Inductive reasoning 1

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:04.080
What is the nth number in
the following sequence?

00:00:04.080 --> 00:00:04.400
Let's see.

00:00:04.400 --> 00:00:07.620
The first term here is a 6.

00:00:07.620 --> 00:00:09.990
So over here I'll
put the term.

00:00:09.990 --> 00:00:12.760
So this is the number.

00:00:12.760 --> 00:00:15.080
I'll just write num there.

00:00:15.080 --> 00:00:16.260
That's the number.

00:00:16.260 --> 00:00:18.970
And this is the term.

00:00:18.970 --> 00:00:21.820
So the first term here is a 6.

00:00:21.820 --> 00:00:25.060
It looks like we add 3 to
that to get to a 9.

00:00:25.060 --> 00:00:26.410
Or maybe there's some
other pattern here.

00:00:26.410 --> 00:00:28.250
So that's the second
term, is 9.

00:00:28.250 --> 00:00:29.350
Then they go from 9 to 12.

00:00:29.350 --> 00:00:30.740
Looks like we add 3 again.

00:00:30.740 --> 00:00:31.480
So we keep adding 3.

00:00:31.480 --> 00:00:35.400
So the third term is 12,
the fourth term is 15.

00:00:35.400 --> 00:00:38.170
And we need to think of a
general expression that will

00:00:38.170 --> 00:00:41.310
give us the value
of the nth term.

00:00:41.310 --> 00:00:44.555
The nth number in the
following sequence.

00:00:44.555 --> 00:00:46.210
So let's think a little
bit about it.

00:00:46.210 --> 00:00:48.500
We know that we're incrementing
by 3 every time

00:00:48.500 --> 00:00:50.460
and we're starting at 6.

00:00:50.460 --> 00:00:52.880
So let's write these in terms
of multiples of 3, since we

00:00:52.880 --> 00:00:56.200
know that that's the
increment level.

00:00:56.200 --> 00:01:00.480
This first term right
here is 3 times 2.

00:01:00.480 --> 00:01:03.360
The second term right
here is 3 times 3.

00:01:03.360 --> 00:01:06.290
This third term is 3 times 4.

00:01:06.290 --> 00:01:09.270
And so it looks like it's 3
times one more than the actual

00:01:09.270 --> 00:01:11.710
term you are dealing with.

00:01:11.710 --> 00:01:12.000
Right?

00:01:12.000 --> 00:01:15.120
Here, if you take your term,
you add 1 and then multiply

00:01:15.120 --> 00:01:16.530
that by 3, you get 6.

00:01:16.530 --> 00:01:19.660
Take your term, add 1, then
multiply it by 3.

00:01:19.660 --> 00:01:22.200
Take your term, add
1, you get 4.

00:01:22.200 --> 00:01:23.300
Then multiply by 3.

00:01:23.300 --> 00:01:26.960
Take your term, add 1-- which
would be 5-- and then multiply

00:01:26.960 --> 00:01:28.480
by 3, you get 15.

00:01:28.480 --> 00:01:31.660
So it looks like any time-- so
if you have your nth term.

00:01:31.660 --> 00:01:34.920
So if you go all the way out
here to your nth term, the

00:01:34.920 --> 00:01:37.050
value here would be
you'd add 1 to it.

00:01:37.050 --> 00:01:41.290
So you'd have n plus 1, and then
you'd multiply that by 3.

00:01:41.290 --> 00:01:44.080
So the nth term is
3 times n plus 1.

00:01:44.080 --> 00:01:50.470
Or if you wanted to distribute
out the 3, it'd be 3n plus 3.

00:01:50.470 --> 00:01:51.790
And just make sure that
works for all of them.

00:01:51.790 --> 00:01:55.600
The first term, 3 times 1
is 3 plus 3, that works.

00:01:55.600 --> 00:01:56.250
That's 6.

00:01:56.250 --> 00:02:00.140
The second term, 3 times
2 plus 3 is 9.

00:02:00.140 --> 00:02:01.480
So it works for all
of these terms. So

00:02:01.480 --> 00:02:02.630
that is the nth term.

00:02:02.630 --> 00:02:05.530
We have to assume that this
pattern continues.


Understanding Logical Statements Commentary

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.670
I want to make a little
clarification for the last

00:00:02.670 --> 00:00:03.200
three videos.

00:00:03.200 --> 00:00:05.600
They've all been of the form:
determine whether the

00:00:05.600 --> 00:00:09.160
statement is always, sometimes,
or never true, and

00:00:09.160 --> 00:00:09.940
explain why.

00:00:09.940 --> 00:00:11.800
And then this was the actual
statement that they gave in

00:00:11.800 --> 00:00:12.860
the last problem.

00:00:12.860 --> 00:00:16.840
And to be frank, I think these
problems are written-- They're

00:00:16.840 --> 00:00:18.680
not completely well
constructed.

00:00:18.680 --> 00:00:23.800
Because when they say the
statement, the assumption is,

00:00:23.800 --> 00:00:28.540
that this whole thing over
here is this statement.

00:00:28.540 --> 00:00:30.010
That this whole thing
is a statement.

00:00:30.010 --> 00:00:31.480
Not just this first part.

00:00:31.480 --> 00:00:35.080
Not just the 3x is greater
than or equal to x.

00:00:35.080 --> 00:00:38.090
So if you assume that this whole
thing is a statement, we

00:00:38.090 --> 00:00:41.150
saw in the last video that
this is never true.

00:00:41.150 --> 00:00:49.230
I mean 3x is greater than or
equal to x only if x is

00:00:49.230 --> 00:00:50.450
greater than or equal to 0.

00:00:50.450 --> 00:00:51.590
We saw this in the last video.

00:00:51.590 --> 00:00:54.880
It is not true for
all real numbers.

00:00:54.880 --> 00:01:00.980
3x is greater than or equal
to x is not true

00:01:00.980 --> 00:01:04.910
for all real numbers.

00:01:04.910 --> 00:01:07.690
This is not true for
all real numbers.

00:01:07.690 --> 00:01:11.230
And the reason why I say that
this was kind of a strangely

00:01:11.230 --> 00:01:18.230
constructed question is I think
the goal of the problem

00:01:18.230 --> 00:01:20.630
writer is that they wanted
us to write sometimes.

00:01:20.630 --> 00:01:22.620
Which is what we did
in the last videos.

00:01:22.620 --> 00:01:26.110
But if you say sometimes, you're
actually interpreting

00:01:26.110 --> 00:01:29.430
this as the statement,
not the whole thing.

00:01:29.430 --> 00:01:34.330
You could say that 3x is greater
than or equal to x

00:01:34.330 --> 00:01:36.620
sometimes for certain
real numbers.

00:01:36.620 --> 00:01:37.380
Sometimes.

00:01:37.380 --> 00:01:38.110
This is true.

00:01:38.110 --> 00:01:40.190
You can say this is
true sometimes.

00:01:40.190 --> 00:01:42.690
But this whole thing, 3x is
greater than or equal to x for

00:01:42.690 --> 00:01:45.610
all or numbers, is not
sometimes true and

00:01:45.610 --> 00:01:46.300
sometimes not true.

00:01:46.300 --> 00:01:47.240
This is never true.

00:01:47.240 --> 00:01:49.620
You can never say it's
for all real numbers.

00:01:49.620 --> 00:01:53.640
It's not like this is true on
Monday and then on Tuesday

00:01:53.640 --> 00:01:54.550
this becomes not true.

00:01:54.550 --> 00:01:56.260
And then on Wednesday
it becomes true.

00:01:56.260 --> 00:01:57.230
This is never true.

00:01:57.230 --> 00:02:00.400
Because the statement for all
real numbers just won't work.

00:02:00.400 --> 00:02:03.040
You can't say for all real
numbers 3x is greater than or

00:02:03.040 --> 00:02:03.490
equal to x.

00:02:03.490 --> 00:02:06.460
I hope I'm not muddying
up the issue here.

00:02:06.460 --> 00:02:07.730
But I want to make
it very clear.

00:02:07.730 --> 00:02:10.570
Because the statement is
a false statement.

00:02:10.570 --> 00:02:15.660
This is a false statement.

00:02:15.660 --> 00:02:17.640
So if you really want to think
about it, if you view this

00:02:17.640 --> 00:02:19.780
whole thing as a statement, this
statement is never true.

00:02:19.780 --> 00:02:22.180
It's not like it's true on
Monday and then all of a

00:02:22.180 --> 00:02:25.660
sudden not true the next month
after that or all of a sudden

00:02:25.660 --> 00:02:26.990
it becomes true again.

00:02:26.990 --> 00:02:28.740
No, it's never going
to be true.

00:02:28.740 --> 00:02:31.420
I don't care how long you wait
or how far in the past you go,

00:02:31.420 --> 00:02:34.015
this statement will never be
true that 3x is greater than

00:02:34.015 --> 00:02:36.380
or equal to x for all
real numbers x.

00:02:36.380 --> 00:02:40.385
The problem writer wanted us to
circle sometimes because as

00:02:40.385 --> 00:02:42.130
I already stated-- and I'm
maybe getting a little

00:02:42.130 --> 00:02:44.710
repetitive here --because I
think they wanted us to think

00:02:44.710 --> 00:02:48.440
about the statement as 3x is
greater than or equal to x.

00:02:48.440 --> 00:02:51.053
So my general opinion-- I mean,
3x is greater than or

00:02:51.053 --> 00:02:55.230
equal to x sometimes for
some real numbers.

00:02:55.230 --> 00:02:56.770
But not for all real numbers.

00:02:56.770 --> 00:02:59.750
So you'd say sometimes if
this was the statement.

00:02:59.750 --> 00:03:02.250
And you would say never
true if this whole

00:03:02.250 --> 00:03:04.510
thing was the statement.

00:03:04.510 --> 00:03:06.060
I hope I'm not muddying up.

00:03:06.060 --> 00:03:08.060
But it's always important
because I think sometimes some

00:03:08.060 --> 00:03:11.210
of the hardest questions in
algebra are, to some degree,

00:03:11.210 --> 00:03:14.410
strangely constructed questions
where the answer

00:03:14.410 --> 00:03:17.480
writer was, or the question
writer was, trying to get you

00:03:17.480 --> 00:03:19.270
to go in some direction.

00:03:19.270 --> 00:03:21.790
And I actually looked at what
they wanted and that's why I

00:03:21.790 --> 00:03:23.140
wrote and said sometimes.

00:03:23.140 --> 00:03:27.540
But, if you really think about
it, the answer is never.


Understanding Logical Statements 4

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.020
Determine whether the
statement is always,

00:00:03.020 --> 00:00:05.990
sometimes, or never true,
and explain why.

00:00:05.990 --> 00:00:09.760
So the statement is, 3x is
greater than or equal to x for

00:00:09.760 --> 00:00:12.380
all real numbers x.

00:00:12.380 --> 00:00:14.090
Well let's try it out with
a couple of numbers.

00:00:14.090 --> 00:00:17.010
If you try it out with-- let's
say x is equal to 0.

00:00:17.010 --> 00:00:24.810
If x is equal to 0, then
3x is equal to 0.

00:00:24.810 --> 00:00:32.460
Which is equal to 0, which
is equal to x.

00:00:32.460 --> 00:00:36.590
So in this situation, if x is
equal to 0, 3x is indeed

00:00:36.590 --> 00:00:38.160
greater than or equal to x.

00:00:38.160 --> 00:00:39.860
In this case, it is equal.

00:00:39.860 --> 00:00:42.500
So we know that it's definitely
not never true.

00:00:42.500 --> 00:00:44.050
And we could try other
situations.

00:00:44.050 --> 00:00:48.510
If you put a 1 here, 3 times 1
is 3, which is greater than or

00:00:48.510 --> 00:00:49.870
equal to 1.

00:00:49.870 --> 00:00:54.040
So it is either always
or sometimes true.

00:00:54.040 --> 00:00:56.630
To cross out always, we just
have to think of a

00:00:56.630 --> 00:00:57.450
counterexample.

00:00:57.450 --> 00:01:00.310
And a good counterexample would
be a negative number.

00:01:00.310 --> 00:01:07.125
If x is negative 1, if x
is equal to negative

00:01:07.125 --> 00:01:09.610
1, then what happens?

00:01:09.610 --> 00:01:14.550
Then 3x is equal to negative 3
and x-- well, we already said

00:01:14.550 --> 00:01:16.070
x is equal to negative 1.

00:01:16.070 --> 00:01:19.300
And in that situation, negative
3 is not greater than

00:01:19.300 --> 00:01:21.810
or equal to negative 1.

00:01:21.810 --> 00:01:28.190
And negative 3 is not greater
than or equal to negative 1.

00:01:28.190 --> 00:01:30.210
It's less than negative 1.

00:01:30.210 --> 00:01:31.430
So this is a counterexample.

00:01:31.430 --> 00:01:33.790
This is not true.

00:01:33.790 --> 00:01:35.180
In general, it's not going
to be true for

00:01:35.180 --> 00:01:37.440
all negative numbers.

00:01:37.440 --> 00:01:39.000
So it's definitely not always.

00:01:39.000 --> 00:01:40.170
It's only sometimes true.

00:01:40.170 --> 00:01:42.240
It's only true in the case
when you're dealing with

00:01:42.240 --> 00:01:44.800
non-negative numbers.

00:01:44.800 --> 00:01:48.590
So it's only sometimes true.


Understanding Logical Statements 3

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.860
Determine whether the
statement is always,

00:00:02.860 --> 00:00:06.390
sometimes, or never true,
and explain why.

00:00:06.390 --> 00:00:09.730
The statement here is b squared
plus b minus 2, all of

00:00:09.730 --> 00:00:13.460
that over b plus 2 is equal
to b minus 1 for all

00:00:13.460 --> 00:00:15.160
real numbers b.

00:00:15.160 --> 00:00:18.440
Well one thing that just jumps
out at you before you even try

00:00:18.440 --> 00:00:20.440
to manipulate this algebraically
is that I can

00:00:20.440 --> 00:00:23.730
think of a b value where this
left expression won't be equal

00:00:23.730 --> 00:00:24.780
to b minus 1.

00:00:24.780 --> 00:00:27.680
A b value that will make this
left expression undefined.

00:00:27.680 --> 00:00:31.410
And it becomes undefined if this
denominator becomes 0.

00:00:31.410 --> 00:00:34.230
And the way the denominator is
going to become 0 is if b is

00:00:34.230 --> 00:00:34.800
negative 2.

00:00:34.800 --> 00:00:36.140
So let's write that down.

00:00:36.140 --> 00:00:45.400
If b is equal to negative 2,
then b squared plus b minus 2

00:00:45.400 --> 00:00:48.920
over b plus 2, what
does that become?

00:00:48.920 --> 00:00:53.220
It equals-- b squared would be 4
minus 2 minus 2, which would

00:00:53.220 --> 00:00:59.380
be 0/0, which is undefined.

00:01:05.319 --> 00:01:10.290
It's not equal to-- let me
write this this way.

00:01:10.290 --> 00:01:22.440
It's not equal to b minus 1,
which is-- negative 2 minus 1,

00:01:22.440 --> 00:01:23.260
negative 3.

00:01:23.260 --> 00:01:26.740
When you put negative 2 in here,
on the left-hand side,

00:01:26.740 --> 00:01:28.060
it doesn't evaluate
to negative 3.

00:01:28.060 --> 00:01:30.520
It evaluates to 0/0,
which is undefined.

00:01:30.520 --> 00:01:34.630
So we already found an example
of a b where this is not true.

00:01:34.630 --> 00:01:37.570
So it's definitely not always
going to be true.

00:01:37.570 --> 00:01:38.600
We found a case.

00:01:38.600 --> 00:01:40.860
Now, let's algebraically
manipulate this and see if

00:01:40.860 --> 00:01:43.330
it's true for pretty much
any other number.

00:01:43.330 --> 00:01:46.490
So let's see if we can maybe
factor the numerator up there.

00:01:46.490 --> 00:01:49.250
So b squared plus b minus 2.

00:01:49.250 --> 00:01:53.070
If you think about it, two
numbers when you take their

00:01:53.070 --> 00:01:56.770
product is negative 2, and when
you add them you get 1.

00:01:56.770 --> 00:01:59.380
Well, that's going to be
positive 2 and negative 1.

00:01:59.380 --> 00:02:03.560
So the numerator we can
rewrite as b plus 2

00:02:03.560 --> 00:02:05.970
times b minus 1.

00:02:05.970 --> 00:02:11.100
And then the denominator
is b plus 2.

00:02:11.100 --> 00:02:12.540
Well we can just leave
it like that.

00:02:12.540 --> 00:02:14.410
Let's just work on the
left-hand side.

00:02:14.410 --> 00:02:19.660
Now, if we assume that b is not
equal to negative 2, then

00:02:19.660 --> 00:02:22.020
b plus 2 divided by b plus
2 is well defined.

00:02:22.020 --> 00:02:23.710
It is 1.

00:02:23.710 --> 00:02:26.020
As long as it's not
0/0 it will be 1.

00:02:26.020 --> 00:02:29.940
And so this expression is the
same thing as b minus 1 if we

00:02:29.940 --> 00:02:32.810
assume that b is not equal
to negative 2.

00:02:32.810 --> 00:02:35.920
So if you assume that b is not
equal to negative 2, this

00:02:35.920 --> 00:02:40.310
thing over here does indeed
equal b minus 1.

00:02:40.310 --> 00:02:44.810
So it is true for all of the
other real numbers except for

00:02:44.810 --> 00:02:46.070
negative 2.

00:02:46.070 --> 00:02:48.110
So this is sometimes true.

00:02:48.110 --> 00:02:51.145
It's true every time except for
the case where b is equal

00:02:51.145 --> 00:02:51.647
to negative 2.

00:02:51.647 --> 00:02:53.340
And you can even try it
with some number here.

00:02:53.340 --> 00:02:54.590
Try it with 0.

00:02:54.590 --> 00:02:59.860
If I were to do 0 squared plus 0
minus 2, you get negative 2.

00:02:59.860 --> 00:03:01.980
Over 0 plus 2 is 2.

00:03:01.980 --> 00:03:05.510
And then that does indeed
equal negative 1.

00:03:05.510 --> 00:03:06.700
So it works with 0,
it works with 1.

00:03:06.700 --> 00:03:09.420
It'll work with any number
other than b is equal to

00:03:09.420 --> 00:03:11.220
negative 2.


Understanding Logical Statements 2

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.630
Identify the hypothesis and
conclusion of the following

00:00:03.630 --> 00:00:06.510
statement, and determine whether
the statement is

00:00:06.510 --> 00:00:10.980
always, sometimes,
or never true.

00:00:10.980 --> 00:00:12.160
So let's see what the
statement is.

00:00:12.160 --> 00:00:16.379
We have x-- well, this should be
an x squared plus 5x plus 7

00:00:16.379 --> 00:00:20.930
is greater than 0 for
all real numbers x.

00:00:20.930 --> 00:00:23.840
So first let's rewrite this
as a proposition,

00:00:23.840 --> 00:00:25.120
as an if/then statement.

00:00:25.120 --> 00:00:28.360
Then it becomes a lot easier to
identify the hypothesis and

00:00:28.360 --> 00:00:29.500
the conclusion.

00:00:29.500 --> 00:00:39.630
So we could rewrite this as
if x is a real number,

00:00:39.630 --> 00:00:42.880
then this is true.

00:00:42.880 --> 00:00:49.430
Then, x squared plus 5x plus
7 is greater than 0.

00:00:49.430 --> 00:00:51.280
And when you write it as an
if/then statement like this,

00:00:51.280 --> 00:00:54.680
it becomes a lot easier to
identify the hypothesis and

00:00:54.680 --> 00:00:56.330
the conclusion.

00:00:56.330 --> 00:00:58.520
Now the hypothesis is literally
just what the if

00:00:58.520 --> 00:01:00.700
part is conjuring up.

00:01:00.700 --> 00:01:02.625
So this right here in pink.

00:01:02.625 --> 00:01:03.700
x is a real number.

00:01:03.700 --> 00:01:04.900
That's the hypothesis.

00:01:04.900 --> 00:01:07.190
Sometimes it's called
the antecedent.

00:01:07.190 --> 00:01:08.556
That is the hypothesis.

00:01:11.980 --> 00:01:14.800
And as we said in previous
videos, it's different,

00:01:14.800 --> 00:01:17.050
although related, to the
notion of a scientific

00:01:17.050 --> 00:01:18.070
hypothesis.

00:01:18.070 --> 00:01:21.060
Here we're dealing with the
hypothesis of a hypothetical

00:01:21.060 --> 00:01:21.445
proposition.

00:01:21.445 --> 00:01:23.980
Don't want to confuse
you too much.

00:01:23.980 --> 00:01:26.600
It's what's inside of the if
part of the statement.

00:01:26.600 --> 00:01:29.010
The conclusion is the conclusion
from the then, or

00:01:29.010 --> 00:01:30.670
it's the consequent
right here.

00:01:30.670 --> 00:01:31.770
Then that.

00:01:31.770 --> 00:01:33.105
That right there is
the conclusion.

00:01:36.320 --> 00:01:38.040
We could have labeled
this too.

00:01:38.040 --> 00:01:41.600
The conclusion here is this
statement right here.

00:01:41.600 --> 00:01:44.770
That right there is
the conclusion.

00:01:44.770 --> 00:01:49.370
That is what you conclude
if x is a real number.

00:01:49.370 --> 00:01:53.410
So you could say for all that is
the hypothesis right there.

00:01:53.410 --> 00:01:57.180
For all real numbers x, that
is the hypothesis.

00:01:57.180 --> 00:01:59.173
Either way you want to do it,
it's always easier to write it

00:01:59.173 --> 00:02:00.820
as an if/then statement.

00:02:00.820 --> 00:02:03.620
Now, the more interesting
question is whether this

00:02:03.620 --> 00:02:06.810
statement is always, sometimes,
or never true.

00:02:06.810 --> 00:02:09.340
And the first thing that might
pop out at you is that the

00:02:09.340 --> 00:02:11.660
coefficient on the highest
degree term here, on the x

00:02:11.660 --> 00:02:13.770
squared term, it's a positive
coefficient.

00:02:13.770 --> 00:02:15.675
So this is going to be an
upward opening parabola.

00:02:18.570 --> 00:02:19.530
Let me draw that.

00:02:19.530 --> 00:02:25.770
So if this is the x-axis,
whether this is always,

00:02:25.770 --> 00:02:29.230
sometimes, or never true rests
on whether the vertex, whether

00:02:29.230 --> 00:02:31.300
the minimum point of this
parabola is above

00:02:31.300 --> 00:02:32.820
or below the x-axis.

00:02:32.820 --> 00:02:33.780
Let me draw it for you.

00:02:33.780 --> 00:02:39.530
So if the parabola looks like
this, then for any x you pick,

00:02:39.530 --> 00:02:42.840
the value of the function, the
value of x squared plus 5x

00:02:42.840 --> 00:02:45.800
plus 7 is always going
to be greater than 0.

00:02:45.800 --> 00:02:53.040
If the function looks like this,
then you're going to

00:02:53.040 --> 00:02:55.540
have some values for x where
it's greater than 0.

00:02:55.540 --> 00:02:58.060
But you're going to have some
values for x where it is equal

00:02:58.060 --> 00:02:59.450
or less than 0.

00:02:59.450 --> 00:03:02.860
So we have to determine whether
it sits above or maybe

00:03:02.860 --> 00:03:05.050
on or below, the x-axis.

00:03:05.050 --> 00:03:07.730
And that's the y-value
of the vertex.

00:03:07.730 --> 00:03:10.000
Now we know that it's
not never true.

00:03:10.000 --> 00:03:12.330
We know that it's not never true
because this is an upward

00:03:12.330 --> 00:03:13.190
opening parabola.

00:03:13.190 --> 00:03:16.400
Even if your vertex is a
negative number, eventually,

00:03:16.400 --> 00:03:18.610
if you get far enough away from
it, you're going to get

00:03:18.610 --> 00:03:21.810
the positive values
of this function.

00:03:21.810 --> 00:03:23.610
So we know it's not never true,
we just have to decide

00:03:23.610 --> 00:03:25.760
between always and sometimes.

00:03:25.760 --> 00:03:28.110
And to figure that out, let's
figure out where the vertex--

00:03:28.110 --> 00:03:30.910
and in particular, where the
y-value of the vertex is.

00:03:30.910 --> 00:03:33.980
But in order to get that
y-value, you need the x value.

00:03:33.980 --> 00:03:39.870
So the x-value of the vertex,
there's a formula here.

00:03:39.870 --> 00:03:41.880
I normally don't like giving
formulas, but we've shown this

00:03:41.880 --> 00:03:43.100
in multiple videos before.

00:03:43.100 --> 00:03:46.140
It's negative b over 2a.

00:03:46.140 --> 00:03:48.800
And I can't help myself, if
you look at the quadratic

00:03:48.800 --> 00:03:50.300
formula, it comes straight
out of that.

00:03:50.300 --> 00:03:53.330
The quadratic formula is
negative b over-- well, I'll

00:03:53.330 --> 00:03:57.110
write it as negative b plus or
minus the square root of b

00:03:57.110 --> 00:03:59.570
squared minus 4ac.

00:03:59.570 --> 00:04:00.970
All of that over 2a.

00:04:00.970 --> 00:04:03.340
Or you could write it as
negative b over 2a plus or

00:04:03.340 --> 00:04:05.040
minus the thing over 2a.

00:04:05.040 --> 00:04:07.660
You normally just see it with
a big 2a in the denominator,

00:04:07.660 --> 00:04:09.430
but you could split
it up like this.

00:04:09.430 --> 00:04:12.110
And notice, this right here is
a formula, I just told you

00:04:12.110 --> 00:04:14.540
it's a formula for the x-value
of the vertex.

00:04:14.540 --> 00:04:17.779
Which makes complete sense
because the vertex is always

00:04:17.779 --> 00:04:21.370
going to be along the line of
symmetry of our parabola.

00:04:21.370 --> 00:04:29.030
And it is going to be equal
distant from each of the 0's.

00:04:29.030 --> 00:04:30.930
So you get a 0 in the
plus direction.

00:04:30.930 --> 00:04:32.400
This is the distance
right here.

00:04:32.400 --> 00:04:34.170
This is one of the 0's in
the plus direction.

00:04:34.170 --> 00:04:36.610
And then you subtract,
that's the other 0.

00:04:36.610 --> 00:04:39.270
So it makes sense that negative
b over 2a is the

00:04:39.270 --> 00:04:41.130
x-value of the line
of symmetry or the

00:04:41.130 --> 00:04:42.650
x-value of the vertex.

00:04:42.650 --> 00:04:44.750
Now with that said, let's
actually figure out what the

00:04:44.750 --> 00:04:48.260
x-value of the vertex
is for this problem.

00:04:48.260 --> 00:04:50.510
So we have x squared--
oh, sorry.

00:04:50.510 --> 00:04:51.790
So negative b over 2a.

00:04:51.790 --> 00:04:56.290
So this is equal to negative
5 because b is 5,

00:04:56.290 --> 00:04:57.790
over 2 times 1.

00:04:57.790 --> 00:04:59.040
Over 2.

00:04:59.040 --> 00:05:00.410
So that's the x-value.

00:05:00.410 --> 00:05:01.730
Let's figure out
where it sits.

00:05:01.730 --> 00:05:04.300
Whether the y-value for
the vertex is below

00:05:04.300 --> 00:05:07.640
or above the x-axis.

00:05:07.640 --> 00:05:10.800
So the y-value, we'll just have
to substitute into this.

00:05:10.800 --> 00:05:15.210
The y-value of the vertex is
going to be equal to negative

00:05:15.210 --> 00:05:16.460
5/2 squared.

00:05:18.900 --> 00:05:22.680
Plus 5 times negative 5/2.

00:05:22.680 --> 00:05:24.230
And then finally, plus 7.

00:05:24.230 --> 00:05:26.620
Let's see if this is
positive or not.

00:05:26.620 --> 00:05:32.340
So negative 5/2 squared
is 25/4.

00:05:32.340 --> 00:05:38.540
Minus 25/2, plus 7.

00:05:38.540 --> 00:05:39.390
So we can add these.

00:05:39.390 --> 00:05:40.720
Let's put everything over 4.

00:05:40.720 --> 00:05:43.700
So this is equal to 25/4.

00:05:43.700 --> 00:05:45.550
25/2 is the same
thing as 50/4.

00:05:45.550 --> 00:05:48.400
Minus 50/4.

00:05:48.400 --> 00:05:52.500
Plus 28/4.

00:05:52.500 --> 00:05:54.790
This is equal to-- well,
we know our denominator

00:05:54.790 --> 00:05:55.690
is going to be 4.

00:05:55.690 --> 00:05:58.500
And you have 25 plus 28 is 53.

00:05:58.500 --> 00:06:00.360
Minus 50 is 3.

00:06:00.360 --> 00:06:03.280
So it equals positive 3/4.

00:06:03.280 --> 00:06:06.070
So the y-value is positive.

00:06:06.070 --> 00:06:07.510
It is above the x-axis.

00:06:07.510 --> 00:06:10.160
So the parabola will look
something like this.

00:06:10.160 --> 00:06:11.590
The parabola will look something
like this where the

00:06:11.590 --> 00:06:14.140
y-value is 3/4 for the vertax.

00:06:14.140 --> 00:06:17.010
The x-value we just saw
was negative 5/2.

00:06:17.010 --> 00:06:19.940
And because the minimum point
of this upward opening

00:06:19.940 --> 00:06:23.260
parabola is a positive number,
the minimum y-value is a

00:06:23.260 --> 00:06:26.890
positive number, it's always
going to have values at that

00:06:26.890 --> 00:06:28.470
level or above it.

00:06:28.470 --> 00:06:30.540
So this is true.

00:06:30.540 --> 00:06:33.470
If x is a real number,
then it is true.

00:06:33.470 --> 00:06:35.130
It is always true.

00:06:35.130 --> 00:06:40.170
It is always true that x squared
plus 5x plus 7 is

00:06:40.170 --> 00:06:42.470
greater than 0.


Understanding Logical Statements 1

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.580
Identify the hypothesis and
conclusion of the following

00:00:03.580 --> 00:00:05.760
statement, and determine whether
the statement is

00:00:05.760 --> 00:00:08.130
always, sometimes,
or never true.

00:00:08.130 --> 00:00:11.970
And the statement here is, the
absolute value of x is greater

00:00:11.970 --> 00:00:15.260
than 0 for all real numbers x.

00:00:15.260 --> 00:00:17.580
Now if I want to decide what the
hypothesis and conclusion

00:00:17.580 --> 00:00:21.610
is here, it's usually a lot
easier to figure them out if

00:00:21.610 --> 00:00:24.110
you rewrite this statement, this
exact statement, as an

00:00:24.110 --> 00:00:25.080
if/then statement.

00:00:25.080 --> 00:00:28.510
So what this statement is saying
is, if x is a real

00:00:28.510 --> 00:00:32.360
number, then the absolute value
of x is greater than 0.

00:00:32.360 --> 00:00:33.710
So let me rewrite that.

00:00:33.710 --> 00:00:47.480
If x is a real number, then
the absolute value of x is

00:00:47.480 --> 00:00:48.420
greater than 0.

00:00:48.420 --> 00:00:50.170
These are equivalent
statements.

00:00:50.170 --> 00:00:52.500
Now when you write it like
this, it's much easier to

00:00:52.500 --> 00:00:55.100
identify the hypothesis
and the conclusion.

00:00:55.100 --> 00:00:58.730
The hypothesis you can view as
it's the if part of this

00:00:58.730 --> 00:00:59.400
proposition.

00:00:59.400 --> 00:01:01.460
So this right here is
the hypothesis.

00:01:01.460 --> 00:01:03.890
x is a real number.

00:01:03.890 --> 00:01:05.274
That is the hypothesis.

00:01:08.910 --> 00:01:11.030
And the conclusion
is the then part.

00:01:11.030 --> 00:01:14.020
Sometimes it's called
the consequent.

00:01:14.020 --> 00:01:16.115
It's this part right here.

00:01:16.115 --> 00:01:17.100
It's this part, x.

00:01:17.100 --> 00:01:19.480
Absolute value of x
is greater than 0.

00:01:19.480 --> 00:01:23.750
This is the conclusion
or the consequent.

00:01:23.750 --> 00:01:25.830
And over here, if you wanted
to circle them, this is the

00:01:25.830 --> 00:01:27.130
conclusion.

00:01:27.130 --> 00:01:29.130
That is the conclusion.

00:01:29.130 --> 00:01:32.940
And the hypothesis is for
all real numbers.

00:01:32.940 --> 00:01:35.430
It's a little bit easier to
identify when you see it

00:01:35.430 --> 00:01:36.270
written like this.

00:01:36.270 --> 00:01:37.730
This is the hypothesis.

00:01:37.730 --> 00:01:40.300
Now I don't want you to get
confused with the idea of a

00:01:40.300 --> 00:01:42.320
scientific hypothesis,
which is related but

00:01:42.320 --> 00:01:43.670
a little bit different.

00:01:43.670 --> 00:01:48.610
A scientific hypothesis is an
explanation for some phenomena

00:01:48.610 --> 00:01:49.860
that can be tested.

00:01:49.860 --> 00:01:54.290
So a scientific hypothesis
might be something like

00:01:54.290 --> 00:01:57.720
saying, blueberries-- which
I'm writing in blue

00:01:57.720 --> 00:01:58.740
intentionally.

00:01:58.740 --> 00:02:02.970
Blueberries help reduce
the risk of cancer.

00:02:08.550 --> 00:02:10.940
This right here is a scientific
hypothesis.

00:02:10.940 --> 00:02:13.430
You could then go feed a bunch
of people blueberries for a

00:02:13.430 --> 00:02:16.210
long period of time and see if
they have a less likelihood of

00:02:16.210 --> 00:02:18.490
developing cancer than people
who never got blueberries and

00:02:18.490 --> 00:02:19.550
everything else is equal.

00:02:19.550 --> 00:02:20.890
And so it would be testable.

00:02:20.890 --> 00:02:22.000
You could see if it's true.

00:02:22.000 --> 00:02:24.430
And the conclusion would be
whether this is true or not.

00:02:24.430 --> 00:02:27.880
But this is in the scientific
hypothesis world, not to be

00:02:27.880 --> 00:02:30.690
confused with the notion
of hypothesis that

00:02:30.690 --> 00:02:31.530
we're using right now.

00:02:31.530 --> 00:02:32.940
And I don't want to confuse
you too much, so I'm just

00:02:32.940 --> 00:02:35.440
going to leave this
on the side here.

00:02:35.440 --> 00:02:38.630
But when you're in Algebra I
and especially-- these are

00:02:38.630 --> 00:02:40.780
according to the California
standards.

00:02:40.780 --> 00:02:45.010
The hypothesis in this context
is the antecedent of this

00:02:45.010 --> 00:02:45.660
proposition.

00:02:45.660 --> 00:02:47.880
Or it's the if part of
this proposition.

00:02:47.880 --> 00:02:49.580
Now, the more interesting
question is

00:02:49.580 --> 00:02:50.520
whether it is true.

00:02:50.520 --> 00:02:52.610
I mean, we can sit and label
statements all day, but that's

00:02:52.610 --> 00:02:53.380
not interesting.

00:02:53.380 --> 00:02:56.430
The more fundamental thing is,
is this a true statement?

00:02:56.430 --> 00:02:59.120
Is it true that if x is a real
number, then the absolute

00:02:59.120 --> 00:03:01.540
value of x is greater than 0?

00:03:01.540 --> 00:03:04.890
Well the one number that pops
out of my mind that disproves

00:03:04.890 --> 00:03:07.190
this is if x is 0.

00:03:07.190 --> 00:03:08.590
0 is a real number.

00:03:12.480 --> 00:03:13.940
That's true.

00:03:13.940 --> 00:03:16.350
But-- and I'll write
but in caps.

00:03:16.350 --> 00:03:22.440
BUT the absolute value
of 0 is equal to 0.

00:03:22.440 --> 00:03:25.320
It's not greater than 0.

00:03:25.320 --> 00:03:27.140
So this disproves
this statement.

00:03:27.140 --> 00:03:29.300
It's definitely not
always true.

00:03:32.130 --> 00:03:35.120
Now, if you pretty much have
any other real number, this

00:03:35.120 --> 00:03:36.010
will be true.

00:03:36.010 --> 00:03:38.580
Because any other real number is
some distance away from 0.

00:03:38.580 --> 00:03:40.300
That's what the absolute
value measures.

00:03:40.300 --> 00:03:43.035
So if you took a 1 or a negative
1 or whatever, the

00:03:43.035 --> 00:03:45.150
absolute value of those
are greater than 0.

00:03:45.150 --> 00:03:47.250
So this is sometimes true.


Number sets 2

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:05.430
What number sets does
the number 3.4028

00:00:05.430 --> 00:00:07.330
repeating belong to?

00:00:07.330 --> 00:00:09.150
And before even answering the
question, let's just think

00:00:09.150 --> 00:00:10.690
about what this represents.

00:00:10.690 --> 00:00:13.000
And especially what this
line on top means.

00:00:13.000 --> 00:00:15.770
So this line on top means
that the 28 just

00:00:15.770 --> 00:00:17.420
keep repeating forever.

00:00:17.420 --> 00:00:25.090
So I could express this number
as 3.4028, but the 28 just

00:00:25.090 --> 00:00:26.110
keep repeating.

00:00:26.110 --> 00:00:29.740
Just keep repeating on and
on and on forever.

00:00:29.740 --> 00:00:32.299
I could just keep writing
them forever and ever.

00:00:32.299 --> 00:00:35.210
And obviously, it's just easier
to write this line over

00:00:35.210 --> 00:00:37.620
the 28 to say that it
repeats forever.

00:00:37.620 --> 00:00:41.290
Now let's think about what
number sets it belongs to.

00:00:41.290 --> 00:00:44.600
Well, the broadest number set
we've dealt with so far is the

00:00:44.600 --> 00:00:45.330
real numbers.

00:00:45.330 --> 00:00:48.420
And this definitely belongs
to the real numbers.

00:00:48.420 --> 00:00:50.300
The real numbers is essentially
the entire number

00:00:50.300 --> 00:00:51.990
line that we're used to using.

00:00:51.990 --> 00:00:55.660
And 3.4028 repeating sits
someplace over here.

00:00:55.660 --> 00:01:01.340
If this is negative 1, this
is 0, 1, 2, 3, 4.

00:01:01.340 --> 00:01:04.730
3.4028 is a little bit more
than 3.4, a little

00:01:04.730 --> 00:01:06.490
bit less than 3.41.

00:01:06.490 --> 00:01:07.760
It would sit right over there.

00:01:07.760 --> 00:01:09.450
So it definitely sits
on the number line.

00:01:09.450 --> 00:01:11.090
It's a real number.

00:01:11.090 --> 00:01:13.870
So it definitely is real.

00:01:13.870 --> 00:01:16.370
It definitely is
a real number.

00:01:16.370 --> 00:01:19.080
But the not so obvious question
is whether it is a

00:01:19.080 --> 00:01:20.180
rational number.

00:01:20.180 --> 00:01:25.040
Remember, a rational number is
one that can be expressed as a

00:01:25.040 --> 00:01:26.890
rational expression
or as a fraction.

00:01:26.890 --> 00:01:34.390
If I were to tell you that p is
rational, that means that p

00:01:34.390 --> 00:01:37.840
can be expressed as the
ratio of two integers.

00:01:37.840 --> 00:01:45.620
That means that p can be
expressed as the ratio of two

00:01:45.620 --> 00:01:47.900
integers, m/n.

00:01:47.900 --> 00:01:50.960
So the question is, can I
express this as the ratio of

00:01:50.960 --> 00:01:51.410
two integers?

00:01:51.410 --> 00:01:52.410
Or another way to think
of it, can I

00:01:52.410 --> 00:01:53.990
express this as a fraction?

00:01:53.990 --> 00:01:58.510
And to do that, let's actually
express it as a fraction.

00:01:58.510 --> 00:02:01.310
Let's define x as being
equal to this number.

00:02:01.310 --> 00:02:09.960
So x is equal to 3.4028
repeating.

00:02:09.960 --> 00:02:12.650
Let's think about
what 10,000x is.

00:02:12.650 --> 00:02:14.470
And the only reason why I want
10,000x is because I want to

00:02:14.470 --> 00:02:16.960
move the decimal point all the
way to the right over here.

00:02:16.960 --> 00:02:21.710
So 10,000x.

00:02:21.710 --> 00:02:23.380
What is that going
to be equal to?

00:02:23.380 --> 00:02:26.350
Well every time you multiply
by a power of 10, you shift

00:02:26.350 --> 00:02:27.420
the decimal one to the right.

00:02:27.420 --> 00:02:29.790
10,000 is 10 to the
fourth power.

00:02:29.790 --> 00:02:31.780
So it's like shifting
the decimal over to

00:02:31.780 --> 00:02:32.830
the right four spaces.

00:02:32.830 --> 00:02:36.400
1, 2, 3, 4.

00:02:36.400 --> 00:02:40.575
So it'll be 34,028.

00:02:40.575 --> 00:02:42.700
But these 28's just
keep repeating.

00:02:42.700 --> 00:02:45.820
So you'll still have the 28's
go on and on, and on and on,

00:02:45.820 --> 00:02:46.720
and on after that.

00:02:46.720 --> 00:02:49.550
They just all got shifted to the
left of the decimal point

00:02:49.550 --> 00:02:50.430
by five spaces.

00:02:50.430 --> 00:02:51.070
You can view it that way.

00:02:51.070 --> 00:02:53.140
That makes sense.

00:02:53.140 --> 00:02:54.670
It's nearly 3 and 1/2.

00:02:54.670 --> 00:02:57.810
If you multiply by 10,000,
you get almost 35,000.

00:02:57.810 --> 00:02:59.490
So that's 10,000x.

00:02:59.490 --> 00:03:00.970
Now, let's also think
about 100x.

00:03:00.970 --> 00:03:04.340
And my whole exercise here is
I want to get two numbers

00:03:04.340 --> 00:03:06.590
that, when I subtract them and
they're in terms of x, the

00:03:06.590 --> 00:03:08.130
repeating part disappears.

00:03:08.130 --> 00:03:10.970
And then we can just treat them
as traditional numbers.

00:03:10.970 --> 00:03:13.260
So let's think about
what 100x is.

00:03:13.260 --> 00:03:15.530
100x.

00:03:15.530 --> 00:03:17.010
That moves this decimal point.

00:03:17.010 --> 00:03:18.370
Remember, the decimal point
was here originally.

00:03:18.370 --> 00:03:20.860
It moves it over to the
right two spaces.

00:03:20.860 --> 00:03:24.830
So 100x would be 300-- Let
me write it like this.

00:03:24.830 --> 00:03:30.750
It would be 340.28 repeating.

00:03:30.750 --> 00:03:32.220
We could have put the 28
repeating here, but it

00:03:32.220 --> 00:03:33.010
wouldn't have made
as much sense.

00:03:33.010 --> 00:03:34.670
You always want to write it
after the decimal point.

00:03:34.670 --> 00:03:37.340
So we have to write 28 again to
show that it is repeating.

00:03:37.340 --> 00:03:39.710
Now something interesting
is going on.

00:03:39.710 --> 00:03:42.400
These two numbers, they're
just multiples of x.

00:03:42.400 --> 00:03:45.790
And if I subtract the bottom one
from the top one, what's

00:03:45.790 --> 00:03:46.710
going to happen?

00:03:46.710 --> 00:03:48.530
Well the repeating part
is going to disappear.

00:03:48.530 --> 00:03:49.170
So let's do that.

00:03:49.170 --> 00:03:52.280
Let's do that on both sides
of this equation.

00:03:52.280 --> 00:03:53.230
Let's do it.

00:03:53.230 --> 00:03:58.210
So on the left-hand side of this
equation, 10,000x minus

00:03:58.210 --> 00:04:03.620
100x is going to be 9,900x.

00:04:03.620 --> 00:04:06.960
And on the right-hand side,
let's see-- The decimal part

00:04:06.960 --> 00:04:08.230
will cancel out.

00:04:08.230 --> 00:04:12.030
And we just have to figure out
what 34,028 minus 340 is.

00:04:12.030 --> 00:04:14.120
So let's just figure this out.

00:04:14.120 --> 00:04:16.010
8 is larger than 0, so we
won't have to do any

00:04:16.010 --> 00:04:16.649
regrouping there.

00:04:16.649 --> 00:04:19.769
2 is less than 4.

00:04:19.769 --> 00:04:22.200
So we will have to do some
regrouping, but we can't

00:04:22.200 --> 00:04:25.510
borrow yet because we
have a 0 over there.

00:04:25.510 --> 00:04:27.710
And 0 is less than 3, so we
have to do some regrouping

00:04:27.710 --> 00:04:29.000
there or some borrowing.

00:04:29.000 --> 00:04:31.770
So let's borrow from
the 4 first.

00:04:31.770 --> 00:04:36.590
So if we borrow from the 4, this
becomes a 3 and then this

00:04:36.590 --> 00:04:38.140
becomes a 10.

00:04:38.140 --> 00:04:40.460
And then the 2 can now
borrow from the 10.

00:04:40.460 --> 00:04:44.090
This becomes a 9 and
this becomes a 12.

00:04:44.090 --> 00:04:45.820
And now we can do
the subtraction.

00:04:45.820 --> 00:04:48.390
8 minus 0 is 8.

00:04:48.390 --> 00:04:51.110
12 minus 4 is 8.

00:04:51.110 --> 00:04:53.880
9 minus 3 is 6.

00:04:53.880 --> 00:04:55.920
3 minus nothing is 3.

00:04:55.920 --> 00:04:57.950
3 minus nothing is 3.

00:04:57.950 --> 00:05:05.320
So 9,900x is equal to 33,688.

00:05:05.320 --> 00:05:09.180
We just subtracted 340
from this up here.

00:05:09.180 --> 00:05:13.110
So we get 33,688.

00:05:13.110 --> 00:05:15.710
Now, if we want to solve
for x, we just divide

00:05:15.710 --> 00:05:21.610
both sides by 9,900.

00:05:21.610 --> 00:05:23.990
Divide the left by 9,900.

00:05:23.990 --> 00:05:26.900
Divide the right by 9,900.

00:05:26.900 --> 00:05:28.000
And then, what are
we left with?

00:05:28.000 --> 00:05:36.850
We're left with x is equal
to 33,688 over 9,900.

00:05:36.850 --> 00:05:38.550
Now what's the big
deal about this?

00:05:38.550 --> 00:05:41.900
Well, x was this number. x was
this number that we started

00:05:41.900 --> 00:05:44.580
off with, this number that
just kept on repeating.

00:05:44.580 --> 00:05:47.500
And by doing a little bit of
algebraic manipulation and

00:05:47.500 --> 00:05:49.660
subtracting one multiple of it
from another, we're able to

00:05:49.660 --> 00:05:52.530
express that same exact
x as a fraction.

00:05:52.530 --> 00:05:55.780
Now this isn't in simplest
terms. I mean they're both

00:05:55.780 --> 00:05:58.900
definitely divisible by 2
and it looks like by 4.

00:05:58.900 --> 00:06:01.960
So you could put this in lowest
common form, but we

00:06:01.960 --> 00:06:02.910
don't care about that.

00:06:02.910 --> 00:06:05.055
All we care about is the fact
that we were able to represent

00:06:05.055 --> 00:06:09.050
x, we were able to represent
this number, as a fraction.

00:06:09.050 --> 00:06:11.620
As the ratio of two integers.

00:06:11.620 --> 00:06:14.720
So the number is
also rational.

00:06:14.720 --> 00:06:16.550
It is also rational.

00:06:16.550 --> 00:06:19.010
And this technique we
did, it doesn't only

00:06:19.010 --> 00:06:20.700
apply to this number.

00:06:20.700 --> 00:06:24.370
Any time you have a number that
has repeating digits, you

00:06:24.370 --> 00:06:25.000
could do this.

00:06:25.000 --> 00:06:27.530
So in general, repeating
digits are rational.

00:06:27.530 --> 00:06:30.090
The ones that are irrational are
the ones that never, ever,

00:06:30.090 --> 00:06:32.860
ever repeat, like pi.

00:06:32.860 --> 00:06:34.590
And so the other things, I think
it's pretty obvious,

00:06:34.590 --> 00:06:35.810
this isn't an integer.

00:06:35.810 --> 00:06:37.410
The integers are the
whole numbers that

00:06:37.410 --> 00:06:38.020
we're dealing with.

00:06:38.020 --> 00:06:40.390
So this is someplace in
between the integers.

00:06:40.390 --> 00:06:43.360
It's not a natural number or a
whole number, which depending

00:06:43.360 --> 00:06:46.240
on the context are viewed
as subsets of integers.

00:06:46.240 --> 00:06:47.360
So it's definitely
none of those.

00:06:47.360 --> 00:06:49.110
So it is real and
it is rational.

00:06:49.110 --> 00:06:51.460
That's all we can
say about it.


Number Sets 1

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.410
What number sets does the
number 8 belong to?

00:00:03.410 --> 00:00:05.140
Now before even answering that
question, let's just think

00:00:05.140 --> 00:00:07.960
about the different number
sets there are.

00:00:07.960 --> 00:00:11.360
There are the real numbers.

00:00:11.360 --> 00:00:13.640
And these are, literally, all
of the numbers we've been

00:00:13.640 --> 00:00:15.240
dealing with to date.

00:00:15.240 --> 00:00:17.750
When we go to more advanced
topics, you'll learn that

00:00:17.750 --> 00:00:20.880
there are such things as
imaginary and complex numbers.

00:00:20.880 --> 00:00:23.620
Or at least in the mathematical
sense they exist.

00:00:23.620 --> 00:00:25.690
But for now, real numbers are
essentially everything that we

00:00:25.690 --> 00:00:26.925
know of on the number line.

00:00:26.925 --> 00:00:30.650
So if I were to draw the number
line right here, a real

00:00:30.650 --> 00:00:33.180
number can be represented as any
point on this number line.

00:00:33.180 --> 00:00:35.300
Or it's the set of all of the
points on this number line if

00:00:35.300 --> 00:00:37.830
you just keep going in the
positive direction and you

00:00:37.830 --> 00:00:41.050
just keep going in the
negative direction.

00:00:41.050 --> 00:00:42.220
It's all of these point.

00:00:42.220 --> 00:00:46.390
And it includes points, it
includes numbers like pi,

00:00:46.390 --> 00:00:49.910
where their decimal
representations, 3.14159, it

00:00:49.910 --> 00:00:52.730
just keeps repeating
on and on and on.

00:00:52.730 --> 00:00:55.900
It cannot be represented as a
fraction or as a rational

00:00:55.900 --> 00:00:56.660
expression.

00:00:56.660 --> 00:00:59.390
It just is this decimal
expansion that goes on and on

00:00:59.390 --> 00:01:00.240
and on and on.

00:01:00.240 --> 00:01:02.580
Square root of 2 is another
one that is also a real.

00:01:02.580 --> 00:01:03.690
But everything else
is also real.

00:01:03.690 --> 00:01:06.430
So real is kind of the broadest
set of numbers we've

00:01:06.430 --> 00:01:07.090
been dealing with.

00:01:07.090 --> 00:01:09.500
Now if we want to be a little
bit more restrictive, we get

00:01:09.500 --> 00:01:10.750
to rational numbers.

00:01:14.310 --> 00:01:16.170
And the best way to think about
rational, there's two

00:01:16.170 --> 00:01:16.920
ways to think about it.

00:01:16.920 --> 00:01:27.170
These are numbers that can be
expressed as-- you could say

00:01:27.170 --> 00:01:35.370
as a rational expression
or as a fraction.

00:01:35.370 --> 00:01:38.640
So for example, negative 1 is
a rational number because

00:01:38.640 --> 00:01:42.300
negative 1 could be written
as negative 1/1.

00:01:42.300 --> 00:01:43.720
0 is a rational number.

00:01:43.720 --> 00:01:45.700
It can be expressed as
0 over anything.

00:01:45.700 --> 00:01:46.360
Could be 0/10.

00:01:46.360 --> 00:01:47.810
It doesn't have to be.

00:01:47.810 --> 00:01:48.920
2 is a rational number.

00:01:48.920 --> 00:01:50.290
1/2 is a rational number.

00:01:50.290 --> 00:01:51.920
All of these are rational
numbers.

00:01:51.920 --> 00:01:55.020
What aren't rational numbers are
numbers like square root

00:01:55.020 --> 00:01:59.660
of 2 or 3.14 or pi or e, these
numbers that cannot be

00:01:59.660 --> 00:02:00.880
represented as a fraction.

00:02:00.880 --> 00:02:03.770
So you can imagine that the
rational numbers, it's kind of

00:02:03.770 --> 00:02:05.510
like the set of real numbers.

00:02:05.510 --> 00:02:06.455
Maybe I'll write it like this.

00:02:06.455 --> 00:02:09.780
It's the set of real numbers
minus-- if you were to take

00:02:09.780 --> 00:02:12.215
out irrational numbers.

00:02:17.150 --> 00:02:22.030
And these are like-- so for
example, we have things like

00:02:22.030 --> 00:02:24.590
pi, e, these are
actual numbers.

00:02:24.590 --> 00:02:29.200
You know, 3.1415, e is
between 2 and 3.

00:02:29.200 --> 00:02:30.120
Square root of 2.

00:02:30.120 --> 00:02:31.300
We can go on and on and on.

00:02:31.300 --> 00:02:33.700
There's an infinite number
of irrational numbers.

00:02:33.700 --> 00:02:34.710
So it's a little abstract.

00:02:34.710 --> 00:02:36.890
But anything that can be
represented as a fraction, you

00:02:36.890 --> 00:02:39.020
definitely know is rational.

00:02:39.020 --> 00:02:40.640
Now, if we want to get a little
bit more restrictive

00:02:40.640 --> 00:02:43.430
than that, if we just want to
deal with the numbers that

00:02:43.430 --> 00:02:45.520
aren't fractions in the
traditional sense.

00:02:45.520 --> 00:02:47.850
That are, I guess we could use
the word whole numbers,

00:02:47.850 --> 00:02:49.420
although we want to be careful
with that word because that

00:02:49.420 --> 00:02:52.710
could also be considered
a number set.

00:02:52.710 --> 00:02:55.670
If we're thinking about just
the 4, the 3, the 2, all of

00:02:55.670 --> 00:03:00.100
the kind of-- I guess
whole numbers.

00:03:00.100 --> 00:03:01.640
Then we're talking about
the integers.

00:03:08.660 --> 00:03:11.520
We could just go from the
negative direction, maybe

00:03:11.520 --> 00:03:14.380
negative 2, negative
1, then 0.

00:03:14.380 --> 00:03:16.610
Then we get 1, 2, 3.

00:03:16.610 --> 00:03:17.770
I think you get the idea.

00:03:17.770 --> 00:03:21.130
It's all of the positive and
negative whole numbers and it

00:03:21.130 --> 00:03:21.910
includes 0.

00:03:21.910 --> 00:03:24.000
So if you were to do the
integers on this line, it

00:03:24.000 --> 00:03:27.110
would be everything that
I'm doing in orange.

00:03:27.110 --> 00:03:30.830
Now, there's two other sets that
are subsets of integers

00:03:30.830 --> 00:03:33.780
depending on the interpretation
you look at.

00:03:33.780 --> 00:03:36.320
So sometimes you'll hear people
say, whole numbers.

00:03:40.090 --> 00:03:42.250
And this is a bit of an
ambiguous statement.

00:03:42.250 --> 00:03:44.880
Because sometimes when people
say whole numbers, they're

00:03:44.880 --> 00:03:46.790
referring to integers.

00:03:46.790 --> 00:03:49.220
Sometimes when they're referring
to whole numbers,

00:03:49.220 --> 00:03:51.830
they're referring to
non-negative integers.

00:03:51.830 --> 00:03:53.810
So 0, 1, 2, 3.

00:03:53.810 --> 00:03:56.900
Sometimes they're referring
to just positive integers.

00:03:56.900 --> 00:03:59.750
So 1, 2, 3, so on
and so forth.

00:03:59.750 --> 00:04:01.175
So this is a little
bit ambiguous.

00:04:05.880 --> 00:04:08.770
I've seen it used in all of
those different ways.

00:04:08.770 --> 00:04:11.190
But if someone says a whole
number, I generally think of

00:04:11.190 --> 00:04:12.590
non-negative numbers.

00:04:12.590 --> 00:04:15.970
It's ambiguous, but for our
purposes, let's think of it as

00:04:15.970 --> 00:04:19.130
0, 1, 2, so on and so forth.

00:04:19.130 --> 00:04:22.040
Now the last one to consider
is natural numbers.

00:04:22.040 --> 00:04:23.946
And once again, this one
is also ambiguous.

00:04:27.410 --> 00:04:29.430
And the reason why it's called
natural numbers is these are

00:04:29.430 --> 00:04:31.640
numbers that you would actually
encounter if you were

00:04:31.640 --> 00:04:33.500
to either try to order
things or actually

00:04:33.500 --> 00:04:37.340
count things in nature.

00:04:37.340 --> 00:04:38.730
So this is also ambiguous.

00:04:41.630 --> 00:04:50.880
But it definitely excludes
negative numbers.

00:04:53.540 --> 00:04:55.250
And so if it definitely excludes
negative numbers,

00:04:55.250 --> 00:04:56.340
you're like, why is
it ambiguous?

00:04:56.340 --> 00:04:58.620
Well, it's ambiguous whether
it includes 0.

00:04:58.620 --> 00:05:01.220
So sometimes 0 is natural.

00:05:01.220 --> 00:05:08.430
So 0 may or may not be natural,

00:05:08.430 --> 00:05:11.830
depending on the context.

00:05:11.830 --> 00:05:14.170
Now, let's answer our
actual question.

00:05:14.170 --> 00:05:16.700
What number sets does
8 belong to?

00:05:16.700 --> 00:05:19.450
Well, 8 definitely belongs
to the real numbers.

00:05:19.450 --> 00:05:20.880
It is on the number line.

00:05:20.880 --> 00:05:22.540
It is definitely a
rational number.

00:05:22.540 --> 00:05:28.690
We can express it as 8/1, or
16/2, or even negative 8 over

00:05:28.690 --> 00:05:29.300
negative 1.

00:05:29.300 --> 00:05:32.050
I mean you could actually have
an infinite number of ways to

00:05:32.050 --> 00:05:33.760
express it as a fraction.

00:05:33.760 --> 00:05:35.760
So it's definitely a
rational number.

00:05:35.760 --> 00:05:37.450
It's definitely an integer.

00:05:37.450 --> 00:05:39.110
It's a whole number.

00:05:39.110 --> 00:05:40.270
It's definitely an integer.

00:05:40.270 --> 00:05:41.300
It is a whole number.

00:05:41.300 --> 00:05:43.960
I mean we're ambiguous what to
include in whole numbers, but

00:05:43.960 --> 00:05:46.550
positive 8 is unambiguously
a whole number.

00:05:46.550 --> 00:05:48.040
It's definitely included.

00:05:48.040 --> 00:05:50.080
And it's unambiguously
a natural number.

00:05:50.080 --> 00:05:53.580
If I go out and count trees,
I can count 8 trees.

00:05:53.580 --> 00:05:55.530
So it is a positive.

00:05:55.530 --> 00:05:57.710
If the number was 0, I'd be
like, well, depending on who

00:05:57.710 --> 00:05:59.360
you talk to it might be
a natural number.

00:05:59.360 --> 00:06:01.860
But 8 definitely is
a natural number.

00:06:01.860 --> 00:06:05.680
So it's a part of all of
these number sets.


Applying Rational Equations 1 Extra Commentary

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.200
I want to make a quick note in
this video because I realized

00:00:03.200 --> 00:00:08.270
that some of these ratio and
rate algebraic problems can

00:00:08.270 --> 00:00:09.470
get a little bit confusing.

00:00:09.470 --> 00:00:11.000
So I just want to make
sure you get the

00:00:11.000 --> 00:00:12.080
intuition behind it.

00:00:12.080 --> 00:00:17.090
So if I were to tell you that
Sal takes-- I don't know--

00:00:17.090 --> 00:00:19.530
let's say he takes 1
hour to eat a pie.

00:00:19.530 --> 00:00:21.420
Let me pick a number
better than 1.

00:00:21.420 --> 00:00:23.190
Let's say he takes 2 hours.

00:00:23.190 --> 00:00:24.340
2 hours to eat a pie.

00:00:24.340 --> 00:00:30.280
So he takes 2 hours per pie.

00:00:30.280 --> 00:00:37.320
And let's say Dean takes
3 hours to eat a pie.

00:00:37.320 --> 00:00:42.500
So he takes 3 hours per pie.

00:00:42.500 --> 00:00:46.210
And if I were to ask you, how
long does it take for both of

00:00:46.210 --> 00:00:48.670
them, if they're both working
together, how long will it

00:00:48.670 --> 00:00:50.630
take them to eat a pie?

00:00:50.630 --> 00:00:52.880
You might be wondering, well,
why can't I just add the 2

00:00:52.880 --> 00:00:55.060
hours per pie and the
3 hours per pie?

00:00:55.060 --> 00:01:05.170
Why can't I just do 2 hours per
pie plus 3 hours per pie?

00:01:05.170 --> 00:01:08.980
Why would this be incorrect?

00:01:08.980 --> 00:01:12.420
Why wouldn't this be their
combined rate?

00:01:12.420 --> 00:01:14.610
I guess the best way to think
about it is think about what's

00:01:14.610 --> 00:01:15.200
happening here.

00:01:15.200 --> 00:01:18.620
You're saying that Sal takes
2 hours a pie, Dean takes 3

00:01:18.620 --> 00:01:19.470
hours a pie.

00:01:19.470 --> 00:01:23.020
If they're both working on the
same pie, it's going to take

00:01:23.020 --> 00:01:23.900
them less time.

00:01:23.900 --> 00:01:25.670
They're going to do it faster.

00:01:25.670 --> 00:01:27.610
But when you add the hours,
what are you doing?

00:01:27.610 --> 00:01:29.410
You're going to get
5 hours per pie.

00:01:32.220 --> 00:01:35.880
This is not answering the
question, how many hours will

00:01:35.880 --> 00:01:38.230
it take for them to
finish one pie?

00:01:38.230 --> 00:01:42.680
This is answering the question,
how much total time

00:01:42.680 --> 00:01:46.040
would be spent if they each
independently ate a pie?

00:01:46.040 --> 00:01:48.330
Sal would spend 2 and then
Dean would spend 3.

00:01:48.330 --> 00:01:50.210
Their total time would be 5.

00:01:50.210 --> 00:01:52.110
So we're just adding
times here.

00:01:52.110 --> 00:01:54.490
If you really want to combine
their effort, you want to

00:01:54.490 --> 00:01:57.600
combine their rates, you have
to express both of these in

00:01:57.600 --> 00:01:59.000
terms of rates.

00:01:59.000 --> 00:02:01.690
And if you want to express 2
hours per pie in terms of

00:02:01.690 --> 00:02:05.220
rate, you would say that this
is the same thing as-- you

00:02:05.220 --> 00:02:07.410
would just write the ratio
the other way around.

00:02:07.410 --> 00:02:13.200
Instead of 2 hours per pie you
would say, 1 pie per 2 hours,

00:02:13.200 --> 00:02:14.660
for 2 hours.

00:02:14.660 --> 00:02:17.810
Or you could view it as
1/2 pie per hours.

00:02:17.810 --> 00:02:20.510
Now it is expressed as a rate.

00:02:20.510 --> 00:02:22.700
Do the same thing for Dean.

00:02:22.700 --> 00:02:25.530
Instead of 3 hours per pie, you
could say that he would do

00:02:25.530 --> 00:02:30.000
1 pie every 3 hours.

00:02:30.000 --> 00:02:32.200
Or another way to think about
it is he does 1/3

00:02:32.200 --> 00:02:34.030
of a pie per hour.

00:02:38.260 --> 00:02:42.080
Now we're speaking in
a language of rates.

00:02:42.080 --> 00:02:44.620
How many pies per hour I can
do, how many Dean can do.

00:02:44.620 --> 00:02:47.670
If we combine it, then we
have our combined rate.

00:02:47.670 --> 00:02:56.640
So if we add 1/2 plus 1/3, that
is our combined rate in

00:02:56.640 --> 00:02:57.560
pies per hour.

00:02:57.560 --> 00:02:58.350
And what is this equal to?

00:02:58.350 --> 00:03:00.540
1/2 is 3/6.

00:03:00.540 --> 00:03:05.460
Plus 1/3 is 2/6, which
is equal to 5/6.

00:03:05.460 --> 00:03:10.720
All the units here are
in pies per hour.

00:03:10.720 --> 00:03:12.580
And then if we wanted to
say, how many hours?

00:03:12.580 --> 00:03:16.590
So we would do literally 5/6 of
a pie per hour when we're

00:03:16.590 --> 00:03:17.980
eating together.

00:03:17.980 --> 00:03:20.210
Now if you want to say how many
hours per pie, you would

00:03:20.210 --> 00:03:21.260
just write this ratio.

00:03:21.260 --> 00:03:22.320
You would just flip it.

00:03:22.320 --> 00:03:29.040
So this is the same thing
as 6/5 hours per pie.

00:03:29.040 --> 00:03:30.290
So hopefully that makes sense.

00:03:30.290 --> 00:03:32.700
I don't want this to seem like
some type of voodoo where I

00:03:32.700 --> 00:03:35.680
just decide arbitrarily to flip
both of these numbers.

00:03:35.680 --> 00:03:38.300
I'm flipping them so I can
express them as rates.

00:03:38.300 --> 00:03:40.020
Which hopefully makes intuitive
sense because you

00:03:40.020 --> 00:03:41.300
can only add rates.

00:03:41.300 --> 00:03:42.850
If you add the times, you're
not saying that they're

00:03:42.850 --> 00:03:44.190
working on the same pie.

00:03:44.190 --> 00:03:47.000
You're saying how much total
time is it taking them.

00:03:47.000 --> 00:03:49.240
Only when you're flipping them
and combining you're saying,

00:03:49.240 --> 00:03:51.990
hey, their rates are being
combined on the same effort.

00:03:51.990 --> 00:03:53.280
I can combine them.

00:03:53.280 --> 00:03:55.450
And that'll tell you their
combined rate.

00:03:55.450 --> 00:03:57.900
And then you can take the flip
of that and say, on a combined

00:03:57.900 --> 00:03:59.530
basis, how many hours?

00:03:59.530 --> 00:04:00.990
And it should make
intuitive sense.

00:04:00.990 --> 00:04:04.150
If we're both eating the same
pie, it's going to take fewer

00:04:04.150 --> 00:04:05.530
hours than either of us.

00:04:05.530 --> 00:04:09.640
And 6/5 is 1 and 1/5
hours or 1.2 hours.

00:04:09.640 --> 00:04:13.580
Or 1 hour and 12 minutes,
which makes sense.

00:04:13.580 --> 00:04:15.740
It's a little bit more than
if you had two Sals.

00:04:15.740 --> 00:04:17.930
Two Sals would take 1 hour.

00:04:17.930 --> 00:04:21.000
And two Deans would take
an hour and a half.

00:04:21.000 --> 00:04:23.700
So it's someplace in between
those two numbers.

00:04:23.700 --> 00:04:26.180
Which hopefully, make sense.


Applying rational equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.810
Two different hoses are being
used to fill a fish pond.

00:00:03.810 --> 00:00:06.220
Used together, the
two hoses take 12

00:00:06.220 --> 00:00:08.800
minutes to fill the pond.

00:00:08.800 --> 00:00:12.510
If used alone, one hose is
able to fill the pond 10

00:00:12.510 --> 00:00:14.090
minutes faster than the other.

00:00:14.090 --> 00:00:17.930
So 10 minutes faster than
the other hose.

00:00:17.930 --> 00:00:22.740
How long does each hose take
to fill the pond by itself?

00:00:22.740 --> 00:00:24.260
So let's think about
each of the hoses.

00:00:24.260 --> 00:00:30.480
We have a faster hose, and then
we have a slower hose.

00:00:30.480 --> 00:00:34.220
And let's say that the faster
hose fills the pond.

00:00:34.220 --> 00:00:40.760
Let's say it takes him
f minutes per pond.

00:00:43.840 --> 00:00:46.100
Now how long is it going to
take the slower hose?

00:00:46.100 --> 00:00:49.900
Well, the faster hose does
it in 10 minutes less.

00:00:49.900 --> 00:00:51.590
It's 10 minutes faster.

00:00:51.590 --> 00:00:54.400
So the slower hose is going
to take 10 minutes more.

00:00:54.400 --> 00:01:02.570
So the slower hose is going to
take f plus 10 minutes per

00:01:02.570 --> 00:01:05.019
fish pond or per pond.

00:01:05.019 --> 00:01:07.630
Now this is in minutes per pond,
but if we want to be

00:01:07.630 --> 00:01:09.880
able to add rates together, we
should really think about in

00:01:09.880 --> 00:01:11.580
terms of ponds per minute.

00:01:11.580 --> 00:01:13.280
So let's rewrite each
of these statements

00:01:13.280 --> 00:01:14.050
as ponds per minute.

00:01:14.050 --> 00:01:18.900
You could write this as f
minutes per 1 pond or f plus

00:01:18.900 --> 00:01:20.530
10 minutes per 1 pond.

00:01:20.530 --> 00:01:22.710
And if you just take the inverse
of each of these

00:01:22.710 --> 00:01:30.250
statements, these ratios are
equivalent to saying 1 pond

00:01:30.250 --> 00:01:32.000
per f minutes.

00:01:32.000 --> 00:01:33.820
So it's really not saying
anything else.

00:01:33.820 --> 00:01:35.020
I'm just inverting the ratio.

00:01:35.020 --> 00:01:38.905
Or you could think of it as
1/f ponds per minute.

00:01:41.590 --> 00:01:43.550
Same logic right here.

00:01:43.550 --> 00:01:52.260
We could essentially rewrite
this ratio as 1 over f plus 10

00:01:52.260 --> 00:01:53.910
ponds per minute.

00:01:53.910 --> 00:01:57.400
So now we have the rate
of the faster hose.

00:01:57.400 --> 00:01:59.840
We have the rate of
the slower hose.

00:01:59.840 --> 00:02:01.710
How many pounds per minute
for the faster hose?

00:02:01.710 --> 00:02:03.820
How many pounds per minute
for the slower hose?

00:02:03.820 --> 00:02:06.870
If we add these two rates,
we'll know the ponds per

00:02:06.870 --> 00:02:09.240
minute when they're
acting together.

00:02:09.240 --> 00:02:15.870
So if we have 1/f ponds per
minute plus 1 over f plus 10

00:02:15.870 --> 00:02:16.670
ponds per minute.

00:02:16.670 --> 00:02:18.690
This is the faster hose, this
is the slower hose.

00:02:18.690 --> 00:02:20.600
This'll tell us how
many ponds for

00:02:20.600 --> 00:02:22.590
minute they can do together.

00:02:22.590 --> 00:02:24.540
Now, we know that information.

00:02:24.540 --> 00:02:31.450
They say the two hoses
take 12 minutes.

00:02:31.450 --> 00:02:32.630
So let me write that
over here.

00:02:32.630 --> 00:02:34.440
So combined.

00:02:34.440 --> 00:02:42.100
The combined take 12
minutes per pond.

00:02:42.100 --> 00:02:45.370
So what is their combined rate
in terms of ponds per minute?

00:02:45.370 --> 00:02:47.660
So you could view this as
12 minutes per one pond.

00:02:47.660 --> 00:02:50.500
You could take the inverse of
this or take the ratio in

00:02:50.500 --> 00:02:52.170
terms of ponds per minute
instead and

00:02:52.170 --> 00:02:55.380
you get 1/12 minutes.

00:02:55.380 --> 00:03:01.140
Sorry, 1/12 ponds per minute.

00:03:01.140 --> 00:03:05.790
In one minute combined, they'll
fill 1/12 of a pond.

00:03:05.790 --> 00:03:06.880
Which makes complete sense.

00:03:06.880 --> 00:03:08.190
Because it takes them
12 minutes to

00:03:08.190 --> 00:03:08.920
fill the whole thing.

00:03:08.920 --> 00:03:11.050
So in one minute they'll
only do 1/12 of it.

00:03:11.050 --> 00:03:14.600
So this is their combined rate
in ponds per minute.

00:03:14.600 --> 00:03:18.860
This is also their combined
rate in ponds per minute.

00:03:18.860 --> 00:03:22.610
So this is going to
be equal to 1/12.

00:03:22.610 --> 00:03:25.430
And now we just have to solve
for f and then f plus 10 is

00:03:25.430 --> 00:03:29.770
going to be how long it
takes the slower hose.

00:03:29.770 --> 00:03:31.020
So let's multiply.

00:03:31.020 --> 00:03:31.850
Let's see what we could do.

00:03:31.850 --> 00:03:35.320
We could multiply both sides of
this equation times f and

00:03:35.320 --> 00:03:37.050
times f plus 10.

00:03:37.050 --> 00:03:38.290
So let's do that.

00:03:38.290 --> 00:03:41.630
So I'm going to multiply both
sides of this equation times f

00:03:41.630 --> 00:03:46.480
and f plus 10 times both
sides of this equation.

00:03:46.480 --> 00:03:50.720
So f and f plus 10.

00:03:50.720 --> 00:03:51.990
Scroll down a little bit.

00:03:51.990 --> 00:03:55.420
Scroll to the left, so we
have some real estate.

00:03:55.420 --> 00:03:58.210
So let's distribute this
f times f plus 10.

00:03:58.210 --> 00:04:02.190
So if we multiply f times f plus
10 times 1/f, that f and

00:04:02.190 --> 00:04:04.690
that f will cancel out and we're
just going to be left

00:04:04.690 --> 00:04:06.700
with an f plus 10.

00:04:06.700 --> 00:04:09.190
That's when you multiply
that term times the f

00:04:09.190 --> 00:04:10.750
times f plus 10.

00:04:10.750 --> 00:04:14.735
Now, when you multiply this
term, when you multiply 1 over

00:04:14.735 --> 00:04:19.310
f plus 10 times f times f plus
10, this and this will cancel

00:04:19.310 --> 00:04:20.310
out and you're just
left with an f.

00:04:20.310 --> 00:04:25.650
So you have plus f is equal
to-- and you have over 12.

00:04:25.650 --> 00:04:27.880
Actually, well in a second, let
me multiply all sides of

00:04:27.880 --> 00:04:28.890
this equation by 12.

00:04:28.890 --> 00:04:30.230
I'll do that next in a second.

00:04:30.230 --> 00:04:34.650
So let's just say this is going
to be equal to 1/12

00:04:34.650 --> 00:04:36.930
times f squared.

00:04:39.640 --> 00:04:40.880
f times f is f squared.

00:04:40.880 --> 00:04:43.640
Plus 10f.

00:04:43.640 --> 00:04:45.230
And now let's just multiply
both sides of

00:04:45.230 --> 00:04:46.280
this equation by 12.

00:04:46.280 --> 00:04:50.700
I could've done it in the last
step so that we don't have any

00:04:50.700 --> 00:04:51.950
fractions here.

00:04:54.740 --> 00:04:57.750
And so the left-hand side
we get 12 times f.

00:04:57.750 --> 00:05:02.770
We get 12f plus 120 plus 12f.

00:05:02.770 --> 00:05:06.070
The right-hand side, that and
that cancels out and you are

00:05:06.070 --> 00:05:09.840
left with f squared plus 10f.

00:05:09.840 --> 00:05:11.060
And now we have a quadratic.

00:05:11.060 --> 00:05:13.390
We just have to get it into
a form that we know how to

00:05:13.390 --> 00:05:15.230
manipulate or deal with.

00:05:15.230 --> 00:05:16.860
Before that, we can
simplify it.

00:05:16.860 --> 00:05:20.070
We have a 12f and a 12f.

00:05:20.070 --> 00:05:26.400
So this becomes 24f plus
120 is equal to f

00:05:26.400 --> 00:05:29.420
squared plus 10f.

00:05:29.420 --> 00:05:32.120
And then let's get all
of this stuff out of

00:05:32.120 --> 00:05:32.900
the left-hand side.

00:05:32.900 --> 00:05:34.540
They'll get all on the
right-hand side.

00:05:34.540 --> 00:05:38.940
So from both sides of this
equation, let's subtract 24f

00:05:38.940 --> 00:05:41.270
and a negative 120
or minus 120.

00:05:41.270 --> 00:05:45.965
So you have minus
24f minus 120.

00:05:45.965 --> 00:05:48.020
The left-hand side
just becomes 0.

00:05:48.020 --> 00:05:49.530
That was the whole point.

00:05:49.530 --> 00:05:52.470
Right-hand side is f squared.

00:05:52.470 --> 00:05:58.380
10 minus 24f is negative 14f.

00:05:58.380 --> 00:06:02.550
Minus 120.

00:06:02.550 --> 00:06:04.170
Now we could factor this.

00:06:04.170 --> 00:06:06.850
Let's see, if you do 20 times--
yeah, that looks like

00:06:06.850 --> 00:06:07.330
it would work.

00:06:07.330 --> 00:06:11.930
So negative 20 and 6, when you
take their product, give you

00:06:11.930 --> 00:06:13.420
negative 120.

00:06:13.420 --> 00:06:17.460
And negative 20 plus
6 is negative 14.

00:06:17.460 --> 00:06:20.240
So we could factor this
right-hand side as 0 is equal

00:06:20.240 --> 00:06:26.810
to f minus 20 times f plus 6.

00:06:26.810 --> 00:06:30.170
When you multiply negative 20
times 6, you get negative 120.

00:06:30.170 --> 00:06:33.180
Negative 20 plus 6
is negative 14.

00:06:33.180 --> 00:06:36.830
And the only way that that's
going to be equal to 0 is if f

00:06:36.830 --> 00:06:45.140
minus 20 is equal to 0 or
f plus 6 is equal to 0.

00:06:45.140 --> 00:06:47.470
Add 20 to both sides
of this equation.

00:06:47.470 --> 00:06:48.680
You get f is equal to 20.

00:06:48.680 --> 00:06:50.720
Remember, f is how many minutes
does it take for the

00:06:50.720 --> 00:06:52.650
fast hose to fill the pond.

00:06:52.650 --> 00:06:54.130
And then, if you take
this one, you

00:06:54.130 --> 00:06:55.410
subtract 6 from both sides.

00:06:55.410 --> 00:06:57.670
You get f is equal
to negative 6.

00:06:57.670 --> 00:06:59.730
Now, when we're talking about
how many minutes does it take

00:06:59.730 --> 00:07:02.460
for the fast hose to fill the
pond, it doesn't make any

00:07:02.460 --> 00:07:04.650
sense to say that it
takes it negative 6

00:07:04.650 --> 00:07:06.040
minutes to fill the pond.

00:07:06.040 --> 00:07:07.790
So we can't use this answer.

00:07:07.790 --> 00:07:09.270
We need a positive answer.

00:07:09.270 --> 00:07:11.810
So this is how many minutes it
takes the fast hose to fill

00:07:11.810 --> 00:07:14.780
the pond. f is equal to 20.

00:07:14.780 --> 00:07:17.530
So this right here, the
faster hose takes

00:07:17.530 --> 00:07:22.000
20 minutes per pond.

00:07:22.000 --> 00:07:23.080
I'll write it here.

00:07:23.080 --> 00:07:27.610
20 minutes per pond
is the fast hose.

00:07:27.610 --> 00:07:29.950
And then the slower hose, it
takes 10 minutes more.

00:07:29.950 --> 00:07:31.860
It's f plus 10.

00:07:31.860 --> 00:07:36.160
So it takes 30 minutes
per pond.

00:07:36.160 --> 00:07:36.650
And we're done.

00:07:36.650 --> 00:07:37.950
I don't want to confuse
you with this stuff.

00:07:37.950 --> 00:07:39.570
The faster hose takes
20 minutes.

00:07:39.570 --> 00:07:42.010
Slower hose takes 30
minutes per pond.

00:07:42.010 --> 00:07:45.150
If they were to do it together,
it would take 12

00:07:45.150 --> 00:07:47.410
minutes, which is a little
bit more than half.

00:07:47.410 --> 00:07:49.930
If you had two faster hoses,
it would take 10 minutes.

00:07:49.930 --> 00:07:51.780
But this guy's a little bit
slower, so it's taking you a

00:07:51.780 --> 00:07:53.610
little bit more than
10 minutes.

00:07:53.610 --> 00:07:54.760
So it makes sense.

00:07:54.760 --> 00:07:57.940
It takes you 12 minutes when
they're working together.


Applying rational equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.670
Working together Anya and
Bill stained a large

00:00:03.670 --> 00:00:05.930
porch deck in 8 hours.

00:00:05.930 --> 00:00:08.920
Last year Anya stained
the deck by herself.

00:00:08.920 --> 00:00:12.550
The year before Bill painted it
by himself, but took twice

00:00:12.550 --> 00:00:14.350
as long as Anya did.

00:00:14.350 --> 00:00:18.770
How long did Anya and Bill take
when each was painting

00:00:18.770 --> 00:00:19.960
the deck alone?

00:00:19.960 --> 00:00:22.060
So let's define some
variables here.

00:00:22.060 --> 00:00:24.440
Let's define A.

00:00:24.440 --> 00:00:32.619
Let's define A as number
of hours for

00:00:32.619 --> 00:00:36.310
Anya to paint a deck.

00:00:39.940 --> 00:00:45.770
And so we could say that
Anya paints, or has A

00:00:45.770 --> 00:00:49.800
hours for one deck.

00:00:49.800 --> 00:00:53.850
Or we can invert this and
say that she can do

00:00:53.850 --> 00:00:59.910
1/A decks per hour.

00:00:59.910 --> 00:01:03.420
Now, let's do another variable
for Bill just like that.

00:01:03.420 --> 00:01:10.470
Let's define B as the
number of hours for

00:01:10.470 --> 00:01:13.730
Bill to paint a deck.

00:01:17.860 --> 00:01:24.710
And so for Bill, he can paint--
it takes him B hours

00:01:24.710 --> 00:01:26.530
per deck, per one deck.

00:01:26.530 --> 00:01:28.440
We could put it over
1 there if we want.

00:01:28.440 --> 00:01:29.710
We don't have to.

00:01:29.710 --> 00:01:36.650
And this is the same thing as
saying that he can do 1 deck

00:01:36.650 --> 00:01:40.220
per every B hours.

00:01:40.220 --> 00:01:41.890
Or another way to think
of it, is he can do

00:01:41.890 --> 00:01:45.280
1/B deck per hour.

00:01:45.280 --> 00:01:48.970
Now, when they work together,
Anya and Bill stained a large

00:01:48.970 --> 00:01:52.280
porch deck in 8 hours.

00:01:52.280 --> 00:01:53.320
So let's write this down.

00:01:53.320 --> 00:01:58.850
Anya plus Bill-- I'll
do it in orange.

00:01:58.850 --> 00:02:03.490
Anya and Bill.

00:02:03.490 --> 00:02:04.640
This information right here.

00:02:04.640 --> 00:02:09.759
Anya and Bill stained a large
porch deck in 8 hours.

00:02:09.759 --> 00:02:15.410
So we could say 8
hours per deck.

00:02:15.410 --> 00:02:20.510
Or 8 hours per 1 deck, which is
the same thing as saying 1

00:02:20.510 --> 00:02:26.150
deck per 8 hours.

00:02:26.150 --> 00:02:28.640
And this is going to be
the combination of

00:02:28.640 --> 00:02:30.000
each of their rates.

00:02:30.000 --> 00:02:34.020
So this 1 deck of per 8 hours,
this is going to be equal to

00:02:34.020 --> 00:02:44.290
Anya's rate, 1/A decks per
hour, plus Bill's rate.

00:02:44.290 --> 00:02:47.150
Plus-- do that same color.

00:02:47.150 --> 00:02:50.880
Plus 1/B decks per hour.

00:02:53.960 --> 00:02:55.540
So we have one equation
set up.

00:02:55.540 --> 00:02:56.870
Let me scroll down
a little bit.

00:02:56.870 --> 00:02:58.360
And I won't write the
units any more.

00:02:58.360 --> 00:03:05.050
We have 1/8 is equal
to 1/A plus 1/B.

00:03:07.970 --> 00:03:10.910
Now, we have two unknowns, so
we need another equation if

00:03:10.910 --> 00:03:12.820
we're going to solve these.

00:03:12.820 --> 00:03:13.200
Let's see.

00:03:13.200 --> 00:03:16.450
It tells us, it tell this right
here the year before

00:03:16.450 --> 00:03:19.120
Bill painted it by himself,
but took twice as

00:03:19.120 --> 00:03:21.350
long as Anya did.

00:03:21.350 --> 00:03:24.160
So the number of hours it takes
Bill to paint the deck

00:03:24.160 --> 00:03:27.720
is twice as long as the number
of hours it takes Anya to

00:03:27.720 --> 00:03:28.560
paint a deck.

00:03:28.560 --> 00:03:31.360
So B is equal to 2A.

00:03:31.360 --> 00:03:34.260
The number of hours Bill takes
is twice as the number of

00:03:34.260 --> 00:03:37.010
hours Anya takes per deck.

00:03:37.010 --> 00:03:39.340
So B is equal to 2A.

00:03:39.340 --> 00:03:43.250
So we can rewrite this equation
as-- I'll stick to

00:03:43.250 --> 00:03:44.140
the colors for now.

00:03:44.140 --> 00:03:49.660
We could say 1/8 is equal
to 1 over Anya.

00:03:49.660 --> 00:03:53.650
Instead of writing 1 over Bill
we would write, so plus 1 over

00:03:53.650 --> 00:03:55.460
Bill is 2 times Anya.

00:03:55.460 --> 00:03:58.210
The number of hours Bill takes
is two times the number of

00:03:58.210 --> 00:03:59.970
hours Anya takes.

00:03:59.970 --> 00:04:01.290
So 2 times Anya.

00:04:01.290 --> 00:04:04.050
And now we have one equation
and one unknown.

00:04:04.050 --> 00:04:05.170
And we can solve for A.

00:04:05.170 --> 00:04:08.150
And the easiest way to solve for
A right here is if we just

00:04:08.150 --> 00:04:11.040
multiply both sides of
the equation by 2A.

00:04:11.040 --> 00:04:14.890
So let's multiply both
sides of this by 2A.

00:04:14.890 --> 00:04:16.920
Multiply the left side by 2A.

00:04:16.920 --> 00:04:19.380
Actually, let's multiply both
sides by 8A, so we get rid of

00:04:19.380 --> 00:04:20.630
this 1/8 as well.

00:04:23.680 --> 00:04:27.610
And then multiply the right-hand
side by 8A as well.

00:04:27.610 --> 00:04:32.050
The left-hand side, 8A divided
by 8 is just A.

00:04:32.050 --> 00:04:37.060
A is equal to-- 1/A times 8A
is just going to be 8.

00:04:37.060 --> 00:04:39.340
1/2A times 8A.

00:04:39.340 --> 00:04:41.930
8A divided by 2A is 4.

00:04:45.100 --> 00:04:48.930
So A, the number of hours it
takes Anya to paint a deck--

00:04:48.930 --> 00:04:52.290
and I made it lowercase, which
I shouldn't have. Well, these

00:04:52.290 --> 00:04:53.650
should all be capital A's.

00:04:56.460 --> 00:05:01.270
This is A is equal to
8A over A is 8.

00:05:01.270 --> 00:05:03.810
8A over 2A is 4.

00:05:03.810 --> 00:05:06.390
So the number of hours it takes
Anya to paint a deck, or

00:05:06.390 --> 00:05:09.050
A, is 12 hours.

00:05:09.050 --> 00:05:11.300
Now what are they asking
over here?

00:05:11.300 --> 00:05:14.940
They're asking, how long did
Anya and Bill take when each

00:05:14.940 --> 00:05:15.810
was painting alone?

00:05:15.810 --> 00:05:16.650
So we figured out Anya.

00:05:16.650 --> 00:05:18.100
It takes her 8 hours.

00:05:18.100 --> 00:05:21.360
And then we know that it takes
Bill twice as long as Anya.

00:05:21.360 --> 00:05:23.170
Bill is two times A.

00:05:23.170 --> 00:05:25.120
So bill is going to
be 2 times 12.

00:05:25.120 --> 00:05:30.020
So Bill is equal to 2 times
Anya, which is equal to 2

00:05:30.020 --> 00:05:31.350
times 12 hours.

00:05:31.350 --> 00:05:34.190
Which is equal to 24 hours.

00:05:34.190 --> 00:05:37.430
So when they're alone it would
take Anya 12 hours, it would

00:05:37.430 --> 00:05:39.720
take Bill 24 hours.

00:05:39.720 --> 00:05:42.580
When they do it combined,
it takes 8 hours.

00:05:42.580 --> 00:05:43.710
Which makes sense.

00:05:43.710 --> 00:05:47.200
Because if Bill took 12 hours
by himself, combined they

00:05:47.200 --> 00:05:48.940
would take 6 hours, they would
take half as long.

00:05:48.940 --> 00:05:50.690
But Bill isn't that efficient.

00:05:50.690 --> 00:05:52.840
He's not as efficient as
Anya, so it takes them

00:05:52.840 --> 00:05:53.590
a little bit longer.

00:05:53.590 --> 00:05:54.560
Takes them 8 hours.

00:05:54.560 --> 00:05:56.590
But it makes sense that combined
they're going to take

00:05:56.590 --> 00:05:58.920
less time than individually.


Solving rational equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:06.250
Solve the equation: x squared
minus x squared minus 4 over x

00:00:06.250 --> 00:00:07.570
minus 2 is equal to 4.

00:00:07.570 --> 00:00:10.920
And they tell us that x won't
or cannot be 2 because if it

00:00:10.920 --> 00:00:12.670
was 2, then this would
be undefined.

00:00:12.670 --> 00:00:14.460
It would be dividing by 0.

00:00:14.460 --> 00:00:15.535
So let's see what x is.

00:00:15.535 --> 00:00:18.860
Let's see the x that actually
satisfies this equation.

00:00:18.860 --> 00:00:21.530
So you might be tempted to try
to express this with the

00:00:21.530 --> 00:00:24.490
common denominator of x minus
2 and then add these two

00:00:24.490 --> 00:00:25.580
expressions.

00:00:25.580 --> 00:00:28.550
But the thing that jumps out
at me initially is that we

00:00:28.550 --> 00:00:31.230
have x squared minus 4 on the
numerator, which is a

00:00:31.230 --> 00:00:32.439
difference of squares.

00:00:32.439 --> 00:00:36.290
Or if you factor it out, it's
x plus 2 times x minus 2.

00:00:36.290 --> 00:00:38.460
So we should be able to factor
this x minus 2 out.

00:00:38.460 --> 00:00:39.460
So let's do that.

00:00:39.460 --> 00:00:42.580
So if we were to rewrite it,
this is equivalent to x

00:00:42.580 --> 00:00:45.940
squared minus-- instead of
writing x squared minus 4, we

00:00:45.940 --> 00:00:47.230
know that's a difference
of squares.

00:00:47.230 --> 00:00:52.300
That is x plus 2 times
x minus 2.

00:00:52.300 --> 00:00:55.630
All of that over x minus 2.

00:00:55.630 --> 00:00:57.160
And that is equal to 4.

00:00:57.160 --> 00:00:58.533
And this whole time we're
assuming that x

00:00:58.533 --> 00:00:59.760
won't be equal to 2.

00:00:59.760 --> 00:01:02.990
And because x does not equal to
2, x minus 2 divided by x

00:01:02.990 --> 00:01:05.600
minus 2 is going
to be defined.

00:01:05.600 --> 00:01:06.890
And it will be 1.

00:01:06.890 --> 00:01:08.720
So those two will cancel out.

00:01:08.720 --> 00:01:13.870
And so we're left with x squared
minus x plus 2 is

00:01:13.870 --> 00:01:15.020
equal to 4.

00:01:15.020 --> 00:01:16.555
We can distribute the
negative sign.

00:01:16.555 --> 00:01:18.450
And I'll just arbitrarily
switch colors here.

00:01:18.450 --> 00:01:21.880
We can distribute the negative
sign, so we get x squared

00:01:21.880 --> 00:01:26.550
minus x minus 2 is equal to 4.

00:01:26.550 --> 00:01:29.350
And what we want to do is put
this in the form ax squared

00:01:29.350 --> 00:01:31.190
plus bx plus c is equal to 0.

00:01:31.190 --> 00:01:35.190
That allows us to either
factor it or apply the

00:01:35.190 --> 00:01:37.100
quadratic equation or complete
the square, or any of the ways

00:01:37.100 --> 00:01:38.260
that we know how to
solve quadratics.

00:01:38.260 --> 00:01:39.000
So let's do that.

00:01:39.000 --> 00:01:40.970
Let's get a 0 on the
right-hand side.

00:01:40.970 --> 00:01:43.720
The best way to do that is to
subtract 4 from both sides of

00:01:43.720 --> 00:01:44.830
this equation.

00:01:44.830 --> 00:01:49.820
Subtract 4 and we are left
with x squared minus x.

00:01:49.820 --> 00:01:52.330
Negative 2 minus 4
is negative 6.

00:01:52.330 --> 00:01:54.460
And then 4 minus 4 is 0.

00:01:54.460 --> 00:01:55.270
That was the whole point.

00:01:55.270 --> 00:01:58.170
So we have x squared minus
x minus 6 is equal to 0.

00:01:58.170 --> 00:01:59.050
Let me write it up here.

00:01:59.050 --> 00:02:03.330
x squared minus x minus
6 is equal to 0.

00:02:03.330 --> 00:02:05.730
And this looks factorable.

00:02:05.730 --> 00:02:07.500
We just have to think of two
numbers that when we multiply

00:02:07.500 --> 00:02:09.669
them give us negative 6.

00:02:09.669 --> 00:02:10.660
So they're going to have
different signs.

00:02:10.660 --> 00:02:13.570
When I add them I'll
get negative 1.

00:02:13.570 --> 00:02:16.420
So it looks like negative
3 and positive 2 work.

00:02:16.420 --> 00:02:21.430
So if we do x minus
3 times x plus 2.

00:02:21.430 --> 00:02:23.200
It's a little bit of trial and
error, but 6 doesn't have that

00:02:23.200 --> 00:02:25.630
many factors to deal with and
3 and 2 are one apart.

00:02:25.630 --> 00:02:27.780
They have different signs, so
that's how you can think of

00:02:27.780 --> 00:02:29.700
how we get to that conclusion.

00:02:29.700 --> 00:02:31.360
Negative 3 times 2
is negative 6.

00:02:31.360 --> 00:02:33.630
Negative 3 plus 2
is negative 1.

00:02:33.630 --> 00:02:35.760
So that is equal to 0.

00:02:35.760 --> 00:02:39.620
So we have two possible
ways to get 0.

00:02:39.620 --> 00:02:47.730
Either x minus 3 is equal to 0
or x plus 2 is equal to 0.

00:02:47.730 --> 00:02:50.610
And then if we take x minus 3 is
equal to 0, if we add 3 to

00:02:50.610 --> 00:02:55.290
both sides of that equation, we
get x is equal to 3 or, if

00:02:55.290 --> 00:02:59.190
we subtract 2 from both sides of
this equation, we get x is

00:02:59.190 --> 00:03:00.640
equal to negative 2.

00:03:00.640 --> 00:03:02.910
So both of these
are solutions.

00:03:02.910 --> 00:03:05.320
And let's apply them into
this equation to make

00:03:05.320 --> 00:03:07.400
sure that they work.

00:03:07.400 --> 00:03:10.200
Because these are solutions to
essentially, the situation

00:03:10.200 --> 00:03:11.940
where we got rid of
the x minus 2.

00:03:11.940 --> 00:03:13.540
Maybe it had some type
of side effects.

00:03:13.540 --> 00:03:16.140
So let's just make sure that
both of these actually work in

00:03:16.140 --> 00:03:17.110
the original equation.

00:03:17.110 --> 00:03:19.400
So let's try x is equal
to 3 first.

00:03:19.400 --> 00:03:24.810
So you get 3 squared minus
3 squared over 4.

00:03:24.810 --> 00:03:28.130
Sorry, 3 squared minus
4 over 3 minus 2.

00:03:28.130 --> 00:03:33.730
So this is equal to 9 minus--
3 squared is 9 --minus 4,

00:03:33.730 --> 00:03:35.160
which is 5, over 1.

00:03:35.160 --> 00:03:37.720
So 9 minus 5, which
is equal to 4.

00:03:37.720 --> 00:03:39.805
Which is exactly what we
needed it to equal.

00:03:39.805 --> 00:03:42.440
And let's try it with
negative 2.

00:03:42.440 --> 00:03:50.000
So if I take negative 2 squared
and I have a minus

00:03:50.000 --> 00:03:54.290
negative 2 squared minus
4, all of that over

00:03:54.290 --> 00:03:56.210
negative 2 minus 2.

00:03:56.210 --> 00:03:58.720
So negative 2 squared is 4.

00:03:58.720 --> 00:04:02.680
Minus negative 2 squared,
which is 4.

00:04:02.680 --> 00:04:04.320
Minus 4.

00:04:04.320 --> 00:04:06.230
All of that over negative
2 minus negative 2

00:04:06.230 --> 00:04:07.310
over negative 4.

00:04:07.310 --> 00:04:09.480
Well 4 minus 4 is 0, so this
whole thing is just

00:04:09.480 --> 00:04:10.740
going to become 0.

00:04:10.740 --> 00:04:12.260
So this whole thing is
going to equal 4.

00:04:12.260 --> 00:04:14.710
So both of these
solutions work.

00:04:14.710 --> 00:04:16.940
And that makes sense, because
when we actually canceled this

00:04:16.940 --> 00:04:20.060
out, we actually didn't
fundamentally change anything

00:04:20.060 --> 00:04:20.779
about the equation.

00:04:20.779 --> 00:04:23.550
Only if you had the situation
where the x would have been

00:04:23.550 --> 00:04:24.000
equal to 2.

00:04:24.000 --> 00:04:26.070
That's the only thing that
you're really changing.

00:04:26.070 --> 00:04:28.140
So that's why it makes
sense that both of

00:04:28.140 --> 00:04:30.910
these solutions work.


Solving rational equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:05.510
Solve the equation, 5 over
2x minus 4 over 3x is

00:00:05.510 --> 00:00:06.880
equal to 7 over 18.

00:00:06.880 --> 00:00:09.720
And they tell us that x can't to
be equal to 0, because that

00:00:09.720 --> 00:00:13.000
would make these two expressions
here undefined.

00:00:13.000 --> 00:00:15.265
Hopefully the answer here is not
0, and then this becomes--

00:00:15.265 --> 00:00:17.970
this is kind of extra,
unnecessary information.

00:00:17.970 --> 00:00:19.990
So let's figure out
how to solve this.

00:00:19.990 --> 00:00:22.970
So a good place to start-- I
don't like having x's in my

00:00:22.970 --> 00:00:23.950
denominators.

00:00:23.950 --> 00:00:26.450
So let's multiply-- and in fact,
in general, I don't like

00:00:26.450 --> 00:00:27.785
having fractions in
my equations.

00:00:27.785 --> 00:00:30.330
So let's see if we can multiply
both sides of this

00:00:30.330 --> 00:00:32.920
equation by some things that
will get rid of the fractions.

00:00:32.920 --> 00:00:34.450
So let me just rewrite it
so we have some space.

00:00:34.450 --> 00:00:41.200
5 over 2x, and then we have
minus 4 over 3x is

00:00:41.200 --> 00:00:46.580
equal to 7 over 18.

00:00:46.580 --> 00:00:51.990
Now, if we want to get rid of
the 2 in the denominator here,

00:00:51.990 --> 00:00:54.450
we could multiply
everything by 2.

00:00:54.450 --> 00:00:56.900
If we want to get rid of this 3
in the denominator, we could

00:00:56.900 --> 00:00:59.380
multiply everything by 3.

00:00:59.380 --> 00:01:01.570
If we want to get rid of this
18 in the denominator, we

00:01:01.570 --> 00:01:03.980
could multiply everything
by 18.

00:01:03.980 --> 00:01:06.960
And 18 also includes
a 2 and a 3.

00:01:06.960 --> 00:01:09.850
The prime factorization
of 18 is 2 times 9,

00:01:09.850 --> 00:01:12.200
which is 3 times 3.

00:01:12.200 --> 00:01:14.360
So when you're multiplying both
sides of the equation by

00:01:14.360 --> 00:01:17.740
18, you're actually multiplying
it by a 2 and a 3

00:01:17.740 --> 00:01:19.230
and another 3.

00:01:19.230 --> 00:01:24.410
So let's just multiply both
sides of this equation by 18.

00:01:24.410 --> 00:01:28.150
So I'll multiply this term
right here by 18.

00:01:28.150 --> 00:01:30.600
And then, this term
right here by 18.

00:01:30.600 --> 00:01:32.850
That'll get rid of all of
these numbers in the

00:01:32.850 --> 00:01:33.960
denominator.

00:01:33.960 --> 00:01:35.700
18 divided by 3 is 6.

00:01:35.700 --> 00:01:37.410
18 divided by 2 is 9.

00:01:37.410 --> 00:01:39.360
But we don't just have numbers
in the denominator, we also

00:01:39.360 --> 00:01:41.020
have these x's in
the denominator.

00:01:41.020 --> 00:01:44.120
So let's also multiply both
sides of the equation by x, so

00:01:44.120 --> 00:01:45.510
that we get rid of these.

00:01:45.510 --> 00:01:47.400
So we're essentially going to
multiply both sides of the

00:01:47.400 --> 00:01:51.390
equation by 18x.

00:01:51.390 --> 00:01:53.870
We're taking, essentially,
the least common multiple

00:01:53.870 --> 00:01:56.430
of 2x, 3x and 18.

00:01:56.430 --> 00:01:59.490
This is the smallest number that
is divisible by all three

00:01:59.490 --> 00:02:00.610
of these characters.

00:02:00.610 --> 00:02:04.012
Now, when we do that our
denominators will disappear.

00:02:04.012 --> 00:02:06.350
x divided by x is 1.

00:02:06.350 --> 00:02:08.490
18 divided by 2 is 9.

00:02:08.490 --> 00:02:13.390
So this term becomes 9
times 5, which is 45.

00:02:13.390 --> 00:02:17.690
And then this term right here,
x divided by x is 1.

00:02:17.690 --> 00:02:20.660
18 divided by 3 is 6.

00:02:20.660 --> 00:02:23.890
So you have 6 times 4 is 24,
but you have a subtraction

00:02:23.890 --> 00:02:29.770
sign here so minus 24 is equal
to-- let me do that in that

00:02:29.770 --> 00:02:33.470
yellow-- is equal to-- and then
you have this term, 7

00:02:33.470 --> 00:02:35.310
over 18 times 18x.

00:02:35.310 --> 00:02:38.010
Well, the 18's cancel out and
you're just left with 7 times

00:02:38.010 --> 00:02:40.810
x is equal to 7x.

00:02:40.810 --> 00:02:44.150
And now this becomes a much,
much simpler equation.

00:02:44.150 --> 00:02:46.980
Now, what's 45 minus 24?

00:02:46.980 --> 00:02:52.430
Let's see, 45 minus 20
would be would be 25.

00:02:52.430 --> 00:02:54.435
Then you subtract
4 more, it's 21.

00:02:54.435 --> 00:02:58.580
So you get 21 is equal to 7x.

00:02:58.580 --> 00:03:04.786
Divide both sides by 7 and
you get x is equal to 3.

00:03:07.930 --> 00:03:10.070
And let's verify that
that works.

00:03:10.070 --> 00:03:12.410
So we have 5 over 2x.

00:03:12.410 --> 00:03:17.190
So that's the same thing as
5 over 2 times 3, minus 4

00:03:17.190 --> 00:03:19.670
over 3 times 3.

00:03:19.670 --> 00:03:28.620
So this is 5/6 minus 4 over
18-- sorry, 4 over 9.

00:03:28.620 --> 00:03:30.410
We want to find a common
denominator.

00:03:30.410 --> 00:03:33.900
18 is the least common multiple
of 6 and 9, so let's

00:03:33.900 --> 00:03:37.070
put it over 18.

00:03:37.070 --> 00:03:40.510
5/6 is the same thing
as 15 over 18.

00:03:40.510 --> 00:03:42.950
Multiply the numerator
and denominator by 3.

00:03:42.950 --> 00:03:46.130
4/9 is the same thing
as 8 over 18.

00:03:46.130 --> 00:03:49.030
Multiply the numerator
and denominator by 2.

00:03:49.030 --> 00:03:50.230
15 minus 8.

00:03:50.230 --> 00:03:54.260
So this becomes 15 minus
8 over 18, which is

00:03:54.260 --> 00:03:56.820
equal to 7 over 18.

00:03:56.820 --> 00:03:57.690
So it works out.

00:03:57.690 --> 00:04:02.190
5 over 2x when x is equal to 3,
minus 4 over 3x when x is

00:04:02.190 --> 00:04:06.150
equal to 3 is indeed
equal to 7/18.

00:04:06.150 --> 00:04:07.690
So we're done.


Solving rational equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.490
Solve the equation, 5 over
x minus 1 is equal to

00:00:04.490 --> 00:00:06.130
3 over x plus 3.

00:00:06.130 --> 00:00:07.930
And they give us two constraints
here. x can't be

00:00:07.930 --> 00:00:09.410
negative 3 or 1.

00:00:09.410 --> 00:00:11.570
And that's because if x was
negative 3, this expression

00:00:11.570 --> 00:00:14.100
right here, you'd be dividing
by 0, it'd be undefined.

00:00:14.100 --> 00:00:16.420
If x was 1, this expression
right here would be dividing

00:00:16.420 --> 00:00:17.480
by 0 and it would
be undefined.

00:00:17.480 --> 00:00:19.520
So let's see what
we can do here.

00:00:19.520 --> 00:00:21.000
So let me just rewrite
it so we have some

00:00:21.000 --> 00:00:22.010
space to work with.

00:00:22.010 --> 00:00:26.940
So 5 over x minus 1 is equal
to 3 over x plus 3.

00:00:26.940 --> 00:00:28.230
We're going to assume
these constraints

00:00:28.230 --> 00:00:30.370
throughout this video.

00:00:30.370 --> 00:00:34.690
Now, I don't like having my x
terms in the denominator, so

00:00:34.690 --> 00:00:37.330
let's see if we can get them
out of the denominator.

00:00:37.330 --> 00:00:40.670
Well, a good place to start, if
we don't want this x plus 3

00:00:40.670 --> 00:00:43.930
in the denominator right here,
we can multiply both sides of

00:00:43.930 --> 00:00:45.695
this equation by x plus 3.

00:00:49.310 --> 00:00:51.510
Anything you do to one side
you have to do the other.

00:00:51.510 --> 00:00:54.190
Now, by the same argument,
I don't like having

00:00:54.190 --> 00:00:56.250
this x minus 1 here.

00:00:56.250 --> 00:00:59.330
So let's multiply both sides of
the equation by x minus 1.

00:01:05.710 --> 00:01:08.010
Now, when we do this, what's
going to happen?

00:01:08.010 --> 00:01:10.310
Well, the whole point of
multiplying by x minus 1 is so

00:01:10.310 --> 00:01:12.500
that that cancels
out with that.

00:01:12.500 --> 00:01:16.150
And the whole reason behind
multiplying by x plus 3 is so

00:01:16.150 --> 00:01:19.010
that this cancels
out with this.

00:01:19.010 --> 00:01:31.410
So we end up with 5 times
x plus 3 is equal to 3

00:01:31.410 --> 00:01:34.680
times x minus 1.

00:01:37.230 --> 00:01:40.600
And all we did is we multiplied
both sides of the

00:01:40.600 --> 00:01:42.260
equation by both denominators.

00:01:42.260 --> 00:01:45.940
Both sides of the equation by
x plus 3 times x minus 1.

00:01:45.940 --> 00:01:48.140
And this is where the whole
notion of cross multiplying

00:01:48.140 --> 00:01:49.440
comes from.

00:01:49.440 --> 00:01:52.100
When we did that, it looks like
we just took 5 times x

00:01:52.100 --> 00:01:56.460
plus 3 is equal to 3
times x minus 1.

00:01:56.460 --> 00:01:58.690
Which is a legitimate thing to
do, but it just comes from the

00:01:58.690 --> 00:02:02.000
idea of multiplying both sides
by both denominators,

00:02:02.000 --> 00:02:03.520
essentially in one step.

00:02:03.520 --> 00:02:04.420
But now this is a pretty

00:02:04.420 --> 00:02:05.750
straightforward linear equation.

00:02:05.750 --> 00:02:08.300
We can just distribute the
numbers and get the x's on one

00:02:08.300 --> 00:02:11.060
term-- on one side and just
solve for things.

00:02:11.060 --> 00:02:13.550
So let's see, we have
5 times x plus 3.

00:02:13.550 --> 00:02:16.500
That's the same thing as
5x plus 15 when you

00:02:16.500 --> 00:02:17.780
distribute the 5.

00:02:17.780 --> 00:02:22.780
And that's going to be equal
to 3x minus 3 when you

00:02:22.780 --> 00:02:24.300
distribute the 3.

00:02:24.300 --> 00:02:26.170
Now, let's get all the x's
on the left-hand side.

00:02:26.170 --> 00:02:28.080
So let's subtract 3x
from both sides.

00:02:31.730 --> 00:02:36.770
And we get 2x plus 15 is equal
to-- those cancel out-- is

00:02:36.770 --> 00:02:38.710
equal to negative 3.

00:02:38.710 --> 00:02:44.840
And then we can subtract
15 from both sides.

00:02:44.840 --> 00:02:49.290
And we're left with 2x, this 2x,
is equal to-- negative 3

00:02:49.290 --> 00:02:51.320
minus 15 is negative 18.

00:02:51.320 --> 00:02:54.005
Divide both sides by 2.

00:02:54.005 --> 00:02:57.700
Divide both sides of the
equation by 2, and we're left

00:02:57.700 --> 00:03:01.680
with x is equal to negative 18
over 2, which is negative 9.

00:03:01.680 --> 00:03:05.390
We get x is equal
to negative 9.

00:03:05.390 --> 00:03:09.400
And, of course, we say well,
it's good it doesn't equal one

00:03:09.400 --> 00:03:10.010
of these things.

00:03:10.010 --> 00:03:11.970
That would have messed
things up.

00:03:11.970 --> 00:03:14.600
And now let's test. Let's make
sure that x does not equal

00:03:14.600 --> 00:03:17.660
negative 9 satisfies
this equation.

00:03:17.660 --> 00:03:19.010
Let's make sure it
satisfies it.

00:03:19.010 --> 00:03:25.410
So 5 over negative 9 minus 1
is equal to-- let's see, 5

00:03:25.410 --> 00:03:27.840
over negative 9 minus
1 is negative 10.

00:03:27.840 --> 00:03:30.170
So it's equal to 5
over negative 10,

00:03:30.170 --> 00:03:31.410
which is negative 1/2.

00:03:31.410 --> 00:03:32.550
That's where I substitute
it into the

00:03:32.550 --> 00:03:34.370
left-hand side of the equation.

00:03:34.370 --> 00:03:36.160
Now, what happens when I
substitute it into the

00:03:36.160 --> 00:03:37.410
right-hand side of
the equation?

00:03:40.760 --> 00:03:47.540
3 over negative 9 plus 3 is
equal to 3 over negative 6,

00:03:47.540 --> 00:03:49.610
which is equal to
negative 1/2.

00:03:49.610 --> 00:03:53.110
So the left-hand side does equal
the right-hand side when

00:03:53.110 --> 00:03:57.620
x is equal to negative 9.

00:03:57.620 --> 00:04:00.400
And to some degree, they didn't
even have to write this

00:04:00.400 --> 00:04:03.760
condition here, because the only
x for which this is true

00:04:03.760 --> 00:04:06.070
is x is equal to negative 9.

00:04:06.070 --> 00:04:09.090
That's the only thing that you
can legitimately put in here

00:04:09.090 --> 00:04:11.260
and this equality holding.