.360 --> 00:00:16.600
and this bottom expression.

00:00:16.600 --> 00:00:18.741
And there's probably
common factors

00:00:18.741 --> 00:00:20.240
that we might be
able to cancel out.

00:00:20.240 --> 00:00:22.406
But we have to be very
careful about how we do them.

00:00:22.406 --> 00:00:24.080
So let's factor each of these.

00:00:24.080 --> 00:00:26.310
So you may or may
not remember how

00:00:26.310 --> 00:00:30.440
to factor quadratic
expressions that

00:00:30.440 --> 00:00:33.790
have a non-one
coefficient out here.

00:00:33.790 --> 00:00:35.470
So this will be
practice in that.

00:00:35.470 --> 00:00:39.051
And we go into the rationale
for why it works in those videos

00:00:39.051 --> 00:00:41.300
where we first learned to
factor rational expressions.

00:00:41.300 --> 00:00:43.290
And they're in this playlist.

00:00:43.290 --> 00:00:45.540
But just as a reminder,
what you want to do is you

00:00:45.540 --> 00:00:47.390
want to split this middle term.

00:00:47.390 --> 00:00:50.410
You want to split the first
degree term into two terms.

00:00:50.410 --> 00:00:52.950
You want to group it, one
of them with this 2x squared

00:00:52.950 --> 00:00:54.370
and one of it with the 20.

00:00:54.370 --> 00:00:56.060
And you want to group
it into two terms

00:00:56.060 --> 00:00:57.910
so that the coefficients
of the two terms,

00:00:57.910 --> 00:01:02.230
when you take their product, is
the same thing as 2 times 20.

00:01:02.230 --> 00:01:04.420
So you want to break--
let me write it over here.

00:01:04.420 --> 00:01:08.000
You want to break 13x,
you want to break it

00:01:08.000 --> 00:01:16.660
into an ax plus a bx, where a
times b is equal to 2 times 20.

00:01:16.660 --> 00:01:19.780
And we show why this
works in a previous video.

00:01:19.780 --> 00:01:22.820
So a times b needs
to be equal to 40.

00:01:22.820 --> 00:01:26.580
And a plus b needs
to be equal to 13.

00:01:26.580 --> 00:01:28.982
So let's just think
about the factors of 40.

00:01:28.982 --> 00:01:32.210
1 and 40 are not going to work.

00:01:32.210 --> 00:01:35.110
2 and 20 are not going to
work, they don't add up to 13.

00:01:35.110 --> 00:01:38.981
3 and 3-- well, 3
doesn't go into 40.

00:01:38.981 --> 00:01:42.390
4 and 10 does not work.

00:01:42.390 --> 00:01:51.360
5 and 8-- so if we take 5
times 8, that does equal 40.

00:01:51.360 --> 00:01:54.280
And then 5 plus 8 does equal 13.

00:01:54.280 --> 00:01:55.580
So that's looking pretty good.

00:01:55.580 --> 00:01:58.150
So let's split up the
13x along those lines.

00:01:58.150 --> 00:02:08.870
So we get 2x squared plus 5
plus 5x plus 8x is plus 20.

00:02:08.870 --> 00:02:12.980
So all I did is I split up
this 13x into a 5x plus an 8x.

00:02:12.980 --> 00:02:17.470
And now we can group
these two terms.

00:02:17.470 --> 00:02:22.680
And then let's go down over
here and do the same thing

00:02:22.680 --> 00:02:24.130
in the denominator.

00:02:24.130 --> 00:02:28.760
So let's do the same exact
thing in the denominator.

00:02:28.760 --> 00:02:31.820
Once again, we want to find
two numbers that add up to 17

00:02:31.820 --> 00:02:35.110
and whose product is
2 times 30, or 60.

00:02:35.110 --> 00:02:38.880
So if we look at the factors
of 60, 1 and 60 don't work.

00:02:38.880 --> 00:02:39.960
They don't add up to 17.

00:02:39.960 --> 00:02:42.140
Or if you took the negative
1 and negative 60--

00:02:42.140 --> 00:02:43.300
you're not going to have
negative numbers because they

00:02:43.300 --> 00:02:46.070
would have to be the same
sign since this is positive.

00:02:46.070 --> 00:02:48.630
And then they would give you
a negative value over here.

00:02:48.630 --> 00:02:51.080
So we have to deal
with positive numbers.

00:02:51.080 --> 00:02:54.900
2 and 30 don't work.

00:02:54.900 --> 00:02:58.750
3 and 15-- remember, we
want to get to 2 times 30,

00:02:58.750 --> 00:03:00.006
we want to get to 60.

00:03:00.006 --> 00:03:04.160
3 times 15-- sorry,
not 3 times 15.

00:03:04.160 --> 00:03:09.150
3 times 20 doesn't work.

00:03:09.150 --> 00:03:11.950
Their product is 60, but
it doesn't add up to 17.

00:03:11.950 --> 00:03:16.110
4 times 15 does not work.

00:03:16.110 --> 00:03:21.530
5 times 12-- 5 times 12
is 60, so that works.

00:03:21.530 --> 00:03:23.075
That's the same
thing as 2 times 30.

00:03:23.075 --> 00:03:24.380
It equals 60.

00:03:24.380 --> 00:03:26.226
And 5 plus 12 is 17.

00:03:26.226 --> 00:03:27.475
So that's looking pretty good.

00:03:27.475 --> 00:03:29.930
So let's separate
this out into 2x

00:03:29.930 --> 00:03:37.260
squared plus 5x
plus 12x plus 30.

00:03:37.260 --> 00:03:40.370
And then we can
group them like this.

00:03:40.370 --> 00:03:41.962
And then let's see
how we can simplify

00:03:41.962 --> 00:03:43.045
each of these expressions.

00:03:43.045 --> 00:03:47.046
This top expression over here,
we have a common factor of x.

00:03:47.046 --> 00:03:48.420
So this top
expression over here,

00:03:48.420 --> 00:03:50.110
we can factor out
an x out of this.

00:03:50.110 --> 00:03:56.310
So this becomes x times-- and
2x squared divided by x is 2x.

00:03:56.310 --> 00:03:58.830
5x divided by x, when you
have the x factored out, that

00:03:58.830 --> 00:04:00.460
is just 5.

00:04:00.460 --> 00:04:03.180
So this first term
is x times 2x plus 5.

00:04:03.180 --> 00:04:06.375
And then plus-- over here
we can factor out a 4.

00:04:06.375 --> 00:04:09.470
4 is divisible into
both 8x and into 20.

00:04:09.470 --> 00:04:11.300
So let's factor out a 4.

00:04:11.300 --> 00:04:16.930
So this is 4 times-- 8x with a
4 factored out is now just 2x.

00:04:16.930 --> 00:04:20.660
And then 20 with a 4
factored out is just a 5.

00:04:20.660 --> 00:04:22.910
And then let's do
the bottom over here.

00:04:22.910 --> 00:04:26.390
I'm kind of doing them both
simultaneously in parallel.

00:04:26.390 --> 00:04:28.840
The bottom over here,
we can factor out

00:04:28.840 --> 00:04:30.880
an x out of this expression.

00:04:30.880 --> 00:04:36.060
So this is x times--
once again, 2x plus 5.

00:04:36.060 --> 00:04:41.070
Plus, this expression over
here, we can factor out a 6

00:04:41.070 --> 00:04:43.960
out of both 12x and 30.

00:04:43.960 --> 00:04:46.690
So let's factor out a 6.

00:04:46.690 --> 00:04:52.110
And then 12x divided by 6 is 2x.

00:04:52.110 --> 00:04:55.270
And then 30 divided by 6 is 5.

00:04:55.270 --> 00:04:56.970
2x plus 5.

00:04:56.970 --> 00:04:58.820
And so we can simplify this.

00:04:58.820 --> 00:05:00.420
We can factor out,
from this term

00:05:00.420 --> 00:05:02.560
we can factor out a 2x
plus 5, from this term we

00:05:02.560 --> 00:05:04.440
can factor out a 2x plus 5.

00:05:04.440 --> 00:05:08.130
So if we factor out a 2x
plus 5 from both of these--

00:05:08.130 --> 00:05:10.870
so we're factoring
out a 2x plus 5,

00:05:10.870 --> 00:05:14.400
you're left with
an x and a plus 4.

00:05:14.400 --> 00:05:17.320
You're left with an x plus 4.

00:05:17.320 --> 00:05:21.330
And then down here we can
factor out a 2x plus 5.

00:05:21.330 --> 00:05:24.110
Down here we can
factor out a 2x--

00:05:24.110 --> 00:05:26.650
let me make it in
that same color.

00:05:26.650 --> 00:05:29.540
We can factor out a 2x plus 5.

00:05:29.540 --> 00:05:31.890
So we're literally just
factoring out a 2x plus 5,

00:05:31.890 --> 00:05:33.800
and we're left with an x plus 6.

00:05:33.800 --> 00:05:35.550
We're just doing the
distributive property

00:05:35.550 --> 00:05:37.640
in reverse over here. x plus 6.

00:05:37.640 --> 00:05:39.270
Now, you might be
tempted to say,

00:05:39.270 --> 00:05:41.350
hey look, I have a 2x plus
5 in the numerator, 2x

00:05:41.350 --> 00:05:43.170
plus 5 in the denominator,
let me just cancel them out

00:05:43.170 --> 00:05:45.461
really fast and I would have
simplified the expression.

00:05:45.461 --> 00:05:47.370
We have to be very
careful, because we first

00:05:47.370 --> 00:05:48.500
need to think about the domain.

00:05:48.500 --> 00:05:50.333
Because if we just
simplify this expression,

00:05:50.333 --> 00:05:53.506
and we include things in
the simplified domain that

00:05:53.506 --> 00:05:55.630
weren't included in the
original function's domain,

00:05:55.630 --> 00:05:57.520
we've actually changed
the expression.

00:05:57.520 --> 00:05:59.020
Or we've changed the function.

00:05:59.020 --> 00:06:01.560
So we have to be very careful
that even when we simplify it,

00:06:01.560 --> 00:06:03.590
we keep the domain the same.

00:06:03.590 --> 00:06:05.570
So what's the domain
of this thing?

00:06:05.570 --> 00:06:08.090
Well, it's going to be all real
numbers except the ones that

00:06:08.090 --> 00:06:10.344
make the denominator equal to 0.

00:06:10.344 --> 00:06:11.760
And what are the
numbers that make

00:06:11.760 --> 00:06:13.410
the denominator equal to 0?

00:06:13.410 --> 00:06:17.820
Well, the denominator is equal
to 0 if x plus 6 is equal to 0,

00:06:17.820 --> 00:06:21.310
or 2x plus 5 is equal to 0.

00:06:21.310 --> 00:06:25.880
And if x plus 6 is 0, then that
means that x is negative 6.

00:06:25.880 --> 00:06:27.750
Subtract 6 from both sides.

00:06:27.750 --> 00:06:29.890
Over here we could
subtract 5 from both sides.

00:06:29.890 --> 00:06:32.420
So you get 2x is
equal to negative 5.

00:06:32.420 --> 00:06:36.590
Divide both sides by 2, you
get x is equal to negative 5/2.

00:06:36.590 --> 00:06:38.290
So the domain of
this expression right

00:06:38.290 --> 00:06:44.066
over here, the domain of this
expression-- if we could call

00:06:44.066 --> 00:06:45.440
this a function,
if we defined it

00:06:45.440 --> 00:06:50.610
as f of x is equal to all
of this-- the domain of it

00:06:50.610 --> 00:07:01.480
is going to be all real
numbers, except negative 6

00:07:01.480 --> 00:07:02.580
and negative 5/2.

00:07:07.670 --> 00:07:09.980
These will make this
expression undefined.

00:07:09.980 --> 00:07:12.630
It will make this
denominator equal to 0.

00:07:12.630 --> 00:07:15.980
Now we can simplify it, as long
as we keep this denominator.

00:07:15.980 --> 00:07:17.590
So now, if we assume
that we're never

00:07:17.590 --> 00:07:20.550
going to see a negative
5/2, then 2x plus 5

00:07:20.550 --> 00:07:22.400
is never going to be
equal to 0, and so we

00:07:22.400 --> 00:07:24.850
can divide the numerator and
denominator by that value,

00:07:24.850 --> 00:07:26.130
canceling it out.

00:07:26.130 --> 00:07:28.850
And so the simplified
expression is what's left over.

00:07:28.850 --> 00:07:30.480
x plus 4 over x plus 6.

00:07:33.930 --> 00:07:35.762
But we have to keep
the same domain.

00:07:35.762 --> 00:07:37.220
If we didn't write
this here, you'd

00:07:37.220 --> 00:07:38.636
say, OK, the only
thing that would

00:07:38.636 --> 00:07:40.780
make this thing 0 is x
is equal to negative 6.

00:07:40.780 --> 00:07:43.720
We have to remember, this is
a simplified version of that.

00:07:43.720 --> 00:07:46.750
And in order for these to be
identical expressions, or maybe

00:07:46.750 --> 00:07:49.270
identical function
definitions, they

00:07:49.270 --> 00:07:52.230
need to have the
exact same domain.

00:07:52.230 --> 00:07:54.300
So the domain for this
simplified expression

00:07:54.300 --> 00:07:55.950
has to be the same
as the original.

00:07:55.950 --> 00:08:01.090
All real numbers except
negative 6 and negative 5/2.


Static and kinetic friction example

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.680
So I have got this block of wood here that has a mass of 5 kilograms

00:00:04.680 --> 00:00:08.830
and it is sitting on some dirt and we are near the surface of the earth

00:00:08.830 --> 00:00:16.720
and the coefficient of static friction between this type of wood and this type of dirt is 0.60

00:00:16.720 --> 00:00:23.030
and the coefficient of kinetic friction between this type of wood and this type of dirt is 0.55

00:00:23.030 --> 00:00:25.630
This was measured by someone else long ago

00:00:25.630 --> 00:00:28.060
or you found it in some type of a book someplace

00:00:28.060 --> 00:00:32.300
And let's say we push on this side of the block with a force of a 100 N

00:00:32.300 --> 00:00:34.440
What is going to happen?

00:00:34.440 --> 00:00:37.250
So the first thing you might realize is if there is no friction

00:00:37.250 --> 00:00:39.310
if this was a completely frictionless boundary and there is

00:00:39.310 --> 00:00:42.640
no air resistance, we are assuming that there is no air resistance in this example

00:00:42.640 --> 00:00:45.320
That in this dimension, in the horizontal dimension

00:00:45.320 --> 00:00:48.220
there would only be one force here, this 100 N force

00:00:48.220 --> 00:00:51.810
It would be completely unbalanced and that would be the net force

00:00:51.810 --> 00:00:56.350
and so you would have a force going in that direction of a 100 N on a mass of 5 kilograms

00:00:56.350 --> 00:00:59.540
Force = Mass times acceleration

00:00:59.540 --> 00:01:01.580
acceleration and force are vector quantities

00:01:01.580 --> 00:01:04.110
So you would have the force divided by the mass

00:01:04.110 --> 00:01:08.640
would give you 20 meters per second of acceleration in the rightward direction

00:01:08.640 --> 00:01:10.950
That is if there were no friction

00:01:10.950 --> 00:01:13.040
but there is friction in this situation

00:01:13.040 --> 00:01:15.160
So let's think about how we'll deal with it

00:01:15.160 --> 00:01:17.950
So the coefficient of friction tells us

00:01:17.950 --> 00:01:22.740
So this right here is the ratio between the magnitude of the force

00:01:22.740 --> 00:01:24.460
that I have called the budging force

00:01:24.460 --> 00:01:28.820
The amount of force you need to apply to get this thing to budge

00:01:28.820 --> 00:01:32.670
to get this thing to start moving. So we can start using the coefficient of kinetic friction

00:01:32.670 --> 00:01:38.200
It's the ratio between that and the magnitude of the force of contact

00:01:38.200 --> 00:01:41.100
between this block and the floor or ground here

00:01:41.100 --> 00:01:44.040
And the magnitude of that force of contact is the same thing

00:01:44.040 --> 00:01:48.410
as the normal force that the ground is applying on the block

00:01:48.410 --> 00:01:52.810
the magnitude of the normal force the ground is applying on the block

00:01:52.810 --> 00:01:54.150
Then once its moving

00:01:54.150 --> 00:01:58.650
then we can say that this is going to be--this will then be equal to

00:01:58.650 --> 00:02:01.710
this over here will be equal to the force of friction

00:02:01.710 --> 00:02:05.140
So this is the force that really overcome friction

00:02:05.140 --> 00:02:08.360
and this over here will be equal to the force of friction

00:02:08.360 --> 00:02:13.280
The magnitude of the force of friction over the force of contact

00:02:13.280 --> 00:02:16.720
the contact force between those two, so over the normal force

00:02:16.720 --> 00:02:18.140
and it makes sense

00:02:18.140 --> 00:02:21.030
that the larger the contact force

00:02:21.030 --> 00:02:23.050
the more that these are being pressed together

00:02:23.050 --> 00:02:26.210
the little at the atomic level, they kind of really get into each others grooves

00:02:26.210 --> 00:02:28.430
the more budging force you would need

00:02:28.430 --> 00:02:31.620
or the more friction force would go against your motion

00:02:31.620 --> 00:02:33.650
And in either situation

00:02:33.650 --> 00:02:35.920
the force of friction is going against your motion

00:02:35.920 --> 00:02:37.320
So even if you push it in that way

00:02:37.320 --> 00:02:40.420
sounds like force of friction is all of a sudden going to help you

00:02:40.420 --> 00:02:44.160
So let's think about what the necessary force will we need

00:02:44.160 --> 00:02:47.500
to overcome the force of friction right here in the static situation

00:02:47.500 --> 00:02:51.960
So the force of gravity on this block

00:02:51.960 --> 00:02:57.760
is going to be the gravitational field which is 9.8 m/s^2 times 5 kilograms

00:02:57.760 --> 00:03:05.550
9.8 m/s times 5 kilograms gives 49 kilogram meters per second or 49 newtons down

00:03:05.550 --> 00:03:08.390
This is the force, the magnitude of the force due to gravity

00:03:08.390 --> 00:03:11.540
the direction is straight down towards the center of the earth

00:03:11.540 --> 00:03:17.140
The normal force, and that force is there because this block is not accelerating downwards

00:03:17.140 --> 00:03:21.390
So there must be some force that completely balances off the force of gravity

00:03:21.390 --> 00:03:26.030
And in this example, it is the normal force

00:03:26.030 --> 00:03:30.100
So it is acting 49 newtons upward

00:03:30.100 --> 00:03:35.060
and so these net out. And that's why this block does not accelerate upwards or downwards

00:03:35.060 --> 00:03:36.970
So what we have is the budge the

00:03:36.970 --> 00:03:41.050
magnitude of the budging force, needs to be equal to, over the magnitude of the normal force

00:03:41.050 --> 00:03:43.620
well this thing right over here is going to be 49 newtons

00:03:43.620 --> 00:03:46.810
Is equal to 0.60

00:03:46.810 --> 00:03:52.370
Or we could say that the magnitude of the budging force

00:03:52.370 --> 00:03:56.700
is equal to 49 newtons times the coefficient of static fiction

00:03:56.700 --> 00:04:06.710
Or that's 49 newtons times 0.60

00:04:06.710 --> 00:04:09.910
And remember coefficient of friction are unitless

00:04:09.910 --> 00:04:13.040
So the units here are still going to be in newtons

00:04:13.040 --> 00:04:22.580
So this 49 times .6 gives us 29.4 newtons

00:04:22.580 --> 00:04:30.550
This is equal to 29.4 newtons

00:04:30.550 --> 00:04:34.520
So that's the force that's started to overcome static friction

00:04:34.520 --> 00:04:36.790
which we are applying more than enough of

00:04:36.790 --> 00:04:39.920
so with a 100 newtons, we would just start to budge it

00:04:39.920 --> 00:04:42.100
and right when we are in just in that moment

00:04:42.100 --> 00:04:44.190
where that thing is just starting to move

00:04:44.190 --> 00:04:45.580
the net force--

00:04:45.580 --> 00:04:47.670
so we have a 100 newtons going in that direction

00:04:47.670 --> 00:04:51.090
and the force of static friction is going to go in this direction--

00:04:51.090 --> 00:04:53.580
maybe I could draw it down here to show it's coming from right over here

00:04:53.580 --> 00:04:57.970
The force of static friction is going to be 29.4 newtons that way

00:04:57.970 --> 00:05:00.650
and so right when I am just starting to budge this

00:05:00.650 --> 00:05:03.030
just when that little movement--

00:05:03.070 --> 00:05:04.930
because once I do that, then all of a sudden it's moving

00:05:05.030 --> 00:05:09.180
and then kinetic friction starts to matter, but just for that moment

00:05:09.650 --> 00:05:15.930
just for that moment I'll have a net force of 100 - 29.4

00:05:16.360 --> 00:05:28.660
to the right, so I have a net force of 70.6 N

00:05:28.660 --> 00:05:32.820
for just a moment while I budge it

00:05:32.820 --> 00:05:35.250
So just exactly while I'm budging it

00:05:35.250 --> 00:05:42.380
While we're overcoming the static friction, we have a 70.6 N net force in the right direction

00:05:42.690 --> 00:05:47.980
And so just for that moment, you divide it by 5 kg mass

00:05:48.000 --> 00:05:52.430
So just for that moment, it will be accelerating at 14.12 m/s^2

00:05:53.050 --> 00:06:00.280
So you'll have an acceleration of 14.1 m/s^2 to the right

00:06:00.300 --> 00:06:04.250
but that will just be for that absolute moment, because once I budge it

00:06:04.360 --> 00:06:07.420
all of a sudden the block will start to be moving

00:06:07.440 --> 00:06:10.880
And once it's moving, the coefficient of kinetic friction starts to matter

00:06:10.880 --> 00:06:12.880
We've got the things out of their little grooves

00:06:12.900 --> 00:06:17.160
and so they're kind of gliding past each other on the top, although there still is resistant

00:06:17.200 --> 00:06:20.550
So once we budge it, we'll have that acceleration for just a moment

00:06:20.560 --> 00:06:23.550
Now all of a sudden, the coefficient of kinetic friction comes to play

00:06:23.930 --> 00:06:27.080
And the force of friction, assuming we're moving

00:06:27.410 --> 00:06:31.120
the magnitude of the force of friction will always go against our movement

00:06:31.550 --> 00:06:35.460
is going to be--remember, our normal force is 49 N

00:06:35.700 --> 00:06:38.850
So we can multiply both sides of this times 49

00:06:39.000 --> 00:06:57.780
We get 49 N times 0.55 which is equal to 26.95 N

00:06:57.800 --> 00:07:00.400
This is the force of friction; this is the magnitude

00:07:00.520 --> 00:07:02.400
and it's going to go against our motions

00:07:02.410 --> 00:07:05.690
So as soon as we start to move in that direction, the force of friction

00:07:05.890 --> 00:07:09.480
is going to be going in that direction

00:07:09.740 --> 00:07:15.050
So once we start moving, assuming that I'm continuing to apply this 100 newtons of force

00:07:15.130 --> 00:07:18.280
what is the net force? So I have 100 N going that way

00:07:18.380 --> 00:07:23.180
and I have 26.95 going that way

00:07:23.200 --> 00:07:25.080
Remember, with vectors, I don't have to draw them here

00:07:25.080 --> 00:07:28.890
I can draw all of their tails start at the center of mass of the

00:07:28.890 --> 00:07:32.810
object. I can draw them whatever, but remember this is acting on the object

00:07:32.810 --> 00:07:37.140
If we want to be precise, we can show it on the center of mass because

00:07:37.140 --> 00:07:39.940
we can view all of these atoms as one collective object

00:07:39.950 --> 00:07:42.150
But anyway, what is the net force now?

00:07:42.160 --> 00:07:46.420
We have 100 N to the right; we have 26.95 to the left

00:07:47.080 --> 00:07:51.750
100 minus 26.95

00:07:51.750 --> 00:07:53.600
100 N that I'm applying to the right

00:07:53.600 --> 00:07:59.260
- 26.95 N which is the force of friction to the left always acting against us

00:07:59.410 --> 00:08:03.410
means that there's a net force to the right of 73.05

00:08:03.430 --> 00:08:10.840
So once we're moving, we have a net force to the right of 73.05 N

00:08:10.840 --> 00:08:18.070
This is the net force and it's acting to the right

00:08:18.260 --> 00:08:22.850
Right after we budge it, how quickly will this accelerate?

00:08:22.850 --> 00:08:32.700
Well, 73.05 divided by the mass, divided by 5 kg, gives us 14.61

00:08:33.220 --> 00:08:40.430
So the acceleration once we're moving is going to be 14.61 m/s squared

00:08:40.630 --> 00:08:44.450
to the right

00:08:44.490 --> 00:08:47.190
So I really want to make sure you understand what's happening here

00:08:47.220 --> 00:08:51.270
We always have enough force to start budging it

00:08:51.560 --> 00:08:53.510
but right when we budged it

00:08:53.510 --> 00:08:56.430
we overcome the static friction for just a moment

00:08:56.430 --> 00:08:59.810
our acceleration was slower

00:08:59.820 --> 00:09:01.900
because we're overcoming that static friction

00:09:01.900 --> 00:09:04.900
but once we budged it and once it's moving

00:09:04.920 --> 00:09:08.600
and assuming that we're continuing to apply a constant force over here

00:09:08.790 --> 00:09:10.480
then all of a sudden, the force of friction since

00:09:10.480 --> 00:09:14.400
we're kind of bump it along the top now and not stuck in their grooves

00:09:14.920 --> 00:09:17.570
we're now using the coefficient of kinetic friction

00:09:17.590 --> 00:09:22.780
And so once it's moving, the net force becomes greater in the rightward direction because

00:09:22.930 --> 00:09:27.590
you can kind of view that force of friction will become less once it starts moving

00:09:27.790 --> 00:09:31.820
And so now the force of friction went down a little bit to 26.95 N

00:09:31.870 --> 00:09:38.630
And so now we're accelerating to right at a slightly faster rate 14.61 m/s^2

00:09:38.640 --> 00:09:41.860
So right when you budge it, it accelerates at 14.1 m/s^2

00:09:41.860 --> 00:09:44.870
but just for a moment, almost unnoticeable moment once it starts moving

00:09:44.920 --> 00:09:48.780
Then you're going to be going to the right with this constant acceleration