Ex 2: Distributive property to simplify

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:01.860
Solve for x.

00:00:01.860 --> 00:00:06.675
And we have x minus 8 is
equal to x/3 plus 1/6.

00:00:06.675 --> 00:00:08.300
Now the first thing
I want to do here--

00:00:08.300 --> 00:00:10.740
and there's multiple
ways to do this problem--

00:00:10.740 --> 00:00:13.140
but what I want to do is just
to simplify the fraction.

00:00:13.140 --> 00:00:17.180
I'm going to multiply everything
times the least common multiple

00:00:17.180 --> 00:00:18.870
of all of these
guys' denominators.

00:00:18.870 --> 00:00:20.040
This is essentially x/1.

00:00:20.040 --> 00:00:23.510
This is 8/1, x/3, 1/6.

00:00:23.510 --> 00:00:28.140
The least common multiple
of 1, 3, and 6 is 6.

00:00:28.140 --> 00:00:30.527
So if I multiply
everything times 6,

00:00:30.527 --> 00:00:33.110
what that's going to do is going
to clear out these fractions.

00:00:33.110 --> 00:00:34.690
So these weren't
fractions to begin with,

00:00:34.690 --> 00:00:36.200
so we're just
multiplying them by 6.

00:00:36.200 --> 00:00:42.870
So it becomes 6x minus 6 times
negative 8, or 6 times 8 is 48.

00:00:42.870 --> 00:00:45.600
And we're subtracting
it right over there.

00:00:45.600 --> 00:00:48.251
And then we have x/3 times 6.

00:00:48.251 --> 00:00:49.500
Let me just write it out here.

00:00:49.500 --> 00:00:55.900
So that's going to be 6
times x/3 plus 6 times 1/6.

00:00:55.900 --> 00:01:02.230
Or we get 6x minus 48 is equal
to 6 times something divided

00:01:02.230 --> 00:01:03.210
by 3.

00:01:03.210 --> 00:01:05.360
That's the same thing
as 6 divided by 3 times

00:01:05.360 --> 00:01:06.450
that something.

00:01:06.450 --> 00:01:10.130
That's just going to
be equal to 2x plus 6

00:01:10.130 --> 00:01:14.070
times 1/6 or 6 divided by
6 is just going to be 1.

00:01:14.070 --> 00:01:16.410
So that first step cleared
out all of the fractions

00:01:16.410 --> 00:01:18.640
and now this is just a
straightforward problem

00:01:18.640 --> 00:01:21.370
with all integer
coefficients or integers

00:01:21.370 --> 00:01:22.920
on either side of the equation.

00:01:22.920 --> 00:01:24.295
And what we want
to do is we want

00:01:24.295 --> 00:01:26.480
to isolate all of the x's
on one side or the other.

00:01:26.480 --> 00:01:29.860
And we might as well isolate
them all on the left hand side.

00:01:29.860 --> 00:01:31.872
So let's subtract
2x from both sides.

00:01:31.872 --> 00:01:33.330
We want to get rid
of this 2x here.

00:01:33.330 --> 00:01:35.460
That's why I'm
subtracting the 2x.

00:01:35.460 --> 00:01:39.460
So let's subtract
2x from both sides.

00:01:39.460 --> 00:01:44.330
And on the right hand side,
I have 2x plus 1 minus 2x.

00:01:44.330 --> 00:01:45.160
Those cancel out.

00:01:45.160 --> 00:01:46.201
That was the whole point.

00:01:46.201 --> 00:01:48.620
So I'm left with just
this 1 over here.

00:01:48.620 --> 00:01:51.860
On the left hand side
I have 6x minus 2x.

00:01:51.860 --> 00:01:53.355
Well, that's just
going to be 4x.

00:01:53.355 --> 00:01:55.480
If I have 6 of something
minus 2 of that something,

00:01:55.480 --> 00:01:57.000
I have 4 of that something.

00:01:57.000 --> 00:01:59.630
Minus 48.

00:01:59.630 --> 00:02:02.640
And now I can-- let's see,
I want to get rid of this 48

00:02:02.640 --> 00:02:04.880
on the left hand side
because I want only x's here.

00:02:04.880 --> 00:02:08.714
So let me add 48 to both
sides of the equation.

00:02:08.714 --> 00:02:09.880
I'll do this in a new color.

00:02:09.880 --> 00:02:15.970
So let me add 48 to both
sides of this equation.

00:02:15.970 --> 00:02:19.000
And on the left hand
side 4x minus 48 plus 48,

00:02:19.000 --> 00:02:19.890
those cancel out.

00:02:19.890 --> 00:02:21.770
I'm left with just a 4x.

00:02:21.770 --> 00:02:26.100
And on the right hand side,
1 plus 48 is going to be 49.

00:02:26.100 --> 00:02:29.630
And now I've isolated the x but
it's still multiplied by a 4.

00:02:29.630 --> 00:02:32.950
So to make that a 1 coefficient,
let's multiply both sides

00:02:32.950 --> 00:02:33.810
by 1/4.

00:02:33.810 --> 00:02:38.200
Or you could also say, let's
divide both sides by 4.

00:02:38.200 --> 00:02:41.397
Anything you do to one side
you have to do to the other.

00:02:41.397 --> 00:02:43.230
And so you have-- what
do we have over here?

00:02:43.230 --> 00:02:45.910
4x/4 is just x.

00:02:45.910 --> 00:02:48.550
x is equal to 49 over 4.

00:02:48.550 --> 00:02:51.120
And that's about as far
as we can simplify it

00:02:51.120 --> 00:02:55.280
because these don't have any
common factors, 49 and 4.

00:02:55.280 --> 00:02:58.629
Let's check to see whether
49/4 is indeed the answer.

00:02:58.629 --> 00:03:00.420
So let's put it into
the original equation.

00:03:00.420 --> 00:03:01.880
Remember, the
original equation is

00:03:01.880 --> 00:03:06.450
what we have in green here
before we multiplied it by six.

00:03:06.450 --> 00:03:08.570
But in theory, we
should be able to put it

00:03:08.570 --> 00:03:11.340
into any of these steps
and the x should satisfy.

00:03:11.340 --> 00:03:13.500
But let's do it in
our original equation.

00:03:13.500 --> 00:03:14.790
So we have x minus 8.

00:03:14.790 --> 00:03:25.360
So we have 49/4 minus 8
should be equal to 49/4 over 3

00:03:25.360 --> 00:03:28.000
plus 1/6.

00:03:28.000 --> 00:03:30.490
So let's see what
we can do here.

00:03:30.490 --> 00:03:33.299
So we can multiply.

00:03:33.299 --> 00:03:34.340
Well, like we did before.

00:03:34.340 --> 00:03:37.380
We can multiply both sides
of this equation by 6.

00:03:37.380 --> 00:03:40.890
That'll help simplify a
lot of the fractions here.

00:03:40.890 --> 00:03:42.830
So if we multiply both
sides of this equation

00:03:42.830 --> 00:03:49.350
by 6-- so we're going to
multiply everything by 6--

00:03:49.350 --> 00:03:52.260
what do we get on
the left hand side?

00:03:52.260 --> 00:03:55.290
6/4 is the same
thing as 3/2, right?

00:03:55.290 --> 00:04:00.060
So this is going to
be 3 times 49 over 2.

00:04:00.060 --> 00:04:07.430
3 times 49 over 2 minus 48
will be equal to 6 divided by 3

00:04:07.430 --> 00:04:07.990
is 2.

00:04:07.990 --> 00:04:11.590
So it's going to be
2 times 49 over 4,

00:04:11.590 --> 00:04:15.090
which is the same thing as 49/2.

00:04:15.090 --> 00:04:18.950
And then 6 times 1/6 is plus 1.

00:04:18.950 --> 00:04:20.899
And let's see.

00:04:20.899 --> 00:04:23.790
Well, I'm just going to
actually just evaluate them out.

00:04:23.790 --> 00:04:27.714
I could just essentially have
to subtract 49/2 from both sides

00:04:27.714 --> 00:04:28.880
and that'll simplify things.

00:04:28.880 --> 00:04:32.140
But let me just figure out
what these evaluate to.

00:04:32.140 --> 00:04:36.189
So 3 times 49, 49 times 3.

00:04:36.189 --> 00:04:37.230
You could think about it.

00:04:37.230 --> 00:04:39.480
It's going to be 3
less than 50 times 3.

00:04:39.480 --> 00:04:40.619
So it's 147.

00:04:40.619 --> 00:04:41.910
But let's just multiply it out.

00:04:41.910 --> 00:04:48.040
9 times 3 is 27 4 times
3 is 12, plus 2 is 14.

00:04:48.040 --> 00:04:49.160
So 147.

00:04:49.160 --> 00:04:49.930
So this is 147/2.

00:04:52.620 --> 00:04:55.150
And then let's put this
over a denominator of 2.

00:04:55.150 --> 00:05:00.330
So 48 is equal to 96/2, right?

00:05:00.330 --> 00:05:03.720
96 divided by 2 is 48 I
just multiplied this by 2.

00:05:03.720 --> 00:05:10.600
So this is minus 96/2 needs
to be equal to 49/2 plus--

00:05:10.600 --> 00:05:14.980
and instead of having this
1 let's write that as 2/2.

00:05:14.980 --> 00:05:18.920
And what's 47 minus 96?

00:05:18.920 --> 00:05:25.150
So 147 minus 100
would be 47, but we're

00:05:25.150 --> 00:05:27.270
going to subtract
4 more than that.

00:05:27.270 --> 00:05:30.010
So we're going to
have-- we're going

00:05:30.010 --> 00:05:31.610
to subtract 4 less than 100.

00:05:31.610 --> 00:05:36.940
So it's going to be 147
minus 96 is going to be 51.

00:05:36.940 --> 00:05:43.440
51/2 is equal to--
49 plus 2 is 51/2.

00:05:43.440 --> 00:05:45.970
So it all checks out.


Compound inequalities 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.180
Solve for y.

00:00:02.180 --> 00:00:08.640
We have 3y plus 7 is less
than 2y and 4y plus 8

00:00:08.640 --> 00:00:10.934
is greater than negative 48.

00:00:10.934 --> 00:00:12.600
So we have to find
all the y's that meet

00:00:12.600 --> 00:00:14.260
both of these constraints.

00:00:14.260 --> 00:00:16.860
So let's just solve for y
in each of the constraints

00:00:16.860 --> 00:00:19.480
and just remember that
this "and" is here.

00:00:19.480 --> 00:00:24.200
So we have 3y plus
7 is less than 2y.

00:00:24.200 --> 00:00:27.150
So let's isolate the y's
on the left-hand side.

00:00:27.150 --> 00:00:29.650
So let's get rid of this
2y on the right-hand side,

00:00:29.650 --> 00:00:34.080
and we can do that by
subtracting 2y from both sides.

00:00:34.080 --> 00:00:37.650
So we're going to subtract
2y from both sides.

00:00:37.650 --> 00:00:41.190
The left-hand side, we have
3y minus 2y, which is just y,

00:00:41.190 --> 00:00:44.969
plus 7 is less than 2y minus 2y.

00:00:44.969 --> 00:00:46.260
And there's nothing else there.

00:00:46.260 --> 00:00:48.780
That's just going to be 0.

00:00:48.780 --> 00:00:50.260
And then we can
get rid of this 7

00:00:50.260 --> 00:00:53.620
here by subtracting
7 from both sides.

00:00:53.620 --> 00:00:56.470
So let's subtract
7 from both sides.

00:00:56.470 --> 00:00:58.970
Left-hand side,
y plus 7 minus 7.

00:00:58.970 --> 00:00:59.780
Those cancel out.

00:00:59.780 --> 00:01:05.216
We just have y is less than 0
minus 7, which is negative 7.

00:01:05.216 --> 00:01:06.590
So that's one of
the constraints.

00:01:06.590 --> 00:01:09.000
That's this constraint
right over here.

00:01:09.000 --> 00:01:11.710
Now let's work on
this constraint.

00:01:11.710 --> 00:01:19.400
We have 4y plus 8 is
greater than negative 48.

00:01:19.400 --> 00:01:21.700
So let's get rid of the 8
from the left-hand side.

00:01:21.700 --> 00:01:23.640
So we can subtract
8 from both sides.

00:01:27.740 --> 00:01:30.930
The left-hand side,
we're just left with a 4y

00:01:30.930 --> 00:01:32.400
because these guys cancel out.

00:01:32.400 --> 00:01:36.950
4y is greater than
negative 48 minus 8.

00:01:36.950 --> 00:01:38.830
So we're going to go
another 8 negative.

00:01:38.830 --> 00:01:43.840
So 48 plus 8 would
be a 56, so this

00:01:43.840 --> 00:01:47.090
is going to be negative 56.

00:01:47.090 --> 00:01:49.300
And now to isolate the y,
we can divide both sides

00:01:49.300 --> 00:01:52.120
by positive 4, and we don't
have to swap the inequality

00:01:52.120 --> 00:01:54.790
since we're dividing
by a positive number.

00:01:54.790 --> 00:01:57.800
So it's divide both
sides by 4 over here.

00:01:57.800 --> 00:02:05.210
So we get y is greater than--
what is 56/4, or negative 56/4?

00:02:05.210 --> 00:02:05.710
Let's see.

00:02:05.710 --> 00:02:09.750
40 is 10 times 4, and then we
have another 16 to worry about.

00:02:09.750 --> 00:02:10.970
So it's 14 times 4.

00:02:10.970 --> 00:02:15.467
So y is greater
than negative 14.

00:02:15.467 --> 00:02:16.050
Is that right?

00:02:16.050 --> 00:02:18.512
4 times 10 is 40,
4 times 4 is 16.

00:02:18.512 --> 00:02:19.750
Yep, 56.

00:02:19.750 --> 00:02:24.600
So y is greater than negative
14 and-- let's remember,

00:02:24.600 --> 00:02:29.690
we have this "and" here-- and
y is less than negative 7.

00:02:29.690 --> 00:02:32.670
So we have to meet both of
these constraints over here.

00:02:32.670 --> 00:02:36.940
So let's draw them
on the number line.

00:02:36.940 --> 00:02:40.460
So I have my number
line over here.

00:02:40.460 --> 00:02:45.710
And let's say negative
14 is over here.

00:02:45.710 --> 00:02:49.829
So you have negative
13, 12, 11, 10, 9, 8,

00:02:49.829 --> 00:02:54.252
7-- that's negative 7-- and
then negative 6, 5, 4, 3, 2, 1.

00:02:54.252 --> 00:02:55.710
This would be 0,
and then you could

00:02:55.710 --> 00:02:58.080
keep going up more positive.

00:02:58.080 --> 00:03:00.350
And so we're looking
for all of the y's that

00:03:00.350 --> 00:03:02.820
are less than negative 7.

00:03:02.820 --> 00:03:06.020
So let's look at this,
less than negative 7.

00:03:06.020 --> 00:03:07.910
So not including
negative 7, so we'll

00:03:07.910 --> 00:03:10.190
do an open circle
around negative 7,

00:03:10.190 --> 00:03:12.400
and less than negative 7.

00:03:12.400 --> 00:03:13.900
And if that was the
only constraint,

00:03:13.900 --> 00:03:15.530
we would keep going to the left.

00:03:15.530 --> 00:03:17.720
But we have this other
constraint-- and y

00:03:17.720 --> 00:03:19.610
has to be greater
than negative 14.

00:03:23.860 --> 00:03:26.010
So you make a circle
around negative 14,

00:03:26.010 --> 00:03:27.870
and everything that's
greater than that.

00:03:27.870 --> 00:03:28.960
And if you didn't have
this other constraint,

00:03:28.960 --> 00:03:30.340
you would keep going.

00:03:30.340 --> 00:03:34.060
But the y's that
satisfy both of them

00:03:34.060 --> 00:03:35.920
are all of the y's in between.

00:03:35.920 --> 00:03:41.650
These are the y's that are
both less than negative 7

00:03:41.650 --> 00:03:43.770
and greater than negative 14.

00:03:43.770 --> 00:03:46.440
And we can verify
that things here work.

00:03:46.440 --> 00:03:48.930
So let's try some values out.

00:03:48.930 --> 00:03:50.450
So a value that
would work, well,

00:03:50.450 --> 00:03:52.140
let me just do
negative 10 is right

00:03:52.140 --> 00:03:54.060
here, 8, 9, this is negative 10.

00:03:54.060 --> 00:03:54.910
That should work.

00:03:54.910 --> 00:03:56.140
So let's try it out.

00:03:56.140 --> 00:04:02.130
So we'd have 3 times
negative 10 plus 7 should

00:04:02.130 --> 00:04:05.140
be less than 2
times negative 10.

00:04:05.140 --> 00:04:09.280
So this is negative 30 plus
7 is negative 23, which

00:04:09.280 --> 00:04:11.450
is indeed less than negative 20.

00:04:11.450 --> 00:04:12.500
So that works.

00:04:12.500 --> 00:04:15.610
And negative 10 has to
work for this one as well.

00:04:15.610 --> 00:04:20.329
So you have 4 times negative 10,
which is negative 40, plus 8.

00:04:20.329 --> 00:04:25.220
Negative 40 plus 8 should
be greater than negative 48.

00:04:25.220 --> 00:04:28.870
Well, negative 40
plus 8 is negative 32.

00:04:28.870 --> 00:04:31.200
We're going 8 in the
positive direction,

00:04:31.200 --> 00:04:32.980
so we're getting less negative.

00:04:32.980 --> 00:04:36.740
And negative 32 is
greater than negative 48.

00:04:36.740 --> 00:04:38.750
It's less negative.

00:04:38.750 --> 00:04:40.150
So this works.

00:04:40.150 --> 00:04:42.300
So negative 10 works.

00:04:42.300 --> 00:04:45.000
Now, let's just verify some
things that shouldn't work.

00:04:45.000 --> 00:04:48.370
So 0 should not work.

00:04:48.370 --> 00:04:49.747
It's not in the solution set.

00:04:49.747 --> 00:04:50.580
So let's try it out.

00:04:50.580 --> 00:04:53.280
We've got 3 times is 0 plus 7.

00:04:53.280 --> 00:04:54.550
That would be 7.

00:04:54.550 --> 00:04:58.350
And 7 is not less than 0.

00:04:58.350 --> 00:05:00.980
So it would violate this
condition right over here

00:05:00.980 --> 00:05:02.650
if we put a 0 over here.

00:05:02.650 --> 00:05:05.630
If you put a negative
15 over here,

00:05:05.630 --> 00:05:08.450
it should violate this
condition right over here

00:05:08.450 --> 00:05:10.780
because it wasn't in
this guy's solution set.

00:05:10.780 --> 00:05:13.280
Anyway, hopefully you
found that useful.


Equation special cases

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:01.640
Solve for x.

00:00:01.640 --> 00:00:04.690
We have 8 times the
quantity 3x plus 10

00:00:04.690 --> 00:00:08.940
is equal to 28x
minus 14 minus 4x.

00:00:08.940 --> 00:00:11.000
So like every equation
we've done so far,

00:00:11.000 --> 00:00:13.632
we just want to isolate
all of the x's on one side

00:00:13.632 --> 00:00:14.340
of this equation.

00:00:14.340 --> 00:00:15.923
But before we do
that, we can actually

00:00:15.923 --> 00:00:17.310
simplify each of these sides.

00:00:17.310 --> 00:00:19.130
On the left-hand
side, we can multiply

00:00:19.130 --> 00:00:21.217
the quantity 3x plus 10 times 8.

00:00:21.217 --> 00:00:22.800
So we're essentially
just distributing

00:00:22.800 --> 00:00:25.470
the 8, the distributive
property right here.

00:00:25.470 --> 00:00:29.550
So this is the same thing
as 8 times 3x, which is 24x,

00:00:29.550 --> 00:00:34.460
plus 8 times 10, which
is 80, is equal to--

00:00:34.460 --> 00:00:37.910
and over here, we have
28x minus 14 minus 4x.

00:00:37.910 --> 00:00:41.140
So we can combine the
28x and the minus 4x.

00:00:41.140 --> 00:00:45.540
If we have 28x minus
4x, that is 24x

00:00:45.540 --> 00:00:48.392
And then you have the
minus 14 right over here.

00:00:48.392 --> 00:00:50.350
Now, the next thing we
could-- and it's already

00:00:50.350 --> 00:00:53.220
looking a little bit suspicious,
but just to confirm that it's

00:00:53.220 --> 00:00:55.300
as suspicious as
it looks, let's try

00:00:55.300 --> 00:00:58.790
to subtract 24x from both
sides of this equation.

00:00:58.790 --> 00:01:02.031
And if we do that, we see that
we actually remove the x's

00:01:02.031 --> 00:01:04.530
from both sides of the equation
because we have a 24x there,

00:01:04.530 --> 00:01:05.770
and we have a 24x there.

00:01:05.770 --> 00:01:06.570
You might say,
hey, let's put all

00:01:06.570 --> 00:01:07.850
the x's on the left-hand side.

00:01:07.850 --> 00:01:09.920
So let's get rid of this 24x.

00:01:09.920 --> 00:01:11.929
So you subtract 24x
right over there,

00:01:11.929 --> 00:01:14.095
but you have to do it to
the left-hand side as well.

00:01:16.610 --> 00:01:18.680
On the left-hand side,
these guys cancel out,

00:01:18.680 --> 00:01:24.390
and you're left with just
80-- these guys cancel out

00:01:24.390 --> 00:01:27.670
as well-- is equal
to a negative 14.

00:01:27.670 --> 00:01:29.010
Now, this looks very bizarre.

00:01:29.010 --> 00:01:33.320
It's making a statement that
80 is equal to negative 14,

00:01:33.320 --> 00:01:35.040
which we know is not true.

00:01:35.040 --> 00:01:36.020
This does not happen.

00:01:36.020 --> 00:01:38.190
80 is never equal
to negative 14.

00:01:38.190 --> 00:01:40.240
They're just inherently inequal.

00:01:40.240 --> 00:01:45.300
So this equation right here
actually has no solution.

00:01:45.300 --> 00:01:48.770
This has no solution.

00:01:48.770 --> 00:01:53.920
There is a no x-value that will
make 80 equal to negative 14.


Compound inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:01.880
Solve for z.

00:00:01.880 --> 00:00:07.280
5z plus 7 is less
than 27 or negative 3z

00:00:07.280 --> 00:00:09.710
is less than or equal to 18.

00:00:09.710 --> 00:00:12.200
So this is a
compound inequality.

00:00:12.200 --> 00:00:14.340
We have two conditions here.

00:00:14.340 --> 00:00:19.350
So z can satisfy this or z
can satisfy this over here.

00:00:19.350 --> 00:00:21.600
So let's just solve each
of these inequalities.

00:00:21.600 --> 00:00:24.610
And just know that z can
satisfy either of them.

00:00:24.610 --> 00:00:26.242
So let's just look at this.

00:00:26.242 --> 00:00:27.950
So if we look at just
this one over here,

00:00:27.950 --> 00:00:32.890
we have 5z plus 7
is less than 27.

00:00:32.890 --> 00:00:35.680
Let's isolate the z's
on the left-hand side.

00:00:35.680 --> 00:00:38.520
So let's subtract 7 from both
sides to get rid of this 7

00:00:38.520 --> 00:00:39.920
on the left-hand side.

00:00:39.920 --> 00:00:43.680
And so our left-hand side
is just going to be 5z.

00:00:43.680 --> 00:00:46.550
Plus 7, minus 7--
those cancel out.

00:00:46.550 --> 00:00:52.740
5z is less than
27 minus 7, is 20.

00:00:52.740 --> 00:00:55.300
So we have 5z is less than 20.

00:00:55.300 --> 00:00:59.221
Now we can divide both sides
of this inequality by 5.

00:00:59.221 --> 00:01:01.470
And we don't have to swap
the inequality because we're

00:01:01.470 --> 00:01:03.470
dividing by a positive number.

00:01:03.470 --> 00:01:08.180
And so we get z
is less than 20/5.

00:01:08.180 --> 00:01:10.742
z is less than 4.

00:01:10.742 --> 00:01:12.450
Now, this was only
one of the conditions.

00:01:12.450 --> 00:01:14.580
Let's [? look at ?] the
other one over here.

00:01:14.580 --> 00:01:20.230
We have negative 3z is
less than or equal to 18.

00:01:20.230 --> 00:01:22.020
Now, to isolate the
z, we could just

00:01:22.020 --> 00:01:25.730
divide both sides of this
inequality by negative 3.

00:01:25.730 --> 00:01:28.540
But remember, when
you divide or multiply

00:01:28.540 --> 00:01:30.660
both sides of an inequality
by a negative number,

00:01:30.660 --> 00:01:32.900
you have to swap the inequality.

00:01:32.900 --> 00:01:35.610
So we could write negative 3z.

00:01:35.610 --> 00:01:37.870
We're going to divide
it by negative 3.

00:01:37.870 --> 00:01:38.745
And then you have 18.

00:01:38.745 --> 00:01:40.800
We're going to divide
it by negative 3.

00:01:40.800 --> 00:01:43.000
But we're going to
swap the inequality.

00:01:43.000 --> 00:01:45.750
So the less than or equal
will become greater than

00:01:45.750 --> 00:01:46.730
or equal to.

00:01:46.730 --> 00:01:48.440
And so these guys cancel out.

00:01:48.440 --> 00:01:50.630
Negative 3 divided
by negative 3 is 1.

00:01:50.630 --> 00:01:55.370
So we have z is greater than
or equal to 18 over negative 3

00:01:55.370 --> 00:01:57.020
is negative 6.

00:01:57.020 --> 00:02:00.530
And remember, it's this
constraint or this constraint.

00:02:00.530 --> 00:02:04.220
And this constraint right
over here boils down to this.

00:02:04.220 --> 00:02:06.340
And this one boils down to this.

00:02:06.340 --> 00:02:09.020
So our solution set--
z is less than 4

00:02:09.020 --> 00:02:14.210
or z is greater than
or equal to negative 6.

00:02:14.210 --> 00:02:15.350
So let me make this clear.

00:02:15.350 --> 00:02:16.110
Let me rewrite it.

00:02:16.110 --> 00:02:21.410
So z could be less than
4 or z is greater than

00:02:21.410 --> 00:02:23.360
or equal to negative 6.

00:02:23.360 --> 00:02:25.220
It can satisfy
either one of these.

00:02:25.220 --> 00:02:28.240
And this is kind of
interesting here.

00:02:28.240 --> 00:02:29.525
Let's plot these.

00:02:32.080 --> 00:02:35.500
So there's a number
line right over there.

00:02:35.500 --> 00:02:37.480
Let's say that 0 is over here.

00:02:37.480 --> 00:02:41.690
We have 1, 2, 3, 4
is right over there.

00:02:41.690 --> 00:02:42.680
And then negative 6.

00:02:42.680 --> 00:02:46.440
We have 1, 2, 3, 4, 5, 6.

00:02:46.440 --> 00:02:48.500
That's negative 6 over there.

00:02:48.500 --> 00:02:54.260
Now, let's think about
z being less than 4.

00:02:54.260 --> 00:02:56.760
We would put a circle around
4, since we're not including 4.

00:02:56.760 --> 00:03:00.615
And it'd be everything
less than 4.

00:03:03.130 --> 00:03:05.430
Now let's think
about what z being

00:03:05.430 --> 00:03:07.800
greater than or equal to
negative 6 would mean.

00:03:07.800 --> 00:03:09.960
That means you can
include negative 6.

00:03:09.960 --> 00:03:12.570
And it's everything-- let me
do that in different color.

00:03:12.570 --> 00:03:15.895
It means you can
include negative 6.

00:03:15.895 --> 00:03:18.177
I want to do that--
oh, here we go.

00:03:18.177 --> 00:03:19.510
It means you include negative 6.

00:03:19.510 --> 00:03:21.440
Let me do it in a
more different color.

00:03:21.440 --> 00:03:22.660
Do it in orange.

00:03:22.660 --> 00:03:24.930
So z is greater than
equal to negative 6.

00:03:24.930 --> 00:03:26.440
Means you can
include negative 6.

00:03:26.440 --> 00:03:32.260
And it's everything greater
than that, including 4.

00:03:32.260 --> 00:03:35.180
So it's everything
greater than that.

00:03:35.180 --> 00:03:36.640
So what we see is
we've essentially

00:03:36.640 --> 00:03:38.450
shaded in the
entire number line.

00:03:38.450 --> 00:03:42.130
Every number will meet either
one of these constraints

00:03:42.130 --> 00:03:43.036
or both of them.

00:03:43.036 --> 00:03:44.660
If we're over here,
we're going to meet

00:03:44.660 --> 00:03:46.696
both of the constraints.

00:03:46.696 --> 00:03:48.070
If we're a number
out here, we're

00:03:48.070 --> 00:03:49.180
going to meet this constraint.

00:03:49.180 --> 00:03:50.710
If we're a number
down here, we're

00:03:50.710 --> 00:03:51.800
going to meet this constraint.

00:03:51.800 --> 00:03:54.050
And you could just try it
out with a bunch of numbers.

00:03:54.050 --> 00:03:56.060
0 will work.

00:03:56.060 --> 00:03:59.440
0 plus 7 is 7, which
is less than 27.

00:03:59.440 --> 00:04:02.920
And 3 times 0 is less than 18,
so it meets both constraints.

00:04:02.920 --> 00:04:06.100
If we put 4 here, it should only
meet one of the constraints.

00:04:06.100 --> 00:04:09.756
Negative 3 times 4 is negative
12, which is less than 18.

00:04:09.756 --> 00:04:12.380
So it meets this constraint, but
it won't meet this constraint.

00:04:12.380 --> 00:04:15.250
Because you do 5
times 4 plus 7 is

00:04:15.250 --> 00:04:17.540
27, which is not less than 27.

00:04:17.540 --> 00:04:18.740
It's equal to 27.

00:04:18.740 --> 00:04:20.089
Remember, this is an or.

00:04:20.089 --> 00:04:22.200
So you just have to meet
one of the constraints.

00:04:22.200 --> 00:04:24.020
So 4 meets this constraint.

00:04:24.020 --> 00:04:25.330
So even 4 works.

00:04:25.330 --> 00:04:27.270
So it's really the
entire number line

00:04:27.270 --> 00:04:31.100
will satisfy either one or
both of these constraints.


Rearrange formulas to isolate specific variables

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.850
The formula for the
area of a triangle

00:00:02.850 --> 00:00:08.800
is A is equal to 1/2 b times
h, where A is equal to area,

00:00:08.800 --> 00:00:10.800
b is equal to
length of the base,

00:00:10.800 --> 00:00:13.440
and h is equal to the
length of the height.

00:00:13.440 --> 00:00:15.600
So area is equal to
1/2 times the length

00:00:15.600 --> 00:00:18.760
of the base times the
length of the height.

00:00:18.760 --> 00:00:21.370
Solve this formula
for the height.

00:00:21.370 --> 00:00:23.080
So just to visualize
this a little bit,

00:00:23.080 --> 00:00:26.140
let me draw a triangle here.

00:00:26.140 --> 00:00:29.540
Let me draw a triangle just
so we know what b and h are.

00:00:29.540 --> 00:00:31.050
b would be the
length of the base.

00:00:31.050 --> 00:00:33.760
So this distance
right over here is b.

00:00:33.760 --> 00:00:39.000
And then this distance
right here is our height.

00:00:39.000 --> 00:00:40.580
That is the height
of the triangle--

00:00:40.580 --> 00:00:42.538
let me do that at a lower
case h because that's

00:00:42.538 --> 00:00:44.150
how we wrote it in the formula.

00:00:44.150 --> 00:00:46.820
Now, they want us to solve
this formula for the height.

00:00:46.820 --> 00:00:52.370
So the formula is area is
equal to 1/2 base times height.

00:00:52.370 --> 00:00:54.075
And we want to solve for h.

00:00:54.075 --> 00:00:55.575
We essentially want
to isolate the h

00:00:55.575 --> 00:00:57.140
on one side of the equation.

00:00:57.140 --> 00:00:58.670
It's already on the
right-hand side.

00:00:58.670 --> 00:01:01.600
So let's get rid of everything
else on the right-hand side.

00:01:01.600 --> 00:01:04.140
So we can do it-- well, I'll
do it one step at a time.

00:01:04.140 --> 00:01:06.270
We could kind of skip
steps if we wanted to.

00:01:06.270 --> 00:01:08.550
But let's see if we can
get rid of this 1/2.

00:01:08.550 --> 00:01:10.820
So the best way to get
rid of a 1/2 that's

00:01:10.820 --> 00:01:12.820
being multiplied by
h is if we multiply

00:01:12.820 --> 00:01:15.290
both sides of the equation
by its reciprocal.

00:01:15.290 --> 00:01:19.170
If we multiply both sides of
the equation by 2/1 or by 2.

00:01:19.170 --> 00:01:20.170
So let's do that.

00:01:20.170 --> 00:01:21.711
So let's multiply--
remember anything

00:01:21.711 --> 00:01:23.610
you do to one side
of the equation,

00:01:23.610 --> 00:01:26.790
you also have to do to the
other side of the equation.

00:01:26.790 --> 00:01:28.120
Now, what did this do?

00:01:28.120 --> 00:01:30.480
Well, the whole point
behind multiplying by 2

00:01:30.480 --> 00:01:32.630
is 2 times 1/2 is 1.

00:01:32.630 --> 00:01:34.670
So on the right-hand
side of the equation,

00:01:34.670 --> 00:01:36.530
we're just going to have a bh.

00:01:36.530 --> 00:01:41.280
And on the left-hand side of
the equation, we have a 2A.

00:01:41.280 --> 00:01:45.190
And we're almost there, we
have a b multiplying by an h.

00:01:45.190 --> 00:01:47.275
If we want to just
isolate the h,

00:01:47.275 --> 00:01:51.630
we could divide both sides
of this equation by b.

00:01:51.630 --> 00:01:53.380
We're just dividing both sides.

00:01:53.380 --> 00:01:56.150
You can almost view b as
the coefficient on the h.

00:01:56.150 --> 00:01:58.740
We're just dividing
both sides by b.

00:01:58.740 --> 00:01:59.790
And then what do we get?

00:01:59.790 --> 00:02:01.780
Well, the right-hand
side, the b's cancel out.

00:02:01.780 --> 00:02:04.590
On the left-hand side, we're
just left with 2A over b.

00:02:04.590 --> 00:02:07.860
So we get h-- and I'm just
swapping the sides here.

00:02:07.860 --> 00:02:10.020
h is equal to 2A over b.

00:02:12.680 --> 00:02:13.600
And we're done.

00:02:13.600 --> 00:02:17.635
We have solved this
formula for the height.

00:02:17.635 --> 00:02:19.010
And I guess this
could be useful.

00:02:19.010 --> 00:02:21.931
If someone just gave
you a bunch of areas

00:02:21.931 --> 00:02:23.680
and a bunch of base
lengths, and they said

00:02:23.680 --> 00:02:25.875
keep giving me the
height for those values,

00:02:25.875 --> 00:02:28.260
or for those triangles.


Word problem: solving equations

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:05.190
The perimeter of Tina's
rectangular garden is 60 feet.

00:00:05.190 --> 00:00:09.980
If the length of the
garden is twice the width,

00:00:09.980 --> 00:00:12.580
what are the dimensions
of the garden?

00:00:12.580 --> 00:00:15.190
So let's draw this garden
here, Tina's garden.

00:00:15.190 --> 00:00:16.750
So it's a rectangle.

00:00:16.750 --> 00:00:19.010
They tell us that it's
a rectangular garden.

00:00:19.010 --> 00:00:21.500
So it looks something like this.

00:00:21.500 --> 00:00:23.230
And let's say that
this is the width.

00:00:23.230 --> 00:00:24.646
So if this is the
width, then this

00:00:24.646 --> 00:00:26.110
is also going to be the width.

00:00:26.110 --> 00:00:27.900
And this is the length up here.

00:00:27.900 --> 00:00:30.070
And they tell us that
the length of the garden

00:00:30.070 --> 00:00:31.570
is twice the width.

00:00:31.570 --> 00:00:35.310
So if this is w, then the
length is going to be 2w.

00:00:35.310 --> 00:00:36.910
It's going to be
twice the width.

00:00:36.910 --> 00:00:40.870
This is also going
to be 2w over here.

00:00:40.870 --> 00:00:42.910
Now, what's the
perimeter of this garden?

00:00:42.910 --> 00:00:46.502
Well, it's going to be w
plus w plus 2w plus 2w.

00:00:46.502 --> 00:00:47.460
Let me write this down.

00:00:47.460 --> 00:00:49.001
The perimeter of
this garden is going

00:00:49.001 --> 00:00:57.370
to be equal to w plus
2w plus w plus 2w, which

00:00:57.370 --> 00:00:58.460
is equal to what?

00:00:58.460 --> 00:01:03.300
This is w plus 2w is 3w, 4w, 6w.

00:01:03.300 --> 00:01:05.560
So this is equal to 6w.

00:01:05.560 --> 00:01:07.770
That's the perimeter
in terms of the width.

00:01:07.770 --> 00:01:12.090
But they also tell us that
the actual numerical value

00:01:12.090 --> 00:01:15.000
of the perimeter is 60 feet.

00:01:15.000 --> 00:01:16.870
It is 60 feet.

00:01:16.870 --> 00:01:21.070
So this perimeter 6w
must be equal to 60

00:01:21.070 --> 00:01:23.310
if we assume that we're
dealing with feet.

00:01:23.310 --> 00:01:27.320
So we just have the
equation 6w is equal to 60.

00:01:27.320 --> 00:01:29.330
We can divide both
sides of this equation

00:01:29.330 --> 00:01:33.410
by 6 so that we have just
a w on the left-hand side.

00:01:33.410 --> 00:01:36.380
6w divided by 6 is just w.

00:01:36.380 --> 00:01:39.780
And then 60 divided by 6 is 10.

00:01:39.780 --> 00:01:41.820
So we have w is equal to 10.

00:01:41.820 --> 00:01:44.580
So the width of
the garden is 10.

00:01:44.580 --> 00:01:47.160
So this distance
over here is 10.

00:01:47.160 --> 00:01:49.210
And then what is the
length of the garden?

00:01:49.210 --> 00:01:51.010
Well, it's 2 times the width.

00:01:51.010 --> 00:01:53.000
So this is equal to 20.

00:01:53.000 --> 00:01:56.570
The length is equal to 20.

00:01:56.570 --> 00:01:57.320
And so we're done.

00:01:57.320 --> 00:02:00.567
This is a 20 by 10 garden.


How to evaluate an expression using substitution

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.170
Express 25 degrees
Celsius as a temperature

00:00:04.170 --> 00:00:08.600
in degrees Fahrenheit using
the formula Fahrenheit, or F,

00:00:08.600 --> 00:00:13.510
is equal to 9/5 times the
Celsius degrees plus 32.

00:00:13.510 --> 00:00:17.160
So they're telling us that
we have 25 degrees Celsius.

00:00:17.160 --> 00:00:19.550
So we could put
that in for C here,

00:00:19.550 --> 00:00:22.132
and we'll get the temperature
in Fahrenheit degrees.

00:00:22.132 --> 00:00:22.840
So let's do that.

00:00:22.840 --> 00:00:29.250
So we'll get F is equal
to 9/5-- now for C,

00:00:29.250 --> 00:00:35.490
we're going to put in
25-- times 25 plus 32.

00:00:35.490 --> 00:00:38.640
And now we can simplify this
before we multiply 25 times 9.

00:00:38.640 --> 00:00:42.350
Remember, this is the same
thing as 9/5 times 25/1.

00:00:42.350 --> 00:00:45.780
We can essentially divide the
numerator and the denominator

00:00:45.780 --> 00:00:47.730
of our eventual product by 5.

00:00:47.730 --> 00:00:50.680
If we divide 25 by 5, we get 5.

00:00:50.680 --> 00:00:53.110
If we divide 5 by 5, we get 1.

00:00:53.110 --> 00:00:56.770
So this boils down
to 9 times 5 plus 32.

00:00:56.770 --> 00:00:58.320
So our Fahrenheit
degrees are going

00:00:58.320 --> 00:01:04.790
to be 9 times 5 is
45 plus 32 degrees.

00:01:04.790 --> 00:01:09.190
Or it's equal to
45 plus 32 is 77,

00:01:09.190 --> 00:01:13.988
so this is 77
degrees Fahrenheit.


Absolute value inequalities example 2

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.110
Solve for p.

00:00:02.110 --> 00:00:08.000
We have the absolute value of p minus 12
plus 4 is less than 14.

00:00:08.000 --> 00:00:11.170
So, let's just do this one step at a time.

00:00:11.170 --> 00:00:12.450
The first thing we wanna do is we really

00:00:12.450 --> 00:00:15.980
just wanna isolate the part that has the
absolute value.

00:00:15.980 --> 00:00:18.930
So, to do that we can just get rid of this
positive 4 here.

00:00:18.930 --> 00:00:24.840
We can do that by subtracting 4 from both
sides of the inequality.

00:00:24.840 --> 00:00:28.858
And so the left hand side the positive 4
and the negative 4

00:00:28.858 --> 00:00:33.590
cancel out we're just left with the
absolute value of p minus 12.

00:00:33.590 --> 00:00:36.632
And on the right hand side it, we get 14
minus

00:00:36.632 --> 00:00:40.290
4 is 10 and we still have the less than
sign.

00:00:40.290 --> 00:00:43.908
So we have the absolute value of p minus
12 is less than 10.

00:00:43.908 --> 00:00:46.440
So let's just think about it a little bit.

00:00:46.440 --> 00:00:48.986
If I were to tell you, if I were to tell
you that

00:00:48.986 --> 00:00:52.950
the absolute value of x is less than 10
what does that mean?

00:00:52.950 --> 00:00:57.750
That means that the distance from x to 0
has to be less than 10.

00:00:57.750 --> 00:01:01.734
So if I were to draw a number line, if I
were to draw a number

00:01:01.734 --> 00:01:06.660
line and put 0 here we can only go up 10
away and even that's too far.

00:01:06.660 --> 00:01:08.168
It has to be less than 10.

00:01:08.168 --> 00:01:10.338
So if this is positive 10 it would have to

00:01:10.338 --> 00:01:13.381
be less than positive 10 cuz 10 is exactly
10 away.

00:01:13.381 --> 00:01:15.710
We have to be less than 10 away from 0.

00:01:16.710 --> 00:01:19.974
And then we could go all the way to the
left until negative 10

00:01:19.974 --> 00:01:24.430
and even that we wouldn't be able to
include because its absolute value is 10.

00:01:24.430 --> 00:01:26.344
It's not less than 10 but negative 9

00:01:26.344 --> 00:01:29.282
negative 9.999 we could include all of
those things.

00:01:29.282 --> 00:01:33.610
The absolute value of any of those things
is gonna be less than 10.

00:01:33.610 --> 00:01:38.758
So another way to write this, this
absolute value inequality

00:01:38.758 --> 00:01:43.708
is that x could be, x could be greater
than negative 10, x

00:01:43.708 --> 00:01:47.965
could be greater than negative 10 and x
needs to be

00:01:47.965 --> 00:01:52.170
less than 10 or we can write this as x is
between.

00:01:52.170 --> 00:01:54.080
So negative 10 is kinda the bottom

00:01:54.080 --> 00:01:57.420
boundary and we're not going to include
it.

00:01:57.420 --> 00:02:00.450
X is going to be greater than that and is
less than 10.

00:02:00.450 --> 00:02:04.500
So this is another way to write the
absolute value of x is less than 10.

00:02:04.500 --> 00:02:07.010
Is essentially saying that x has to be
between negative 10

00:02:07.010 --> 00:02:09.930
and 10 and it can't be either negative 10
or 10.

00:02:09.930 --> 00:02:12.080
We're not, there's no equals sign here.

00:02:12.080 --> 00:02:13.517
So, the same exact logic.

00:02:13.517 --> 00:02:16.340
Instead of an x we have a p minus 12 here.

00:02:16.340 --> 00:02:21.444
So, we can write the absolute value of p
minus 12 is less than 10 is

00:02:21.444 --> 00:02:27.020
saying that negative 10 is less than p
minus 12 which is less than 10.

00:02:27.020 --> 00:02:30.380
And we can just solve this compound
inequality all

00:02:30.380 --> 00:02:33.103
at once by isolating the p in the middle.

00:02:33.103 --> 00:02:35.315
The best way to isolate the p in the

00:02:35.315 --> 00:02:38.459
middle we wanna get rid of this negative
12.

00:02:38.459 --> 00:02:43.550
So let's add 12 to all three, all three
sections of this compound inequality.

00:02:45.060 --> 00:02:49.370
And so we get negative 10 plus 12 is
positive 2.

00:02:49.370 --> 00:02:53.620
And then is less than p minus 12 plus 12
is just p.

00:02:53.620 --> 00:02:58.480
And then is less than 10 plus 12 is 22.

00:02:58.480 --> 00:03:02.250
So p is greater than 2 and less than 22.

00:03:02.250 --> 00:03:08.710
So if we were to plot it on a number line
our solution set looks like this.

00:03:08.710 --> 00:03:11.510
So this might be 2 over here.

00:03:11.510 --> 00:03:15.350
This might be 22 over here maybe 0 sitting
right over here.

00:03:15.350 --> 00:03:17.620
P is greater than 2.

00:03:17.620 --> 00:03:19.710
P is greater than 2.

00:03:19.710 --> 00:03:23.060
It's not greater than or equal to so we
have to not fill this in.

00:03:23.060 --> 00:03:26.920
It has to be an open circle it's only
greater than and it's less than 22.

00:03:26.920 --> 00:03:30.390
It's not less than or equal to so we're
not gonna fill that circle in.

00:03:30.390 --> 00:03:34.470
And it's everything, it's everything in
between.

00:03:34.470 --> 00:03:36.510
And we can verify it for ourselves.

00:03:36.510 --> 00:03:39.690
Let's try a value that might work out
well.

00:03:39.690 --> 00:03:41.490
Well, 12 is in between these two numbers.

00:03:41.490 --> 00:03:43.170
It's in our magenta region right here.

00:03:43.170 --> 00:03:44.780
So let's try p is equal to 12.

00:03:44.780 --> 00:03:46.190
So if you have 12 minus 12.

00:03:46.190 --> 00:03:51.720
So it's the absolute value of 12 minus 12
plus 4 which should be less than 14.

00:03:51.720 --> 00:03:58.140
So this is 0 plus 4 which needs to be less
than 14 and 4 is definitely less than 14.

00:03:58.140 --> 00:04:01.150
So 12 worked 0 shouldn't work.

00:04:01.150 --> 00:04:03.210
Lets try 0.

00:04:03.210 --> 00:04:05.440
Zero minus 12 so it's the absolute value
of 0.

00:04:05.440 --> 00:04:07.900
I wanna do this in a different color.

00:04:07.900 --> 00:04:12.230
It is the absolute value of 0 minus 12
plus 4.

00:04:12.230 --> 00:04:14.120
This should not be less than 14.

00:04:14.120 --> 00:04:15.160
This should not work.

00:04:15.160 --> 00:04:17.709
So we get the absolute value of negative
12.

00:04:17.709 --> 00:04:20.005
Plus 4 should not be less than 14.

00:04:20.005 --> 00:04:23.246
Then we should end up with a contradiction
here.

00:04:23.246 --> 00:04:25.941
So we have 12 plus 4 less than 14 we end

00:04:25.941 --> 00:04:29.052
up with 16 is less than 14 which is not
true.

00:04:29.052 --> 00:04:32.900
So 0 does not work so at least we're
feeling pretty good.

00:04:32.900 --> 00:04:35.105
We took something outside of our solution
set didn't

00:04:35.105 --> 00:04:37.440
work something inside of our solution set
it did work.


One-step inequality involving addition

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.740
Solve for x plus 8 is
less than or equal to 6

00:00:04.740 --> 00:00:06.700
and graph the solution.

00:00:06.700 --> 00:00:12.480
So we have x plus 8 is
less than or equal to 6.

00:00:12.480 --> 00:00:15.810
So let's solve this
inequality for x.

00:00:15.810 --> 00:00:18.724
And the easiest way to isolate
an x on-- let's isolate

00:00:18.724 --> 00:00:20.390
on the left-hand side
since it's already

00:00:20.390 --> 00:00:22.130
there-- is to just
get rid of this 8.

00:00:22.130 --> 00:00:24.046
And the best way to get
rid of this positive 8

00:00:24.046 --> 00:00:27.640
is to subtract 8 from both
sides So let's subtract 8

00:00:27.640 --> 00:00:28.480
from both sides.

00:00:28.480 --> 00:00:31.795
That won't change the
direction of the inequality.

00:00:31.795 --> 00:00:34.210
So the left-hand side--
x plus 8 minus 8.

00:00:34.210 --> 00:00:35.690
And you're just left with an x.

00:00:35.690 --> 00:00:39.350
The right-hand side-- 6
minus 8 is negative 2.

00:00:39.350 --> 00:00:41.910
And we still have the
less than or equal.

00:00:41.910 --> 00:00:43.260
So we solved the inequality.

00:00:43.260 --> 00:00:46.440
We have x is less than
or equal to negative 2.

00:00:46.440 --> 00:00:48.940
So let's draw that
on a number line.

00:00:48.940 --> 00:00:50.900
So that's my number line.

00:00:50.900 --> 00:00:52.440
Let's stick 0 over here.

00:00:52.440 --> 00:00:54.370
Maybe if we go 1,
and then we could

00:00:54.370 --> 00:00:58.800
go negative 1,
negative 2, negative 3.

00:00:58.800 --> 00:01:00.612
And we could keep
going to the left.

00:01:00.612 --> 00:01:02.070
And we want all of
the x's that are

00:01:02.070 --> 00:01:04.610
less than or equal
to negative 2.

00:01:04.610 --> 00:01:07.690
Since it can be
equal to negative 2,

00:01:07.690 --> 00:01:11.850
we'll put a filled-in line
right here at negative 2,

00:01:11.850 --> 00:01:14.190
and all of the values
less than that.

00:01:14.190 --> 00:01:16.777
If it was just less than, if
there wasn't the equal sign,

00:01:16.777 --> 00:01:17.860
we would have an open dot.

00:01:17.860 --> 00:01:19.890
But since this is
less than or equal to,

00:01:19.890 --> 00:01:21.390
we've closed this dot.

00:01:21.390 --> 00:01:26.620
And then we want all of
the values below that.

00:01:26.620 --> 00:01:29.380
And you could just sample a
few and verify for yourself

00:01:29.380 --> 00:01:30.280
that they work.

00:01:30.280 --> 00:01:33.010
Based on this,
negative 3 should work.

00:01:33.010 --> 00:01:36.200
And if you took negative
3-- negative 3 plus 8

00:01:36.200 --> 00:01:39.590
is 5, which is definitely
less than 6, so that works.

00:01:39.590 --> 00:01:40.870
And negative 1 shouldn't work.

00:01:40.870 --> 00:01:42.710
It's not included in
this set over here.

00:01:42.710 --> 00:01:44.160
So let's try that out.

00:01:44.160 --> 00:01:49.330
Negative 1 plus 8 is 7, which
is definitely not less than 6.

00:01:49.330 --> 00:01:51.150
So just sampling
some points, it seems

00:01:51.150 --> 00:01:53.820
like we've got the
right solution.


Multiplying and dividing with inequalities example

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.830
Solve for c and
graph the solution.

00:00:02.830 --> 00:00:07.460
We have negative 5c is
less than or equal to 15.

00:00:07.460 --> 00:00:11.510
So negative 5c is less
than or equal to 15.

00:00:11.510 --> 00:00:13.509
I just rewrote it a
little bit bigger.

00:00:13.509 --> 00:00:15.050
So if we want to
solve for c, we just

00:00:15.050 --> 00:00:17.430
want to isolate the c
right over here, maybe

00:00:17.430 --> 00:00:19.190
on the left-hand side.

00:00:19.190 --> 00:00:21.480
It's right now being
multiplied by negative 5.

00:00:21.480 --> 00:00:23.780
So the best way to just have
a c on the left-hand side

00:00:23.780 --> 00:00:27.730
is we can multiply both
sides of this inequality

00:00:27.730 --> 00:00:33.070
by the inverse of negative
5, or by negative 1/5.

00:00:33.070 --> 00:00:37.630
So we want to multiply
negative 1/5 times negative 5c.

00:00:37.630 --> 00:00:41.470
And we also want to multiply
15 times negative 1/5.

00:00:41.470 --> 00:00:43.650
I'm just multiplying both
sides of the inequality

00:00:43.650 --> 00:00:47.430
by the inverse of negative 5,
because this will cancel out

00:00:47.430 --> 00:00:49.495
with the negative 5 and
leave me just with c.

00:00:49.495 --> 00:00:51.490
Now I didn't draw
the inequality here,

00:00:51.490 --> 00:00:54.520
because we have to remember,
if we multiply or divide

00:00:54.520 --> 00:00:58.570
both sides of an inequality
by a negative number,

00:00:58.570 --> 00:01:01.370
you have to flip the inequality.

00:01:01.370 --> 00:01:02.490
And we are doing that.

00:01:02.490 --> 00:01:05.620
We are multiplying both
sides by negative 1/5, which

00:01:05.620 --> 00:01:08.402
is the equivalent of dividing
both sides by negative 5.

00:01:08.402 --> 00:01:09.860
So we need to turn
this from a less

00:01:09.860 --> 00:01:13.550
than or equal to a
greater than or equal.

00:01:13.550 --> 00:01:16.390
And now we can
proceed solving for c.

00:01:16.390 --> 00:01:19.580
So negative 1/5 times
negative 5 is 1.

00:01:19.580 --> 00:01:21.060
So the left-hand
side is just going

00:01:21.060 --> 00:01:26.320
to be c is greater than or
equal to 15 times negative 1/5.

00:01:26.320 --> 00:01:29.490
That's the same thing as
15 divided by negative 5.

00:01:29.490 --> 00:01:31.740
And so that is negative 3.

00:01:31.740 --> 00:01:35.660
So our solution is c is greater
than or equal to negative 3.

00:01:35.660 --> 00:01:37.970
And let's graph it.

00:01:37.970 --> 00:01:40.940
So that is my number line.

00:01:40.940 --> 00:01:47.110
Let's say that is 0, negative
1, negative 2, negative 3.

00:01:47.110 --> 00:01:49.570
And then I could go above, 1, 2.

00:01:49.570 --> 00:01:51.950
And so c is greater than
or equal to negative 3.

00:01:51.950 --> 00:01:53.820
So it can be equal
to negative 3.

00:01:53.820 --> 00:01:55.630
So I'll fill that
in right over there.

00:01:55.630 --> 00:01:57.046
Let me do it in a
different color.

00:01:57.046 --> 00:01:58.610
So I'll fill it in
right over there.

00:01:58.610 --> 00:02:00.630
And then it's
greater than as well.

00:02:00.630 --> 00:02:06.710
So it's all of these values
I am filling in in green.

00:02:06.710 --> 00:02:09.880
And you can verify that it works
in the original inequality.

00:02:09.880 --> 00:02:11.530
Pick something that should work.

00:02:11.530 --> 00:02:12.880
Well, 0 should work.

00:02:12.880 --> 00:02:15.920
0 is one of the numbers
that we filled in.

00:02:15.920 --> 00:02:20.410
Negative 5 times 0 is 0, which
is less than or equal to 15.

00:02:20.410 --> 00:02:21.857
It's less than 15.

00:02:21.857 --> 00:02:23.690
Now let's try a number
that's outside of it.

00:02:23.690 --> 00:02:25.067
And I haven't drawn it here.

00:02:25.067 --> 00:02:27.400
I could continue with the
number line in this direction.

00:02:27.400 --> 00:02:28.800
We would have a negative 4 here.

00:02:28.800 --> 00:02:30.660
Negative 4 should
not be included.

00:02:30.660 --> 00:02:33.180
And let's verify that
negative 4 doesn't work.

00:02:33.180 --> 00:02:37.950
Negative 4 times negative
5 is positive 20.

00:02:37.950 --> 00:02:41.180
And positive 20 is
not less than 15,

00:02:41.180 --> 00:02:43.760
so it's good that we did
not include negative 4.

00:02:43.760 --> 00:02:44.920
So this is our solution.

00:02:44.920 --> 00:02:48.957
And this is that
solution graphed.

00:02:48.957 --> 00:02:51.040
And I wanted to do that
in that other green color.

00:02:51.040 --> 00:02:51.640
Here you go.

00:02:51.640 --> 00:02:53.880
That's what it looks like.


Multi-step inequalities 2

WEBVTT
Kind: captions
Language: en

00:00:00.882 --> 00:00:09.228
Solve for x. And we have 5x plus 7 is greater than 3 times x plus 1.

00:00:09.228 --> 00:00:12.632
So let's just try to isolate "x" on one side of this inequality.

00:00:12.632 --> 00:00:15.733
But before we do that, let's just simplify this righthand side.

00:00:15.733 --> 00:00:20.333
so we get 5x plus 7 is greater than -

00:00:20.333 --> 00:00:27.533
let's distribute this 3. So 3 times x plus 1 is the same thing as 3 times x plus 3 times 1

00:00:27.548 --> 00:00:32.554
so it's going to be 3x plus 3 times 1 is 3.

00:00:32.554 --> 00:00:35.231
Now if we want to put our x's on the lefthand side,

00:00:35.231 --> 00:00:37.802
we can subtract 3x from both sides.

00:00:37.802 --> 00:00:40.282
That will get rid of this 3x on the righthand side.

00:00:40.282 --> 00:00:44.718
So let's do that. Let's subtract 3x from both sides, and

00:00:44.718 --> 00:00:51.759
we get on the lefthand side: 5x minus 3x is 2x plus 7

00:00:52.036 --> 00:00:55.933
is greater than - 3x minus 3x - those cancel out.

00:00:55.933 --> 00:00:59.383
That was the whole point behind subtracting 3x from both sides -

00:00:59.383 --> 00:01:05.867
is greater than 3. Is greater than 3. No we can subtract 7 from both sides

00:01:05.867 --> 00:01:08.267
to get rid of this positive 7 right over here.

00:01:08.267 --> 00:01:14.129
So, let's subtract, let's subtract 7 from both sides. And we get on the lefthand side...

00:01:14.129 --> 00:01:24.080
2x plus 7 minus 7 is just 2x. Is greater than 3 minus 7 which is negative 4.

00:01:24.080 --> 00:01:27.282
And then let's see, we have 2x is greater than negative 4.

00:01:27.282 --> 00:01:31.210
If we just want an x over here, we can just divide both sides by 2.

00:01:31.210 --> 00:01:36.223
Since 2 is a positive number, we don't have to swap the inequality.

00:01:36.223 --> 00:01:46.733
So let's just divide both sides by 2, and we get x is greater than negative 4 divide by 2 is negative 2.

00:01:46.733 --> 00:01:51.425
So the solution will look like this. Draw the number line.

00:01:51.610 --> 00:01:56.195
I can draw a straighter number line than that. There we go. Still not that great,

00:01:56.195 --> 00:01:58.508
but it will serve our purposes.

00:01:58.508 --> 00:02:05.133
Let's say that's -3, -2, -1, 0, 1, 2, 3.

00:02:05.133 --> 00:02:09.662
X is greater than negative 2. It does not include negative 2.

00:02:09.662 --> 00:02:14.367
It is not greater than or equal to negative 2, so we have to exclude negative 2.

00:02:14.367 --> 00:02:20.431
And we exclude negative 2 by drawing an open circle at negative 2, but all the values greater than that

00:02:20.431 --> 00:02:25.435
are valid x's that would solve, that would satisfy this inequality.

00:02:25.435 --> 00:02:29.985
So anything above it - anything above it will work.

00:02:29.985 --> 00:02:33.667
And let's just try, let's try just try something that should work.

00:02:33.667 --> 00:02:35.667
and then let's try something that shouldn't work.

00:02:35.667 --> 00:02:39.887
So 0 should work. It is greater than negative 2. It's right over here.

00:02:39.902 --> 00:02:48.834
So, let's verify that. 5 times 0 plus 7 should be greater than 3 times 0 plus 1.

00:02:48.834 --> 00:02:55.404
So this is 7 - 'cause this is just a 0 - 7 should be greater than 3.

00:02:55.404 --> 00:03:00.236
Right. 3 times 1. So 7 should be greater than 3, and it definitely is.

00:03:00.236 --> 00:03:04.400
Now let's try something that should not work. Let's try negative 3.

00:03:04.400 --> 00:03:10.831
So 5 times negative 3... 5 times negative 3 plus 7, let's see if it is greater than

00:03:10.831 --> 00:03:18.642
3 times negative 3 plus 1. So this is negative 15 plus 7 is negative 8

00:03:18.827 --> 00:03:21.569
That is negative 8.

00:03:21.569 --> 00:03:27.667
Let's see if that is greater than negative 3 plus 1 is negative 2 times 3 is negative 6.

00:03:27.667 --> 00:03:34.200
Negative 8 is not - is not greater than negative 6. Negative 8 is more negative than negative 6.

00:03:34.200 --> 00:03:39.769
It's less than. So, it is good that negative 3 didn't work 'cause we didn't include that in our solution

00:03:39.831 --> 00:03:40.400
set.

00:03:40.400 --> 00:03:43.399
So we tried something that is in our solution set and it did work.

00:03:43.399 --> 00:03:44.900
And something that is not, and it didn't work.

00:03:44.900 --> 00:03:47.069
So we are feeling pretty good.


Compound inequalities 4

WEBVTT
Kind: captions
Language: en

00:00:00.813 --> 00:00:10.533
Solve for x, 5x - 3 is less than 12 "and" 4x plus 1 is greater than 25.

00:00:10.533 --> 00:00:13.867
So let's just solve for X in each of these constraints and keep in mind

00:00:13.867 --> 00:00:16.867
that any x has to satisfy both of them

00:00:16.867 --> 00:00:18.867
because it's an "and" over here

00:00:18.867 --> 00:00:20.600
so first we have this 5 x

00:00:20.600 --> 00:00:23.308
minus 3 is less than 12

00:00:23.308 --> 00:00:25.738
so if we want to isolate the x

00:00:25.738 --> 00:00:28.036
we can get rid of this negative 3

00:00:28.036 --> 00:00:29.200
here by adding 3 to both sides

00:00:29.200 --> 00:00:33.467
so let's add 3 to both sides of this inequality.

00:00:33.467 --> 00:00:38.000
The left-hand side, we're just left with a 5x, the minus 3 and the plus 3 cancel out.

00:00:38.000 --> 00:00:42.113
5x is less than 12 plus 3 is 15.

00:00:42.113 --> 00:00:44.600
Now we can divide both sides by positive 5,

00:00:44.600 --> 00:00:47.467
that won't swap the inequality since 5 is positive.

00:00:47.467 --> 00:00:50.600
So we divide both sides by positive 5

00:00:50.600 --> 00:00:53.046
and we are left with just from this constraint

00:00:53.046 --> 00:00:57.041
that x is less than 15 over 5, which is 3.

00:00:57.041 --> 00:00:59.333
So that constraint over here.

00:00:59.333 --> 00:01:01.667
But we have the second constraint as well.

00:01:01.667 --> 00:01:05.600
We have this one, we have 4x plus 1

00:01:05.600 --> 00:01:07.964
is greater than 25.

00:01:07.964 --> 00:01:11.800
So very similarly we can subtract one from both sides

00:01:11.800 --> 00:01:14.467
to get rid of that one on the left-hand side.

00:01:14.467 --> 00:01:17.667
And we get 4x, the ones cancel out.

00:01:17.667 --> 00:01:22.267
is greater than 25 minus one is 24.

00:01:22.267 --> 00:01:25.000
Divide both sides by positive 4

00:01:25.000 --> 00:01:28.810
Don't have to do anything to the inequality since

00:01:28.810 --> 00:01:30.323
it's a positive number.

00:01:30.323 --> 00:01:32.667
And we get x is greater than

00:01:32.667 --> 00:01:35.600
24 over 4 is 6.

00:01:35.600 --> 00:01:37.800
And remember there was that "and" over here.

00:01:37.800 --> 00:01:39.800
We have this "and".

00:01:39.800 --> 00:01:46.533
So x has to be less than 3 "and" x has to be greater than 6.

00:01:46.533 --> 00:01:49.585
So already your brain might be realizing

00:01:49.585 --> 00:01:51.600
that this is a little bit strange.

00:01:51.600 --> 00:01:56.333
This first constraint says that x needs to be less than 3

00:01:56.333 --> 00:01:58.933
so this is 3 on the number line.

00:01:58.933 --> 00:02:00.867
We're saying x has to be less than 3

00:02:00.867 --> 00:02:05.000
so it has to be in this shaded area right over there.

00:02:05.000 --> 00:02:07.569
This second constraint says that x has to be

00:02:07.569 --> 00:02:08.882
greater than 6.

00:02:08.882 --> 00:02:11.698
So if this is 6 over here, it says that x has to greater than 6.

00:02:11.698 --> 00:02:14.200
It can't even include 6.

00:02:14.200 --> 00:02:16.200
And since we have this "and" here.

00:02:16.200 --> 00:02:19.467
The only x-es that are a solution for this compound inequality

00:02:19.467 --> 00:02:21.333
are the ones that satisfy both.

00:02:21.333 --> 00:02:23.133
The ones that are in the overlap

00:02:23.133 --> 00:02:24.667
of their solution set.

00:02:24.667 --> 00:02:26.267
But when you look at it right over here it's clear that

00:02:26.267 --> 00:02:27.800
there is no overlap.

00:02:27.800 --> 00:02:33.133
There is no x that is both greater than 6 "and" less than 3.

00:02:33.133 --> 99:59:59.999
So in this situation we have no solution.


Absolute value equations example 1

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:05.250
Solve for x, and we have the
absolute value of 2x minus 5

00:00:05.250 --> 00:00:08.150
is equal to 11.

00:00:08.150 --> 00:00:10.340
Now, the big, I
guess, insight you

00:00:10.340 --> 00:00:12.370
need to have whenever you
have an absolute value

00:00:12.370 --> 00:00:13.994
equation like this
is just to remember,

00:00:13.994 --> 00:00:17.640
if I have the
absolute value of a

00:00:17.640 --> 00:00:23.000
is equal to 11, what
do we know about a?

00:00:23.000 --> 00:00:27.460
That means that a is equal to
11, or what else could a be?

00:00:27.460 --> 00:00:28.830
Well, it could be negative 11.

00:00:28.830 --> 00:00:32.110
a could be equal to negative
11 because there's two possible

00:00:32.110 --> 00:00:35.070
a's that are exactly
11 away from 0.

00:00:35.070 --> 00:00:38.660
It could be positive 11 or
it could be negative 11.

00:00:38.660 --> 00:00:41.950
That's the exact same logic you
apply in an equation like this.

00:00:41.950 --> 00:00:46.180
If the absolute value of
2x minus 5 is equal to 11,

00:00:46.180 --> 00:00:49.800
that means that the stuff
inside the absolute value sign

00:00:49.800 --> 00:00:54.550
either equals 11 or the stuff
inside the absolute value sign

00:00:54.550 --> 00:00:56.770
is equal to negative 11.

00:00:56.770 --> 00:01:00.140
Because if this is equal to 11,
the absolute value of 11 is 11.

00:01:00.140 --> 00:01:02.140
If this thing over here
is equal to negative 11,

00:01:02.140 --> 00:01:03.514
when you take its
absolute value,

00:01:03.514 --> 00:01:05.360
it'll still be equal to 11.

00:01:05.360 --> 00:01:06.940
So let's write that down.

00:01:06.940 --> 00:01:14.270
So 2x minus 5 could
be equal to 11 or 2x

00:01:14.270 --> 00:01:18.470
minus 5 could be equal
to negative 11, right?

00:01:18.470 --> 00:01:20.720
Because in either case, if
we took its absolute value,

00:01:20.720 --> 00:01:22.240
you would get 11.

00:01:22.240 --> 00:01:24.550
Now, let's just solve
each of these equations.

00:01:24.550 --> 00:01:31.550
Over here, we can add 5 to
both sides of this equation.

00:01:31.550 --> 00:01:33.820
We get 11 plus 5 is 16.

00:01:33.820 --> 00:01:37.440
2x minus 5 plus 5 is just 2x.

00:01:37.440 --> 00:01:38.770
2x is equal to 16.

00:01:38.770 --> 00:01:44.630
Divide both sides by 2,
we get x is equal to 8.

00:01:44.630 --> 00:01:46.760
Now, let's do this situation.

00:01:46.760 --> 00:01:49.140
2x minus 5 is equal
to negative 11.

00:01:49.140 --> 00:01:53.410
Let's add 5 to both
sides of this equation.

00:01:53.410 --> 00:01:57.210
The left-hand side we just
get a 2x, the 5's cancel out.

00:01:57.210 --> 00:02:03.370
2x is equal to negative 5
plus 11, that's negative 6,

00:02:03.370 --> 00:02:05.130
and then we can divide
both sides by 2.

00:02:08.660 --> 00:02:11.820
And we get x is
equal to negative 3.

00:02:11.820 --> 00:02:13.760
So we've gotten two
solutions depending

00:02:13.760 --> 00:02:15.770
on whether this thing
in here is either

00:02:15.770 --> 00:02:17.450
equal to 11 or negative 11.

00:02:17.450 --> 00:02:20.770
We got x could be equal to
8, in which case 2x minus 5

00:02:20.770 --> 00:02:24.040
would be 11, or we have x is
equal to negative 3, in which

00:02:24.040 --> 00:02:26.930
case 2x minus 5 is
equal to negative 11.

00:02:26.930 --> 00:02:28.260
Let's verify it.

00:02:28.260 --> 00:02:29.940
So if x is equal
to 8, let's show

00:02:29.940 --> 00:02:32.180
that it satisfies this equation.

00:02:32.180 --> 00:02:35.890
So we'd get 2 times 8 minus 5.

00:02:35.890 --> 00:02:39.290
The absolute value of 2 times 8
minus 5 should be equal to 11.

00:02:39.290 --> 00:02:42.026
2 times 8 is 16
minus 5 is 11, so we

00:02:42.026 --> 00:02:44.540
have the absolute value of
11, should be equal to 11.

00:02:44.540 --> 00:02:45.970
And we know that's true.

00:02:45.970 --> 00:02:48.130
The absolute value of 11 is 11.

00:02:48.130 --> 00:02:49.740
So it all works out.

00:02:49.740 --> 00:02:51.740
Now, let's try it with x
is equal to negative 3.

00:02:51.740 --> 00:02:57.530
We get the absolute value of
2 times negative 3 minus 5

00:02:57.530 --> 00:03:00.050
should be equal to 11.

00:03:00.050 --> 00:03:02.250
So what's 2 times negative 3?

00:03:02.250 --> 00:03:05.440
2 times negative 3 is
negative 6 minus 5.

00:03:05.440 --> 00:03:08.740
The absolute value of that
should be equal to 11.

00:03:08.740 --> 00:03:11.810
Well, negative 6 minus
5 is negative 11.

00:03:11.810 --> 00:03:13.500
So the absolute
value of negative 11

00:03:13.500 --> 00:03:15.170
should be equal
to 11, and it is.

00:03:15.170 --> 00:03:17.600
The absolute value
of negative 11 is 11.

00:03:17.600 --> 00:03:19.480
So we get 11 is equal to 11.

00:03:19.480 --> 00:03:21.605
So they both work.


First living things on land clarification

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.059
In the video on the Cambrian
explosion, I talk about how

00:00:04.059 --> 00:00:06.450
surprisingly, or
somewhat surprisingly,

00:00:06.450 --> 00:00:11.010
that animals were the first to
colonize or to move on land.

00:00:11.010 --> 00:00:13.059
They did that before plants did.

00:00:13.059 --> 00:00:14.600
And someone brought
up what I thought

00:00:14.600 --> 00:00:15.641
was a very good question.

00:00:15.641 --> 00:00:18.860
If the animals were the
first, what did they eat?

00:00:18.860 --> 00:00:21.020
So I thought that was
one, a good question.

00:00:21.020 --> 00:00:23.770
So it justified a whole
video on clarifying

00:00:23.770 --> 00:00:27.410
exactly who was
first on the land.

00:00:27.410 --> 00:00:30.820
So right here, this is a
picture of algae on the coast.

00:00:30.820 --> 00:00:33.140
This is kind of algal
scum right over here.

00:00:33.140 --> 00:00:35.920
So this right here is algae.

00:00:35.920 --> 00:00:38.640
And just to be clear,
sometimes cyanobacteria,

00:00:38.640 --> 00:00:42.690
which we talked about as the
first photosynthetic organism,

00:00:42.690 --> 00:00:45.590
sometimes that's called
blue-green algae.

00:00:45.590 --> 00:00:46.930
But that's really bacteria.

00:00:46.930 --> 00:00:49.030
Algae is considered
to be eukaryotic.

00:00:49.030 --> 00:00:52.070
And it just doesn't have the
structures of modern plants.

00:00:52.070 --> 00:00:53.710
So this is algae right here.

00:00:53.710 --> 00:00:57.320
And our best estimate
is that algae actually

00:00:57.320 --> 00:01:02.070
colonized kind of coastal rocks
about 1.2 billion years ago.

00:01:02.070 --> 00:01:08.420
So this is 1.2 billion
years ago, "g" giga,

00:01:08.420 --> 00:01:10.110
billion years ago.

00:01:10.110 --> 00:01:12.740
So if you wanted the first
thing that even resembled or was

00:01:12.740 --> 00:01:16.500
close to plants or animals,
and if you consider algae close

00:01:16.500 --> 00:01:18.950
to a plant, then this
would be the winner

00:01:18.950 --> 00:01:20.160
of who got on land first.

00:01:20.160 --> 00:01:23.500
This is 1.2 billion years ago.

00:01:23.500 --> 00:01:26.880
Now in the last video where I
talk about animals colonizing

00:01:26.880 --> 00:01:29.280
the land first,
they weren't animals

00:01:29.280 --> 00:01:30.970
that only existed on land.

00:01:30.970 --> 00:01:33.930
They would have been
animals that probably spent

00:01:33.930 --> 00:01:36.664
most of their time in the ocean
collecting food or whatever.

00:01:36.664 --> 00:01:39.080
And then they would show up
on the land maybe to lay eggs.

00:01:39.080 --> 00:01:40.700
And if you think about
it, back then the land

00:01:40.700 --> 00:01:41.750
would have been a
really good place

00:01:41.750 --> 00:01:43.166
to lay eggs because
there wouldn't

00:01:43.166 --> 00:01:44.540
have been much else on the land.

00:01:44.540 --> 00:01:47.130
So you would have been
protected from predators.

00:01:47.130 --> 00:01:52.110
So it might have been
slug-like creatures like this.

00:01:52.110 --> 00:01:54.571
Some people talk about kind
of spider-like creatures.

00:01:54.571 --> 00:01:56.320
But it still would
have been at the coast.

00:01:56.320 --> 00:01:59.380
And these would
have been creatures

00:01:59.380 --> 00:02:01.450
that would have spend a
lot of time in the ocean

00:02:01.450 --> 00:02:03.200
and some time in their land.

00:02:03.200 --> 00:02:07.150
So this is what I was
referring to as kind of animals

00:02:07.150 --> 00:02:10.280
colonizing the
land before plants.

00:02:10.280 --> 00:02:13.200
And this would have happened
about 530 million years ago.

00:02:20.090 --> 00:02:24.490
Now, the first living organisms
to fully live on the land,

00:02:24.490 --> 00:02:28.480
their whole life is on the
land, those would be plants.

00:02:28.480 --> 00:02:31.490
So it depends if you think about
things that part of their life,

00:02:31.490 --> 00:02:32.810
you'd get the animals.

00:02:32.810 --> 00:02:35.080
Things that lived their
whole life on the land,

00:02:35.080 --> 00:02:37.220
then you go back to the plants.

00:02:37.220 --> 00:02:39.410
So this right here,
this is what we

00:02:39.410 --> 00:02:42.840
think the first primitive
plants would have looked like.

00:02:42.840 --> 00:02:44.340
And the evidence--
we actually don't

00:02:44.340 --> 00:02:46.490
have fossils from these
plants themselves.

00:02:46.490 --> 00:02:49.280
We have fossils of their spores.

00:02:49.280 --> 00:02:52.510
But we think that the earliest
fossils of their spores,

00:02:52.510 --> 00:02:54.440
which show that
these existed, were

00:02:54.440 --> 00:02:58.310
about 475 million years ago.

00:02:58.310 --> 00:03:03.050
So this is-- let me do
this in another color--

00:03:03.050 --> 00:03:07.500
this right over here is
475 million years ago.

00:03:07.500 --> 00:03:10.490
So 1.2 billion years
ago, you have the algae.

00:03:10.490 --> 00:03:13.140
530 million years
ago, we have evidence

00:03:13.140 --> 00:03:15.192
of things kind of
oozing out of the ocean

00:03:15.192 --> 00:03:16.900
and maybe laying their
eggs or something.

00:03:16.900 --> 00:03:20.910
475 million years
ago, we have evidence

00:03:20.910 --> 00:03:23.570
of what we would kind
of call really plants.

00:03:23.570 --> 00:03:27.270
But the evidence is really
the fossils of their spores.

00:03:27.270 --> 00:03:29.710
And then the first
evidence of real-- I

00:03:29.710 --> 00:03:31.400
guess you could
call them animals

00:03:31.400 --> 00:03:36.340
that spend their entire life on
the land, the oldest fossil we

00:03:36.340 --> 00:03:39.470
have-- it was discovered
in Scotland fairly

00:03:39.470 --> 00:03:41.660
recently, in 2004.

00:03:41.660 --> 00:03:44.450
And this is the fossil
right over here.

00:03:44.450 --> 00:03:48.490
It was actually discovered by
a bus driver, by Mike Newman.

00:03:48.490 --> 00:03:52.380
Mike Newman, who is a
bus driver in Scotland.

00:03:52.380 --> 00:03:54.355
And they actually named
the thing after him.

00:03:54.355 --> 00:03:56.860
It's called
Pneumodesmus newmani.

00:03:56.860 --> 00:03:59.570
So they got the newmani
from Mike Newman.

00:03:59.570 --> 00:04:02.830
And this fossil is
428 million years old.

00:04:10.450 --> 00:04:12.650
And right now, it's
the oldest fossil

00:04:12.650 --> 00:04:14.850
we have of a true land animal.

00:04:14.850 --> 00:04:18.690
So if you think about true
plants versus true land

00:04:18.690 --> 00:04:21.300
animals, things that spent
their entire life on the land,

00:04:21.300 --> 00:04:22.730
the plants do win out.

00:04:22.730 --> 00:04:25.180
If you think about things
that spent part of their time

00:04:25.180 --> 00:04:28.350
on the land, then the
animals probably won out.

00:04:28.350 --> 00:04:30.850
If you view algae as plants,
then the plants won out.

00:04:30.850 --> 00:04:32.766
So it depends where you
want to draw the line.

00:04:32.766 --> 00:04:36.140
And this first fossil,
this is of a myriapod

00:04:36.140 --> 00:04:38.880
which just means a lot of legs.

00:04:38.880 --> 00:04:42.250
Let me write over
here, myriapod.

00:04:42.250 --> 00:04:43.750
You probably know
the word "myriad."

00:04:43.750 --> 00:04:46.660
Myriad means a bunch of
things or a huge amount.

00:04:46.660 --> 00:04:50.379
So myriapod, a huge
amount of legs.

00:04:50.379 --> 00:04:52.920
And you might be familiar with
the millipedes and centipedes.

00:04:52.920 --> 00:04:53.855
Those are myriapods.

00:04:53.855 --> 00:04:57.690
And so those first primitive
myriapods, 428 million years

00:04:57.690 --> 00:04:58.310
ago.

00:04:58.310 --> 00:05:00.600
And they would have lived
off of plants, and maybe

00:05:00.600 --> 00:05:02.240
other myriapods,
and other slugs,

00:05:02.240 --> 00:05:05.190
and whatever other animals
they might have found.

00:05:05.190 --> 00:05:07.100
They mind have looked
something like that.

00:05:07.100 --> 00:05:09.349
So hopefully, that gives a
little bit of clarification

00:05:09.349 --> 00:05:13.404
over-- it wasn't like you
had dogs sitting on the land

00:05:13.404 --> 00:05:14.570
and they had nothing to eat.

00:05:17.280 --> 00:05:19.980
It's kind of a gray area in what
you define a plant or an animal

00:05:19.980 --> 00:05:22.017
and who gets kind of
the bragging rights

00:05:22.017 --> 00:05:23.350
for being the first on the land.

00:05:23.350 --> 00:05:25.900
And it depends on really what
you consider a plant or animal

00:05:25.900 --> 00:05:28.780
and whether spending
part of your life

00:05:28.780 --> 00:05:32.250
on the land, whether spending
part of your life on the land

00:05:32.250 --> 00:05:33.796
will actually qualify.


Biodiversity flourishes in Phanerozoic eon

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.350
The Earth is now
starting to get closer

00:00:03.350 --> 00:00:07.990
to being hospitable to people
like us or animals like us.

00:00:07.990 --> 00:00:13.000
In the last video, we saw
during the Proterozoic Eon,

00:00:13.000 --> 00:00:15.550
oxygen began to accumulate
in the atmosphere.

00:00:15.550 --> 00:00:17.760
This actually caused
this first snowball Earth

00:00:17.760 --> 00:00:21.210
and this mass extinction of
all the anaerobic species.

00:00:21.210 --> 00:00:24.790
But it made conditions
suitable for eukaryotic cells.

00:00:24.790 --> 00:00:27.560
And maybe even more important,
these eukaryotic cells

00:00:27.560 --> 00:00:30.460
were able to form
multicellular organisms.

00:00:30.460 --> 00:00:33.370
And we see where that starts
right here on this chart,

00:00:33.370 --> 00:00:34.820
on this time clock.

00:00:34.820 --> 00:00:37.262
Multicellular life
starts right over here.

00:00:37.262 --> 00:00:38.220
And I want to be clear.

00:00:38.220 --> 00:00:40.180
All of these things are
a bit moving targets.

00:00:40.180 --> 00:00:42.620
As we discover more things
in the geological record

00:00:42.620 --> 00:00:44.480
and we get more tools
at our disposal,

00:00:44.480 --> 00:00:45.674
these numbers get tweaked.

00:00:45.674 --> 00:00:47.090
But they do give
you a good sense,

00:00:47.090 --> 00:00:48.980
based on our current
understanding,

00:00:48.980 --> 00:00:51.830
of when these things
start to appear.

00:00:51.830 --> 00:00:53.506
And coinciding with
multicelluar life,

00:00:53.506 --> 00:00:55.130
and this is interesting
in it's s right

00:00:55.130 --> 00:00:58.760
because it has its own
metalevel effect on evolution,

00:00:58.760 --> 00:01:01.565
you actually start also
having sexual reproduction.

00:01:06.920 --> 00:01:08.570
And what's interesting
about this,

00:01:08.570 --> 00:01:10.540
why this has such a big
impact on evolution--

00:01:10.540 --> 00:01:13.130
and we talk about it a lot
in the biology playlist--

00:01:13.130 --> 00:01:16.950
is before evolution,
variation in DNA

00:01:16.950 --> 00:01:20.290
had to be completely dependent
really on mutations, and just

00:01:20.290 --> 00:01:24.040
random movement around within
DNA, or maybe some viruses.

00:01:24.040 --> 00:01:26.210
Now with sexual
reproduction, you

00:01:26.210 --> 00:01:30.010
had kind of a
systematic mixing of DNA

00:01:30.010 --> 00:01:32.390
so that you got more variation
in the gene pool, which

00:01:32.390 --> 00:01:34.930
allowed more
selection for-- or I

00:01:34.930 --> 00:01:38.360
guess you had more
variance to select for.

00:01:38.360 --> 00:01:40.700
And so you kind of
had an acceleration

00:01:40.700 --> 00:01:43.390
in the actual pace of evolution.

00:01:43.390 --> 00:01:46.820
So we're talking-- I've
looked at a bunch of sources

00:01:46.820 --> 00:01:49.940
from, they say 1.2
billion, 1.5 billion,

00:01:49.940 --> 00:01:52.150
a little bit over a billion,
if you call a little bit

00:01:52.150 --> 00:01:54.220
several million
years ago, you start

00:01:54.220 --> 00:01:58.430
having these multicellular life
forms and sexual reproduction.

00:01:58.430 --> 00:02:02.310
The other thing that we talked
about in the Proterozoic Eon

00:02:02.310 --> 00:02:05.590
is the accumulation of
oxygen allowed the ozone

00:02:05.590 --> 00:02:06.460
layer to build up.

00:02:06.460 --> 00:02:08.830
Ozone is just
three oxygen atoms.

00:02:08.830 --> 00:02:10.490
It is O3.

00:02:10.490 --> 00:02:14.150
And by the end of
the Proterozoic Eon--

00:02:14.150 --> 00:02:17.920
so we're talking, I don't know,
maybe 550 million years ago,

00:02:17.920 --> 00:02:20.270
give or take tens of,
or hundreds, or maybe

00:02:20.270 --> 00:02:22.360
100 million years--
these are all

00:02:22.360 --> 00:02:25.440
moving targets-- the ozone
layer was dense enough

00:02:25.440 --> 00:02:28.300
to protect the
land from UV rays.

00:02:28.300 --> 00:02:31.320
We talked about that
in the last video,

00:02:31.320 --> 00:02:33.690
that the Earth is being
bombarded with UV rays.

00:02:33.690 --> 00:02:35.390
And the ozone layer
is the only thing

00:02:35.390 --> 00:02:39.430
that really keeps us from being
seriously irradiated by the Sun

00:02:39.430 --> 00:02:41.920
and allows land animals
to actually live.

00:02:41.920 --> 00:02:44.520
And so coinciding
with that time period,

00:02:44.520 --> 00:02:47.480
around 550 million
years ago, you

00:02:47.480 --> 00:02:51.330
start to have life colonizing,
especially significant life,

00:02:51.330 --> 00:02:53.120
colonizing land.

00:02:53.120 --> 00:03:00.017
So life colonizes land,
colonizes the land.

00:03:00.017 --> 00:03:02.100
And this was kind of an
interesting-- when I first

00:03:02.100 --> 00:03:04.710
learned it, it was
kind of an aha moment.

00:03:04.710 --> 00:03:08.110
You always assume that
kind of trees and grasses

00:03:08.110 --> 00:03:09.880
are kind of part
of the background.

00:03:09.880 --> 00:03:12.760
They come part and
parcel with land.

00:03:12.760 --> 00:03:16.060
But it turns out that
animals colonized land

00:03:16.060 --> 00:03:17.870
before plants did.

00:03:17.870 --> 00:03:19.590
Plants didn't come
into the picture

00:03:19.590 --> 00:03:24.240
until about 450 million
years ago, give or take

00:03:24.240 --> 00:03:27.380
a few tens of millions of years.

00:03:27.380 --> 00:03:31.660
And so we're now entering the
end of the Proterozoic Eon.

00:03:31.660 --> 00:03:33.740
Life has started
to colonize land.

00:03:33.740 --> 00:03:35.275
We now have an ozone layer.

00:03:35.275 --> 00:03:36.900
And what happens--
and actually there's

00:03:36.900 --> 00:03:40.490
another snowball glaciation
or a snowball Earth

00:03:40.490 --> 00:03:45.300
near the end of the Proterozoic
Era, Eon, I should say.

00:03:45.300 --> 00:03:47.850
And there's a bunch of theories
about why it came about.

00:03:47.850 --> 00:03:48.980
And then why disappeared.

00:03:48.980 --> 00:03:52.740
Maybe there were volcanoes,
greenhouse gases, who knows.

00:03:52.740 --> 00:03:54.730
But as we enter
the end of that, we

00:03:54.730 --> 00:03:56.980
start seeing life
began to flourish.

00:03:56.980 --> 00:03:59.190
And it starts to
really flourish as we

00:03:59.190 --> 00:04:02.460
enter the Phaner-- I
always have trouble

00:04:02.460 --> 00:04:05.410
saying this-- the
Phanerozoic Eon.

00:04:05.410 --> 00:04:07.120
And it's not even labeled here.

00:04:07.120 --> 00:04:12.651
The Phanerozoic Eon is this
chunk of time right over here.

00:04:12.651 --> 00:04:13.650
And let me write it out.

00:04:13.650 --> 00:04:22.170
So this right over here is the
Phanerozoic, the Phanerozoic

00:04:22.170 --> 00:04:23.070
Eon.

00:04:23.070 --> 00:04:26.780
And so this chart, these
divisions right here are eons.

00:04:26.780 --> 00:04:29.430
And then they jump into,
instead of doing eons here,

00:04:29.430 --> 00:04:31.500
they then break into eras.

00:04:31.500 --> 00:04:33.446
Eras are subsets of eons.

00:04:33.446 --> 00:04:35.070
They are hundreds of
millions of years.

00:04:35.070 --> 00:04:38.450
So this is the Paleozoic
Era, the Mesozoic Era,

00:04:38.450 --> 00:04:39.920
and the Cenozoic Era.

00:04:39.920 --> 00:04:42.220
And that's actually
our current era.

00:04:42.220 --> 00:04:44.420
But perhaps the most
interesting-- well,

00:04:44.420 --> 00:04:45.920
I don't want to
pick favorites here.

00:04:45.920 --> 00:04:47.800
But it's one of the
most interesting times

00:04:47.800 --> 00:04:52.380
in the geologic era--
is the first period

00:04:52.380 --> 00:04:59.070
in the Paleozoic Era,
which is the first era

00:04:59.070 --> 00:05:02.010
in the Phanerozoic Eon.

00:05:02.010 --> 00:05:03.869
And that's the Cambrian period.

00:05:03.869 --> 00:05:06.160
You might have heard of it
before, the Cambrian period.

00:05:06.160 --> 00:05:14.080
That's about this period of
time right over here, Cambrian.

00:05:14.080 --> 00:05:17.360
And during this period
of time, the Earth

00:05:17.360 --> 00:05:20.810
experiences what we call
the Cambrian Explosion.

00:05:20.810 --> 00:05:23.810
And that's because there's
just this explosion

00:05:23.810 --> 00:05:28.790
in the number of species
and genera that existed,

00:05:28.790 --> 00:05:31.225
the biodiversity, on the planet.

00:05:31.225 --> 00:05:33.850
And it might just be that we had
the ozone layer protecting us,

00:05:33.850 --> 00:05:35.490
things were colonizing land.

00:05:35.490 --> 00:05:37.760
It was an oxygen-rich
environment.

00:05:37.760 --> 00:05:42.700
We start seeing complex
multicellular organisms.

00:05:42.700 --> 00:05:45.400
It's about that time, if
you fast forward maybe

00:05:45.400 --> 00:05:46.900
a few tens of
millions of years, you

00:05:46.900 --> 00:05:51.040
start seeing the first fish,
the first kind of preamphibians

00:05:51.040 --> 00:05:52.220
or protoamphibians.

00:05:52.220 --> 00:05:53.550
You fast forward a little bit.

00:05:53.550 --> 00:05:57.120
As we get out of
the Cambrian period,

00:05:57.120 --> 00:05:59.000
we start seeing the plants.

00:05:59.000 --> 00:06:01.290
So they actually draw
it right over here

00:06:01.290 --> 00:06:04.382
on this-- land plants-- or at
this point right over here.

00:06:04.382 --> 00:06:06.090
And, of course, these
are moving targets,

00:06:06.090 --> 00:06:09.960
depending on what we discover
in the fossil record.

00:06:09.960 --> 00:06:14.170
And for me, the big aha moment
here is so many of these things

00:06:14.170 --> 00:06:17.250
that you consider
fundamental to what

00:06:17.250 --> 00:06:20.930
Earth is are relatively
recent phenomena,

00:06:20.930 --> 00:06:26.330
Plants weren't on land until
about 450 million years ago.

00:06:26.330 --> 00:06:31.670
Insects weren't on land-- or did
not even exist until about 400

00:06:31.670 --> 00:06:32.950
million years ago.

00:06:32.950 --> 00:06:36.390
Reptiles didn't exist until
about 300 million years ago.

00:06:36.390 --> 00:06:39.710
So we are about
right over here now.

00:06:39.710 --> 00:06:42.360
Mammals didn't exist until
about 200 million years ago.

00:06:42.360 --> 00:06:45.940
Birds didn't exist until
about 150 million years ago.

00:06:45.940 --> 00:06:48.380
The whole dinosaur age,
which we kind of consider

00:06:48.380 --> 00:06:51.790
in our distant past, that's
essentially the Mesozoic Era

00:06:51.790 --> 00:06:53.040
right here.

00:06:53.040 --> 00:06:56.265
So this is the Age of the
Dinosaurs right over here.

00:06:56.265 --> 00:06:57.640
When you look at
your time clock,

00:06:57.640 --> 00:07:01.960
you can see it's a relatively
recent time period.

00:07:01.960 --> 00:07:04.290
And it actually ends
with, we currently

00:07:04.290 --> 00:07:09.260
believe, a huge rock, a
six-mile in diameter rock,

00:07:09.260 --> 00:07:13.150
colliding with what is now the
Yucatan Peninsula in Mexico,

00:07:13.150 --> 00:07:15.150
or right off the coast
of the Yucatan Peninsula.

00:07:15.150 --> 00:07:17.630
And it destroyed all
of the large land life

00:07:17.630 --> 00:07:20.170
forms, especially the dinosaurs.

00:07:20.170 --> 00:07:22.960
And to put all of
this in perspective--

00:07:22.960 --> 00:07:24.380
and actually the
thing that really

00:07:24.380 --> 00:07:26.790
was an aha moment for
me-- it's, OK, plants

00:07:26.790 --> 00:07:29.400
are 450 million years ago.

00:07:29.400 --> 00:07:32.820
Grass, I kind of view as this
fundamental thing in nature.

00:07:32.820 --> 00:07:35.690
But grass has only been
around for about-- I've

00:07:35.690 --> 00:07:38.550
seen multiple estimates--
40 to 70 million years.

00:07:38.550 --> 00:07:42.670
Grass is a relatively
new thing on the planet.

00:07:42.670 --> 00:07:45.730
Flowers have only been
around for 130 million years.

00:07:45.730 --> 00:07:47.880
So there was a time
where you had dinosaurs,

00:07:47.880 --> 00:07:52.160
but you did not have flowers
and you did not have grass.

00:07:52.160 --> 00:07:53.780
And so you fast
forward all the way.

00:07:53.780 --> 00:07:55.280
And so when you
look at this scale,

00:07:55.280 --> 00:07:57.410
it's kind of funny
to look at this.

00:07:57.410 --> 00:08:00.850
This is the time period where
the dinosaurs showed up.

00:08:00.850 --> 00:08:03.150
This whole brown line is
where the mammals showed up.

00:08:03.150 --> 00:08:06.840
So the dinosaurs started to
show up along with the mammals.

00:08:06.840 --> 00:08:09.130
And then, of course, the
dinosaurs died out here.

00:08:09.130 --> 00:08:12.720
Our ancestors, when the
giant rock hit the Earth,

00:08:12.720 --> 00:08:14.430
must have been
boroughed in holes

00:08:14.430 --> 00:08:17.910
and were able to stash some
food away, or who knows what,

00:08:17.910 --> 00:08:19.190
and didn't get fully affected.

00:08:19.190 --> 00:08:22.720
I'm sure most of the large
mammals were destroyed.

00:08:22.720 --> 00:08:25.559
But what's almost-- it's
humbling, or almost humorous,

00:08:25.559 --> 00:08:27.600
or almost ridiculous, when
you look at this chart

00:08:27.600 --> 00:08:31.224
is they put a little dot--
you can't even see it here,

00:08:31.224 --> 00:08:31.890
a little button.

00:08:31.890 --> 00:08:33.860
They say 2 million years
ago, the first humans--

00:08:33.860 --> 00:08:35.679
and even this is being
pretty generous when

00:08:35.679 --> 00:08:37.179
they say first humans.

00:08:37.179 --> 00:08:39.070
These are really
the first prehumans.

00:08:39.070 --> 00:08:42.520
The first humans that
are the same as us,

00:08:42.520 --> 00:08:44.920
if you took one of those
babies and your brought them up

00:08:44.920 --> 00:08:46.920
in the suburbs and gave
them haircuts and stuff,

00:08:46.920 --> 00:08:48.960
they would be the
same thing as we are,

00:08:48.960 --> 00:08:52.660
those didn't exist until
200,000 years ago, give or take.

00:08:52.660 --> 00:08:56.000
200,000 to 400,000 years
ago, I've seen estimates.

00:08:56.000 --> 00:08:59.717
So this is actually a very
generous period of time

00:08:59.717 --> 00:09:00.550
to say first humans.

00:09:00.550 --> 00:09:02.380
It's actually 200,000 years ago.

00:09:02.380 --> 00:09:05.730
And just to give you
an idea of how new

00:09:05.730 --> 00:09:09.895
we are and how new
evolution is, it

00:09:09.895 --> 00:09:12.020
was only 5 million years
ago-- and I mentioned this

00:09:12.020 --> 00:09:14.399
in a previous video-- it was
only 5 million years ago--

00:09:14.399 --> 00:09:15.690
so this is just to get a sense.

00:09:15.690 --> 00:09:16.890
This is 0 years.

00:09:16.890 --> 00:09:22.060
Homo sapien sapien, only
around for 200,000 years.

00:09:22.060 --> 00:09:24.240
The Neanderthals, they
were cousin species.

00:09:24.240 --> 00:09:25.664
They weren't our ancestors.

00:09:25.664 --> 00:09:26.830
Many people think they were.

00:09:26.830 --> 00:09:27.955
They were a cousin species.

00:09:27.955 --> 00:09:29.380
We come from the same root.

00:09:29.380 --> 00:09:31.500
Although there are
now theories that they

00:09:31.500 --> 00:09:35.140
might have remixed
in with Homo sapiens.

00:09:35.140 --> 00:09:38.340
So maybe some of us have
some Neanderthal DNA.

00:09:38.340 --> 00:09:40.400
And it shouldn't be
viewed as an insult.

00:09:40.400 --> 00:09:41.510
They had big brains.

00:09:41.510 --> 00:09:42.900
Well, they didn't
necessarily have big brains.

00:09:42.900 --> 00:09:43.720
They had big heads.

00:09:43.720 --> 00:09:46.090
But that seems to
imply a big brain.

00:09:46.090 --> 00:09:47.070
But who knows?

00:09:47.070 --> 00:09:49.959
We always tend to portray
them as somehow inferior.

00:09:49.959 --> 00:09:52.500
But I don't want to get into
the political correctness of how

00:09:52.500 --> 00:09:55.060
to portray Neanderthals.

00:09:55.060 --> 00:09:59.440
But anyway, this is a very
small period of time, 200,000.

00:09:59.440 --> 00:10:01.020
If you go 2 million
years, then you

00:10:01.020 --> 00:10:04.640
get to kind of the prehuman,
the prehuman ancestors.

00:10:04.640 --> 00:10:10.210
And our family tree only
diverged from the chimpanzees

00:10:10.210 --> 00:10:11.640
5 million years ago.

00:10:11.640 --> 00:10:15.270
If you draw that on
this clock right here,

00:10:15.270 --> 00:10:17.370
it would be like two
pixels or maybe not

00:10:17.370 --> 00:10:20.630
even two pixels as when we
diverged from the chimpanzees.

00:10:20.630 --> 00:10:23.400
So hopefully, that gives
you a sense of things.

00:10:23.400 --> 00:10:27.064
At least for me, it really
puts things in perspective.