CA Algebra I: Functions

WEBVTT
Kind: captions
Language: en

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All right, problem 79.

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What is the domain of
the function shown

00:00:06.060 --> 00:00:06.920
on the graph below?

00:00:06.920 --> 00:00:08.610
That sounds like fancy word.

00:00:08.610 --> 00:00:10.380
But a function is just a rule.

00:00:10.380 --> 00:00:12.550
It's just a rule that
maps from one set

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of numbers to another.

00:00:13.690 --> 00:00:16.510
And the set of numbers that it
maps from is the domain.

00:00:16.510 --> 00:00:21.210
And the set of numbers that
it maps to is the range.

00:00:21.210 --> 00:00:24.080
So if this is the set of numbers
of the domain, this is

00:00:24.080 --> 00:00:27.310
set of numbers of the range, the
function tells us how do

00:00:27.310 --> 00:00:30.090
we get from this number, some
place in domain, to that

00:00:30.090 --> 00:00:33.470
number, some place
in the range.

00:00:33.470 --> 00:00:37.180
So in general, the domain is all
the numbers for which this

00:00:37.180 --> 00:00:38.900
mapping, or this function,
is defined.

00:00:47.000 --> 00:00:49.840
OK, so let's see, so the
domain is where is this

00:00:49.840 --> 00:00:51.090
function defined?

00:00:55.940 --> 00:00:58.440
And in general, when you plot it
on the xy-axis, the domain

00:00:58.440 --> 00:01:01.710
is all of the x values that
you're defined over and the

00:01:01.710 --> 00:01:03.100
range is all the y values.

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So the function is a mapping
from x to y.

00:01:08.400 --> 00:01:10.700
So let's see where it's defined
when x is equal to 4.

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It tells us we're mapping from
x is equal to 4 to y is equal

00:01:14.030 --> 00:01:14.600
to negative 1.

00:01:14.600 --> 00:01:19.710
So it's defined for x is equal
to 4, for x is equal to 5, for

00:01:19.710 --> 00:01:24.710
x is equal to 2, and it's
defined for x is equal to 1.

00:01:24.710 --> 00:01:26.360
So it's defined for all
of those numbers.

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That's the domain.

00:01:27.320 --> 00:01:28.590
This function is not defined.

00:01:28.590 --> 00:01:30.540
We don't know what happens when
x is equal to minus 2.

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It's not defined.

00:01:31.800 --> 00:01:33.590
It doesn't tell us what
y equals here.

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So these are the only
numbers for which

00:01:36.300 --> 00:01:37.330
this function is defined.

00:01:37.330 --> 00:01:44.290
It tells us that 4 maps to
negative 1, that 5 maps to

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negative 2, that 2 maps
to minus 5, and that 1

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maps to minus 4.

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That's all it tells us.

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And this is actually
the range.

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And that's the domain.

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So let's see, 4, 5, 2, 1.

00:02:07.530 --> 00:02:08.910
This is the domain.

00:02:08.910 --> 00:02:10.919
That's the set of all
numbers for which

00:02:10.919 --> 00:02:12.980
this function is defined.

00:02:12.980 --> 00:02:15.580
This minus 1, 2, 5, and 4.

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This is the range right here.

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Choice B is the range.

00:02:19.470 --> 00:02:21.120
All right.

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I think we're on the
last problem.

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This is going to be a lot
of cutting and pasting.

00:02:24.990 --> 00:02:26.232
I don't know if it's
going to fit.

00:02:26.232 --> 00:02:29.210
I have to shrink it.

00:02:29.210 --> 00:02:31.876
OK, they say which of the
following is not a function?

00:02:31.876 --> 00:02:33.126
That's fascinating.

00:02:35.690 --> 00:02:38.410
OK, I copied it.

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And now I have pasted it.

00:02:40.240 --> 00:02:44.860
And it did all barely
fit on this page.

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So which is not a function?

00:02:46.780 --> 00:02:48.720
And remember, what I said
in the beginning.

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I don't have space below here.

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Let's think about what a
function is and then we could

00:02:51.340 --> 00:02:52.850
look at the graph.

00:02:52.850 --> 00:02:56.760
So a function is a mapping
from one set to another.

00:02:56.760 --> 00:02:59.320
From set A to set B.

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Where A is the domain
and B is the range.

00:03:01.470 --> 00:03:03.300
And this is important thing to
realize about a function.

00:03:03.300 --> 00:03:08.460
So if this is set A, and if this
is set B, that for any

00:03:08.460 --> 00:03:13.590
value in set A, it only maps
to one value in set B.

00:03:13.590 --> 00:03:14.900
You cannot have this.

00:03:14.900 --> 00:03:18.790
You can't have one value in set
A going to this value and

00:03:18.790 --> 00:03:20.040
going to another value.

00:03:22.640 --> 00:03:27.010
So that would not make
it a function.

00:03:27.010 --> 00:03:30.590
If we say x is the domain, if
this is the set of all values

00:03:30.590 --> 00:03:32.960
we can put into x, and this all
the set of values that can

00:03:32.960 --> 00:03:37.600
go into y or that can be y,
then one value of x can't

00:03:37.600 --> 00:03:41.210
produce two values of y.

00:03:41.210 --> 00:03:42.260
You can have it the
other way around.

00:03:42.260 --> 00:03:44.500
You could have two different
values of x producing the same

00:03:44.500 --> 00:03:45.960
value of y.

00:03:45.960 --> 00:03:48.070
And we'll think about
what that is.

00:03:48.070 --> 00:03:49.490
That's completely valid.

00:03:49.490 --> 00:03:50.680
You could have two numbers.

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When x is equal to 2 and x is
equal to 7, you can still have

00:03:53.650 --> 00:03:57.640
y equaling the same thing.

00:03:57.640 --> 00:03:59.700
Say this is x is equal to 5.

00:03:59.700 --> 00:04:02.280
X is equal to 5 could
be y is equal to 7.

00:04:02.280 --> 00:04:04.630
Or x is equal to 5 could
be y is equal to 8.

00:04:04.630 --> 00:04:05.360
We don't know that.

00:04:05.360 --> 00:04:05.860
So let's see.

00:04:05.860 --> 00:04:08.100
Let's think about how we can
visually think about this.

00:04:10.640 --> 00:04:16.800
For every value of x, we only
have one definition for y.

00:04:16.800 --> 00:04:20.010
And people often call that the
vertical line test. At no

00:04:20.010 --> 00:04:22.460
point on this graph can I draw
a vertical line and it

00:04:22.460 --> 00:04:24.580
intersects it twice.

00:04:24.580 --> 00:04:27.210
If a vertical intersects it
twice, that means for a given

00:04:27.210 --> 00:04:30.800
value of x, this function
defines two different y's.

00:04:30.800 --> 00:04:31.890
Not going to happen.

00:04:31.890 --> 00:04:35.490
Now this function does
for any given y have

00:04:35.490 --> 00:04:37.890
two different x's.

00:04:37.890 --> 00:04:39.930
Like that point and
that point.

00:04:39.930 --> 00:04:40.810
That's OK.

00:04:40.810 --> 00:04:41.850
That's completely fine.

00:04:41.850 --> 00:04:45.680
So for example, when x is equal
to minus 1.5 and x is

00:04:45.680 --> 00:04:47.730
equal to 3, they point
to the same number.

00:04:56.280 --> 00:04:57.490
That's completely OK.

00:04:57.490 --> 00:04:58.990
As long as one of these
doesn't point to

00:04:58.990 --> 00:05:01.390
two separate numbers.

00:05:01.390 --> 00:05:04.290
So that magenta is what that
first graph is doing.

00:05:04.290 --> 00:05:08.330
Same thing here, for any
given y we can have two

00:05:08.330 --> 00:05:12.040
x's pointing to it.

00:05:12.040 --> 00:05:12.740
And they're just fine.

00:05:12.740 --> 00:05:14.200
That's completely fine
for a function.

00:05:14.200 --> 00:05:16.726
Now, we talk about function
inverses, well I don't want to

00:05:16.726 --> 00:05:17.730
get too complicated.

00:05:17.730 --> 00:05:18.460
That's completely fine.

00:05:18.460 --> 00:05:21.200
If we do the vertical line test,
at no point can we draw

00:05:21.200 --> 00:05:27.020
vertical line and get, for any
given x, two separate y's.

00:05:27.020 --> 00:05:27.760
Same thing for here.

00:05:27.760 --> 00:05:30.080
Actually here, we have a
complete 1:1 mapping.

00:05:30.080 --> 00:05:32.560
For any given x,
there's one y.

00:05:32.560 --> 00:05:35.110
I think here it should be clear,
that if I were to draw

00:05:35.110 --> 00:05:48.560
vertical line, let's say right
there, for x is equal to 1, it

00:05:48.560 --> 00:05:49.840
has two definitions.

00:05:49.840 --> 00:05:52.520
Even better, for x is equal
to 0, y could be

00:05:52.520 --> 00:05:55.250
minus 2 or plus 2.

00:05:55.250 --> 00:05:56.460
So that's the case
that we said.

00:05:56.460 --> 00:06:02.550
For x is equal to 0, it's
pointing to plus 2 and it's

00:06:02.550 --> 00:06:03.500
pointing to minus 2.

00:06:03.500 --> 00:06:06.470
It's pointing to two different
values which you cannot have

00:06:06.470 --> 00:06:07.070
in a function.

00:06:07.070 --> 00:06:09.310
For a function, for any
given x, you can only

00:06:09.310 --> 00:06:11.650
define one y value.

00:06:11.650 --> 00:06:15.230
So this relation is
not a function.

00:06:15.230 --> 00:06:17.430
D.

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Next question.

00:06:19.790 --> 00:06:24.400
I think this is the last.
It's a page on its own.

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Oh, that was the
last question.

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Did I skip a problem?

00:06:27.320 --> 00:06:30.560
See that was 80.

00:06:30.560 --> 00:06:31.190
That's it.

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We're all done.

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All right, see you soon.


CA Algebra I: More Word Problems

WEBVTT
Kind: captions
Language: en

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Problem 75.

00:00:02.790 --> 00:00:06.300
Two airplanes left the same
airport traveling in opposite

00:00:06.300 --> 00:00:07.090
directions.

00:00:07.090 --> 00:00:11.500
If one plane averages 400 miles
per hour-- so this is

00:00:11.500 --> 00:00:12.330
the airport.

00:00:12.330 --> 00:00:14.430
One plane goes in that
direction at

00:00:14.430 --> 00:00:18.550
400 miles per hour.

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That's a mile not a meter.

00:00:19.970 --> 00:00:22.110
And another plane averages
250 miles per hour.

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And it's going in the
other direction.

00:00:25.250 --> 00:00:26.840
And I do this as vectors.

00:00:26.840 --> 00:00:30.980
So the length of the my
arrow represents its

00:00:30.980 --> 00:00:32.330
speed to some degree.

00:00:32.330 --> 00:00:34.810
And the direction represents
direction.

00:00:34.810 --> 00:00:37.660
250 miles per hour in that
direction, 400 miles per hour

00:00:37.660 --> 00:00:38.170
in that direction.

00:00:38.170 --> 00:00:40.550
In how many hours will the
distance between the two

00:00:40.550 --> 00:00:44.420
planes be 1625 miles?

00:00:44.420 --> 00:00:47.250
So you just have to say, if
you're sitting in this plane

00:00:47.250 --> 00:00:49.400
right here, this plane
to the left.

00:00:49.400 --> 00:00:52.530
And you let's say you think
you're stationary.

00:00:52.530 --> 00:00:56.280
How fast is that plane going to
be moving to the rightwards

00:00:56.280 --> 00:00:59.800
direction, or I guess to
the eastward direction?

00:00:59.800 --> 00:01:01.760
Well from your point of view,
it's going to be going in the

00:01:01.760 --> 00:01:04.790
east direction at 650
miles per hour.

00:01:04.790 --> 00:01:06.510
If you thought that you were
stationary, it would just be

00:01:06.510 --> 00:01:13.200
going away from you at
650 miles per hour.

00:01:13.200 --> 00:01:14.940
Same thing, if you were sitting
in this plane right

00:01:14.940 --> 00:01:17.270
here, and you thought you were
stationary-- and you are,

00:01:17.270 --> 00:01:19.400
relative to yourself-- you'd
say, oh, why this guy is going

00:01:19.400 --> 00:01:22.540
to the west at 650 miles per
hour relative to me.

00:01:22.540 --> 00:01:24.300
And we're starting at
the same point.

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So this just becomes a standard
distance is equal to

00:01:26.610 --> 00:01:28.520
rate times time problem.

00:01:28.520 --> 00:01:31.780
So if we have to cover 1625
miles, because that's what

00:01:31.780 --> 00:01:33.740
they want to know, when
does that happen?

00:01:33.740 --> 00:01:37.150
At what point are the planes
that far from each other?

00:01:37.150 --> 00:01:41.410
That is equal to their relative
velocity, they're

00:01:41.410 --> 00:01:45.870
separating from each other
at 650 miles per hour.

00:01:45.870 --> 00:01:48.250
And we want to know how long
is that going to take?

00:01:48.250 --> 00:01:56.420
So time is going to equal
1625 divided by 650.

00:01:56.420 --> 00:01:58.570
Let's see if we can simplify
that at all.

00:01:58.570 --> 00:02:04.480
Well, we could divide the top
and the bottom by 25.

00:02:04.480 --> 00:02:09.870
If we divide the top one-- let
me just get my calculator.

00:02:09.870 --> 00:02:17.090
So 1625 divided by 650
is equal to 2.5.

00:02:20.190 --> 00:02:21.770
And we were doing everything
in miles per hour.

00:02:21.770 --> 00:02:25.290
So time is going to be
equal to 2.5 hours.

00:02:25.290 --> 00:02:27.700
This choice is A.

00:02:27.700 --> 00:02:30.070
Problem 76.

00:02:30.070 --> 00:02:32.870
I'll just copy and paste all
these word problems. We'll

00:02:32.870 --> 00:02:35.150
parse them together.

00:02:35.150 --> 00:02:41.200
Problem 76 in green.

00:02:41.200 --> 00:02:44.930
Lisa will make punch that is 25%
fruit by adding pure fruit

00:02:44.930 --> 00:02:52.290
juice to a two liter mixture
that is 10% pure fruit juice.

00:02:52.290 --> 00:02:57.060
How many liters of pure fruit
juice does she need to add?

00:02:57.060 --> 00:02:58.080
Pure fruit juice.

00:02:58.080 --> 00:03:04.810
So let P equal liters of pure.

00:03:04.810 --> 00:03:06.810
And that's what they want
us to figure out.

00:03:06.810 --> 00:03:08.830
So we have to figure out P.

00:03:08.830 --> 00:03:09.450
All right.

00:03:09.450 --> 00:03:14.170
And adding it to a two liter
mixture., So so let's say M is

00:03:14.170 --> 00:03:15.700
equal to the liters
of the mixture.

00:03:20.200 --> 00:03:23.670
That is 10% fruit juice.

00:03:23.670 --> 00:03:28.470
So that is this right here, to
a two liter mixture, that is

00:03:28.470 --> 00:03:29.740
10% fruit juice.

00:03:29.740 --> 00:03:31.100
We're calling that M.

00:03:31.100 --> 00:03:35.220
That is the two liter mixture
that is 10% fruit juice.

00:03:35.220 --> 00:03:37.140
OK, how many liters of
pure fruit juice

00:03:37.140 --> 00:03:39.050
does she need to add?

00:03:39.050 --> 00:03:40.985
OK, so let's think about
it a little bit.

00:03:50.520 --> 00:03:54.060
So, punch is going
to be pure juice.

00:03:54.060 --> 00:03:56.870
So let's think about how much
pure juice is going to be in

00:03:56.870 --> 00:03:59.000
the system.

00:03:59.000 --> 00:04:03.120
The liters of pure juice
she adds, plus the

00:04:03.120 --> 00:04:05.030
liters of the mixture.

00:04:05.030 --> 00:04:07.620
Well it's a two liter mixture
that is 10% fruit juice.

00:04:07.620 --> 00:04:10.090
So how much fruit juice
is already there?

00:04:10.090 --> 00:04:12.330
So, it's 0.2 liters
of fruit juice.

00:04:15.080 --> 00:04:18.450
I shouldn't even define M, we
know that M is equal to 2.

00:04:18.450 --> 00:04:19.459
So ignore that.

00:04:19.459 --> 00:04:21.029
You can tell I'm doing these
problems in real time.

00:04:21.029 --> 00:04:23.210
I'm struggling with
them as you would.

00:04:23.210 --> 00:04:26.160
Anyway, so how much fruit juice
is there going to be in

00:04:26.160 --> 00:04:27.110
the mixture?

00:04:27.110 --> 00:04:29.580
There's the amount of
liters of the pure

00:04:29.580 --> 00:04:32.290
fruit juice she adds.

00:04:32.290 --> 00:04:35.390
This is a pure fruit
juice, that's P.

00:04:35.390 --> 00:04:38.340
And then you're going to have
the two liters of the mixture

00:04:38.340 --> 00:04:39.550
that is 10% fruit juice.

00:04:39.550 --> 00:04:41.630
So how much fruit juice
is coming from that?

00:04:41.630 --> 00:04:45.850
Well 10% of two liters is
0.2 liters already.

00:04:45.850 --> 00:04:47.520
And then how many liters
is there going

00:04:47.520 --> 00:04:49.160
to be in the mixture?

00:04:49.160 --> 00:04:51.000
In the combined, how many
total liters are there?

00:04:51.000 --> 00:04:54.460
Well, there's the liters of the
pure fruit juice, plus the

00:04:54.460 --> 00:04:57.210
two liters that we're
adding to.

00:04:57.210 --> 00:05:00.530
This is the amount
of fruit juice.

00:05:00.530 --> 00:05:02.790
This is a total amount
of liquid.

00:05:02.790 --> 00:05:05.950
And we got the 0.2
from 10% times 2.

00:05:05.950 --> 00:05:10.140
And we said that this
has to equal 25%.

00:05:10.140 --> 00:05:12.010
We want this to be
25% of this.

00:05:12.010 --> 00:05:14.100
So this should be 0.25.

00:05:14.100 --> 00:05:17.480
That was an inadvertent
mark right down there.

00:05:17.480 --> 00:05:21.000
So now we just solve for P.

00:05:21.000 --> 00:05:23.740
The hard part is setting
up the problem.

00:05:23.740 --> 00:05:26.770
So let's multiply P plus
2 times both sides.

00:05:26.770 --> 00:05:32.706
So you get P plus 0.2 is equal
to 0.25 times this.

00:05:32.706 --> 00:05:39.880
So 0.25P plus 0.25
times 2 is 0.5.

00:05:39.880 --> 00:05:40.760
Fair enough.

00:05:40.760 --> 00:05:42.490
Let's subtract 0.2
from both sides.

00:05:42.490 --> 00:05:50.300
You get P is equal to 0.25P,
0.5 minus 0.2 is plus 0.3.

00:05:50.300 --> 00:05:52.790
Let's subtract 0.25P
from both sides.

00:05:52.790 --> 00:05:59.440
1P minus 0.25P is 0.75P
is equal to 0.3.

00:06:02.050 --> 00:06:03.410
And let's see where this goes.

00:06:06.130 --> 00:06:07.550
Did I do that right?

00:06:07.550 --> 00:06:12.130
0.75P is equal to 0.3.

00:06:12.130 --> 00:06:13.380
Right.

00:06:19.060 --> 00:06:20.410
The decimals sometimes
get confusing.

00:06:20.410 --> 00:06:22.430
So let's multiply both sides
of this equation by 10.

00:06:22.430 --> 00:06:27.500
You get 7.5P is equal to 0.3.

00:06:27.500 --> 00:06:30.250
Sorry, is equal to 3,
we multiplied by 10.

00:06:30.250 --> 00:06:35.690
So you get P is equal
to 3 over 7.5.

00:06:35.690 --> 00:06:37.760
How many times does
7.5 go into 3?

00:06:37.760 --> 00:06:38.860
So that's the same thing
as how many times

00:06:38.860 --> 00:06:42.280
does 75 go into 30.

00:06:42.280 --> 00:06:43.840
Put a decimal sign.

00:06:43.840 --> 00:06:46.100
Let's see, it goes into
it I think four times.

00:06:51.690 --> 00:06:52.860
It's 0.4.

00:06:52.860 --> 00:06:55.120
So P is equal to 0.4 liters.

00:06:55.120 --> 00:07:01.120
We have to add 0.4 liters of
pure fruit juice to the two

00:07:01.120 --> 00:07:02.310
liter mixture.

00:07:02.310 --> 00:07:04.650
So you add 0.4 to the two liter
mixture, so you end up

00:07:04.650 --> 00:07:08.150
with 0.6 liters of
pure fruit juice.

00:07:08.150 --> 00:07:10.200
So the numerator becomes 0.6.

00:07:10.200 --> 00:07:14.130
And the denominator
becomes 2.4.

00:07:14.130 --> 00:07:18.410
Which, sure enough,
is 1/4 or 25%.

00:07:18.410 --> 00:07:20.450
Fascinating problem.

00:07:20.450 --> 00:07:22.530
Next question.

00:07:22.530 --> 00:07:23.830
And these are good to learn.

00:07:23.830 --> 00:07:26.510
A lot of times in math class
people are like, what do I

00:07:26.510 --> 00:07:27.170
need to learn this for?

00:07:27.170 --> 00:07:29.143
But this type of algebra,
you're going to

00:07:29.143 --> 00:07:30.170
use it all the time.

00:07:30.170 --> 00:07:32.930
Especially if you go into
any type of science or

00:07:32.930 --> 00:07:33.730
engineering.

00:07:33.730 --> 00:07:36.050
Or, for that matter, investing
or finance.

00:07:36.050 --> 00:07:38.780
You're going to use these kind
of, boy can I figure this out

00:07:38.780 --> 00:07:41.370
or not type of equations all
the time when you're mixing

00:07:41.370 --> 00:07:43.700
things and figuring out margins
and all the rest.

00:07:43.700 --> 00:07:44.590
OK, let's see.

00:07:44.590 --> 00:07:47.390
Which relation is a function?

00:07:47.390 --> 00:07:49.320
So I have a little bit of
a review of a function.

00:07:49.320 --> 00:07:52.670
So a function is a relation
that maps from

00:07:52.670 --> 00:07:54.820
a domain to a range.

00:07:59.170 --> 00:08:00.400
I want to say from A to B.

00:08:00.400 --> 00:08:04.390
Or we could say from the
domain to the range.

00:08:04.390 --> 00:08:06.140
Where it takes one set of
numbers, which you call the

00:08:06.140 --> 00:08:09.080
domain, and we have another
set of numbers.

00:08:09.080 --> 00:08:12.000
And for every number here,
there's exactly one

00:08:12.000 --> 00:08:14.080
number in the range.

00:08:14.080 --> 00:08:15.870
And that's the key.

00:08:15.870 --> 00:08:18.510
From here to here, there's
only one number.

00:08:18.510 --> 00:08:21.030
Now two of these could
meet the same number.

00:08:21.030 --> 00:08:23.320
But you can't have one number
here going to two separate

00:08:23.320 --> 00:08:24.310
numbers there.

00:08:24.310 --> 00:08:25.560
That's the key.

00:08:28.930 --> 00:08:30.990
Here we have the same number.

00:08:34.200 --> 00:08:36.570
You have a minus 2 here.

00:08:36.570 --> 00:08:38.460
And you have a minus 2 here.

00:08:38.460 --> 00:08:41.220
So that's like the
same number.

00:08:41.220 --> 00:08:42.780
Let's say this is a minus 2.

00:08:42.780 --> 00:08:47.570
That minus 2 is going
to a 6 and here it's

00:08:47.570 --> 00:08:49.530
going to minus 2.

00:08:49.530 --> 00:08:51.240
So you have the same
number going to

00:08:51.240 --> 00:08:52.320
two different numbers.

00:08:52.320 --> 00:08:53.960
And you can't have that
in a function.

00:08:53.960 --> 00:08:57.330
For any given number in the
domain, you definitely get

00:08:57.330 --> 00:08:58.410
only one in the range.

00:08:58.410 --> 00:09:01.100
So A is not the answer.

00:09:01.100 --> 00:09:03.800
Let's see, does this have
that same problem?

00:09:03.800 --> 00:09:05.440
See every number is there
exactly once.

00:09:05.440 --> 00:09:09.170
You have one minus 2, one 0,
one 1, one 2, so you're not

00:09:09.170 --> 00:09:10.580
going to have that problem.

00:09:10.580 --> 00:09:13.940
My suspicion is it's
already B.

00:09:13.940 --> 00:09:15.200
C has the same problem.

00:09:15.200 --> 00:09:19.100
4 goes to a bunch of
different numbers.

00:09:19.100 --> 00:09:23.760
4 goes to 0, 4 goes to
1, so it's not C.

00:09:23.760 --> 00:09:27.650
And then this defines
two values for 10.

00:09:27.650 --> 00:09:29.280
So that doesn't work either.

00:09:29.280 --> 00:09:30.650
So the answer is B.

00:09:33.720 --> 00:09:38.890
Problem 78.

00:09:38.890 --> 00:09:40.140
All right.

00:09:49.220 --> 00:09:52.120
For which equation graphed
below, are all

00:09:52.120 --> 00:09:54.180
the y values negative?

00:09:54.180 --> 00:09:56.460
So if all the y values are
negative, we have to stay

00:09:56.460 --> 00:09:57.460
below the x-axis.

00:09:57.460 --> 00:10:00.420
This is where all the y values
are negative, right there.

00:10:00.420 --> 00:10:01.520
So this one seems to work.

00:10:01.520 --> 00:10:02.430
A seems to work.

00:10:02.430 --> 00:10:04.205
Here, these aren't negative
y values.

00:10:04.205 --> 00:10:06.330
These are positive y values.

00:10:06.330 --> 00:10:07.660
I didn't even copy and
paste all these.

00:10:07.660 --> 00:10:11.940
But I can already see that these
are positive y values.

00:10:11.940 --> 00:10:13.150
So A was the answer.

00:10:13.150 --> 00:10:14.700
I'm sorry, I ran out of space.

00:10:14.700 --> 00:10:16.670
Anyway, I'll see you
in the next video.

00:10:16.670 --> 00:10:18.310
All of these y values,
they're all negative.

00:10:18.310 --> 00:10:21.090
You're staying in the third
and fourth quadrant.

00:10:21.090 --> 00:10:22.340
All right, next video.


CA Algebra I: Word Problems

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.170
We're on problem 71.

00:00:02.170 --> 00:00:04.680
And they ask us, which fraction
is equivalent to this

00:00:04.680 --> 00:00:09.780
thing, 3x over 5 divided by
x plus 4 over x plus 2?

00:00:09.780 --> 00:00:11.800
Now the best way to do this,
let's simplify the denominator

00:00:11.800 --> 00:00:16.176
first. So that's equal
to 3x over 5.

00:00:16.176 --> 00:00:19.160
All of that over-- let's find
a common denominator.

00:00:19.160 --> 00:00:20.490
The common denominator is 4.

00:00:20.490 --> 00:00:24.430
So if we have a common
denominator of 4, x/4 is of

00:00:24.430 --> 00:00:26.600
course x/4.

00:00:26.600 --> 00:00:29.730
x/2 is the same thing
as 2x over 4.

00:00:29.730 --> 00:00:31.970
So plus 2x over 4.

00:00:31.970 --> 00:00:34.560
2x over 4 is the same
thing as x/2.

00:00:34.560 --> 00:00:39.580
And now if you divide by a
certain fraction, that's the

00:00:39.580 --> 00:00:41.840
same thing as multiplying
buy its inverse.

00:00:41.840 --> 00:00:46.640
So this is going to be equal to
3x over 5 times the inverse

00:00:46.640 --> 00:00:52.500
of this, times 4
over x plus 2x.

00:00:52.500 --> 00:00:53.410
Let's see what we could do.

00:00:53.410 --> 00:00:54.875
Well we could factor
out an x here.

00:00:54.875 --> 00:01:03.700
So this becomes x
times 1 plus 2.

00:01:03.700 --> 00:01:05.390
Actually, 1 plus 2 is 3.

00:01:05.390 --> 00:01:07.910
I should have said x plus
2x is equal to 3x.

00:01:07.910 --> 00:01:09.540
I should have done.

00:01:09.540 --> 00:01:10.710
That's silly, I shouldn't
have done that.

00:01:10.710 --> 00:01:12.750
I should have said x plus
2x is equal to 3x.

00:01:16.640 --> 00:01:17.550
My bad.

00:01:17.550 --> 00:01:20.110
Anyway, 3x and 3x cancel out.

00:01:20.110 --> 00:01:20.610
Numerator and denominator.

00:01:20.610 --> 00:01:24.370
And you're left with 4/5.

00:01:24.370 --> 00:01:26.660
And that's choice C.

00:01:26.660 --> 00:01:28.080
It's getting late.

00:01:28.080 --> 00:01:29.330
Next question.

00:01:31.810 --> 00:01:35.090
72.

00:01:35.090 --> 00:01:36.170
Let me copy and paste this.

00:01:36.170 --> 00:01:39.090
Word problems, it's good to
parse these together.

00:01:41.610 --> 00:01:47.970
All right, OK, a pharmacist
mixed some 10% saline solution

00:01:47.970 --> 00:01:52.780
with some 15% saline solution to
obtain 100 milliliters of a

00:01:52.780 --> 00:01:56.550
12% saline solution.

00:01:56.550 --> 00:01:59.790
How much of the 10% saline
solution did the pharmacist

00:01:59.790 --> 00:02:02.070
use in the mixture?

00:02:02.070 --> 00:02:04.040
OK, so 10% and 15%.

00:02:04.040 --> 00:02:12.370
So let T equal the amount
of 10% percent.

00:02:12.370 --> 00:02:19.450
And let's say that F is equal to
the amount of 15% solution.

00:02:19.450 --> 00:02:22.390
I picked T for ten,
F for fifteen.

00:02:22.390 --> 00:02:26.170
OK, so the amount of 10%
solution plus the amount of

00:02:26.170 --> 00:02:29.860
15% solution, that has to
equal 100 milliliters.

00:02:29.860 --> 00:02:32.170
I'll assume we're dealing
in milliliters.

00:02:32.170 --> 00:02:33.930
That equals 100.

00:02:33.930 --> 00:02:37.220
And then, how much saline are
we going to end up with?

00:02:37.220 --> 00:02:44.010
So, if we have the 10% saline
solution, however much saline

00:02:44.010 --> 00:02:45.330
we have, how much saline
is that going to

00:02:45.330 --> 00:02:46.200
contribute to the whole?

00:02:46.200 --> 00:02:49.570
Well it's 10% saline, so we're
going to contribute 10% of

00:02:49.570 --> 00:02:51.910
that as saline.

00:02:51.910 --> 00:02:55.480
So 10% times the amount of 10%
solution tells me how much

00:02:55.480 --> 00:02:57.850
saline I'm contributing.

00:02:57.850 --> 00:03:04.380
Plus 15% times the amount of
the 15% saline solution I'm

00:03:04.380 --> 00:03:05.200
contributing.

00:03:05.200 --> 00:03:07.820
This is the amount of saline
from that solution.

00:03:07.820 --> 00:03:08.820
And that equals what?

00:03:08.820 --> 00:03:10.680
Now this is interesting.

00:03:10.680 --> 00:03:13.260
You end up with a 12%
saline solution.

00:03:13.260 --> 00:03:16.540
100 milliliters of a 12%
saline solution.

00:03:16.540 --> 00:03:19.360
So how much saline total am I
going to have when I mix it

00:03:19.360 --> 00:03:20.470
all together?

00:03:20.470 --> 00:03:22.020
I had 100 milliliters.

00:03:22.020 --> 00:03:23.595
It's a 12% saline solution.

00:03:23.595 --> 00:03:27.230
So that means 12% of this 100
milliliters is saline.

00:03:27.230 --> 00:03:28.840
So that equals 12.

00:03:28.840 --> 00:03:30.960
There's 12 milliliters
of saline.

00:03:30.960 --> 00:03:34.920
With this much contributed from
the 10% solution and that

00:03:34.920 --> 00:03:38.010
much contributed from
the 15% solution.

00:03:38.010 --> 00:03:39.480
Let's see what we can do.

00:03:39.480 --> 00:03:41.840
So now we have two linear
equations and two unknowns.

00:03:41.840 --> 00:03:45.250
We're ready to solve.

00:03:45.250 --> 00:03:45.890
What do they want to know?

00:03:45.890 --> 00:03:48.690
They want to know how much of
the 10% saline solution.

00:03:48.690 --> 00:03:51.090
So the best thing, we want to
cancel out the F's somehow and

00:03:51.090 --> 00:03:52.135
solve for T.

00:03:52.135 --> 00:03:55.480
So let's multiply this top
equation times minus 0.15,

00:03:55.480 --> 00:03:57.720
because I want to cancel
this out with this.

00:03:57.720 --> 00:03:58.420
So let's do that.

00:03:58.420 --> 00:04:03.220
If you multiply times minus
0.15, we end up with minus

00:04:03.220 --> 00:04:14.832
0.15T minus 0.15F is
equal to minus 15.

00:04:14.832 --> 00:04:19.970
So we add these two equations,
0.1 plus minus 0.15 that's

00:04:19.970 --> 00:04:26.330
minus 0.05T.

00:04:26.330 --> 00:04:27.916
These two cancel out.

00:04:27.916 --> 00:04:31.740
It's equal to 12 minus
15 is minus 3.

00:04:31.740 --> 00:04:34.490
OK, we can multiply both sides
by negative 1, you get plus

00:04:34.490 --> 00:04:38.310
0.05 times T is equal to 3.

00:04:38.310 --> 00:04:40.280
Sorry, I'm getting messy here.

00:04:40.280 --> 00:04:44.010
So if you divide both sides by
0.05, you have T is equal to 3

00:04:44.010 --> 00:04:49.220
divided by 0.05.

00:04:49.220 --> 00:04:51.770
0.05, that's the same
thing is 1/20.

00:04:51.770 --> 00:04:55.340
So this is equal to 3 times
20, which is equal to 60.

00:04:55.340 --> 00:04:59.330
So we contributed 60 milliliters
of the 10% percent

00:04:59.330 --> 00:05:00.190
saline solution.

00:05:00.190 --> 00:05:02.480
So T is equal to 60 and
that's choice A.

00:05:02.480 --> 00:05:05.300
And this is a little bit
confusing, because you're like

00:05:05.300 --> 00:05:07.680
what variable do I use for the
10% solution and how do I set

00:05:07.680 --> 00:05:08.810
up a second equation?

00:05:08.810 --> 00:05:11.070
And a good place to start is to
say, oh well they want to

00:05:11.070 --> 00:05:13.710
know how much of the 10%
saline solution did the

00:05:13.710 --> 00:05:16.135
pharmacist use, so let me
set that is a variable.

00:05:16.135 --> 00:05:17.550
Let me set another
variable for the

00:05:17.550 --> 00:05:18.860
other type of solution.

00:05:18.860 --> 00:05:21.170
If I add them together I have
to get 100 milliliters.

00:05:21.170 --> 00:05:23.900
And the tricky part was just
probably figuring out how much

00:05:23.900 --> 00:05:26.130
saline did each of the
solutions contribute?

00:05:26.130 --> 00:05:27.890
And then when you add that
up, it equals 12

00:05:27.890 --> 00:05:29.380
milliliters of saline.

00:05:29.380 --> 00:05:31.110
Anyway, next question.

00:05:33.820 --> 00:05:44.950
73.

00:05:44.950 --> 00:05:48.560
Andy's average driving speed
for a 4 hour trip was

00:05:48.560 --> 00:05:50.590
45 miles per hour.

00:05:50.590 --> 00:05:53.800
During the first 3 hours he
drove 40 miles per hour.

00:05:53.800 --> 00:05:57.250
What was his average speed for
the last hour of his trip?

00:05:57.250 --> 00:05:59.860
OK, so distance is equal
to rate times time.

00:06:05.190 --> 00:06:11.690
OK, so his total for a 4 hour
trip was 45 miles per hour.

00:06:11.690 --> 00:06:16.150
Or you could say rate is equal
to distance divided by time.

00:06:16.150 --> 00:06:18.560
Just divide both sides
of that by time.

00:06:18.560 --> 00:06:21.530
So they said, the average
driving speed for a 4 hour

00:06:21.530 --> 00:06:23.550
trip was 45 miles per hour.

00:06:23.550 --> 00:06:28.560
So 45 miles per hour is equal
to the distance he traveled

00:06:28.560 --> 00:06:30.590
divided by his time.

00:06:30.590 --> 00:06:31.870
Divided by 4 hours.

00:06:31.870 --> 00:06:34.360
So we can figure out the
distance right off the bat.

00:06:34.360 --> 00:06:35.570
So let's actually do that.

00:06:35.570 --> 00:06:41.850
The distance is equal
to 4 times 45.

00:06:41.850 --> 00:06:43.100
That's what?

00:06:45.610 --> 00:06:54.270
It's equal to 180 miles.

00:06:54.270 --> 00:06:55.370
Fair enough.

00:06:55.370 --> 00:06:57.590
They say during the first
3 hours, he drove

00:06:57.590 --> 00:07:01.800
40 miles per hour.

00:07:01.800 --> 00:07:05.040
What was his average speed for
the last hour of his trip?

00:07:05.040 --> 00:07:06.540
So we just have to figure
out how much ground

00:07:06.540 --> 00:07:07.430
did he have to cover?

00:07:07.430 --> 00:07:12.200
So, during the first 3 hours,
he drove 40 per hour.

00:07:12.200 --> 00:07:15.740
So how far did he go?

00:07:15.740 --> 00:07:17.540
So that's a different distance,
that's just the

00:07:17.540 --> 00:07:18.250
first 3 hours.

00:07:18.250 --> 00:07:21.410
So the distance over the first
3 hours is equal to the rate

00:07:21.410 --> 00:07:25.030
of the first 3 hours, it's equal
to 40, times the time,

00:07:25.030 --> 00:07:27.010
times 3 hours.

00:07:27.010 --> 00:07:32.070
So his distance over the first 3
hours is equal to 120 miles.

00:07:32.070 --> 00:07:35.060
So how far did he have to travel
in that last hour?

00:07:35.060 --> 00:07:37.750
Well, in the first 3 hours
he went 120 miles.

00:07:37.750 --> 00:07:39.840
He went a total of 180 miles.

00:07:39.840 --> 00:07:43.620
So his distance in that
last hour is going to

00:07:43.620 --> 00:07:44.710
be 180 minus 120.

00:07:44.710 --> 00:07:49.190
He had to go 60 miles
in that last hour.

00:07:49.190 --> 00:07:50.400
So how fast did he go?

00:07:50.400 --> 00:07:52.760
Well he went 60 miles in that
last hour, so he went

00:07:52.760 --> 00:07:54.590
60 miles per hour.

00:07:54.590 --> 00:07:56.880
Choice B.

00:07:56.880 --> 00:07:59.000
So this might seem like a
complicated problem, and it is

00:07:59.000 --> 00:07:59.620
a little tricky.

00:07:59.620 --> 00:08:02.250
But it's all about distance
equals rate times time.

00:08:02.250 --> 00:08:07.190
And then recognizing that you
could use this first statement

00:08:07.190 --> 00:08:09.290
to figure out the total
distance he traveled.

00:08:09.290 --> 00:08:11.930
You could use the second
statement to figure out how

00:08:11.930 --> 00:08:14.990
far he traveled in the
first 3 hours.

00:08:14.990 --> 00:08:17.510
And then you know how far he
had to travel in that last

00:08:17.510 --> 00:08:21.010
hour, or essentially how
fast he had to travel.

00:08:21.010 --> 00:08:23.020
Next question.

00:08:23.020 --> 00:08:31.730
74.

00:08:31.730 --> 00:08:35.179
One pipe can fill a tank
in 20 minutes.

00:08:35.179 --> 00:08:39.210
So let's say pipe 1 its
rate is equal to

00:08:39.210 --> 00:08:45.730
1 tank per 20 minutes.

00:08:45.730 --> 00:08:46.570
That's how I like to write it.

00:08:46.570 --> 00:08:47.010
That's what they told us.

00:08:47.010 --> 00:08:51.270
One pipe can fill a tank in 20
minutes, while another takes

00:08:51.270 --> 00:08:52.960
30 minutes to fill
the same tank.

00:08:52.960 --> 00:08:58.090
So pipe 2, is equal to 1
tank per 30 minutes.

00:08:58.090 --> 00:08:59.720
That's its rate.

00:08:59.720 --> 00:09:01.455
Or 1/30 tank per minute.

00:09:01.455 --> 00:09:03.560
This is 1/20 tank per minute.

00:09:03.560 --> 00:09:04.860
Fair enough.

00:09:04.860 --> 00:09:08.180
How long would it take the two
pipes together to fill it?

00:09:08.180 --> 00:09:10.390
So their combined rate is
just both of these.

00:09:10.390 --> 00:09:14.460
If you have both pipes going in,
you're going to have 1/20

00:09:14.460 --> 00:09:21.630
tank per minute plus 1/60
tank per minute.

00:09:21.630 --> 00:09:22.660
Let's see.

00:09:22.660 --> 00:09:26.570
1/20 plus 1/60, common
denominator is 60.

00:09:26.570 --> 00:09:32.230
1/20 is 3/60 plus 1/60 is
1/60 tanks per minute.

00:09:32.230 --> 00:09:34.510
It's just adding fractions.

00:09:34.510 --> 00:09:41.390
So that equals 4/60
tanks per minute.

00:09:41.390 --> 00:09:46.850
4/60, that's 1/15 tanks
per minute.

00:09:49.760 --> 00:09:54.820
Or you view this as equal to
one tank per 15 minutes.

00:09:54.820 --> 00:09:56.650
So how long does it take
to fill one tank?

00:09:56.650 --> 00:09:58.110
15 minutes.

00:09:58.110 --> 00:09:59.490
Choice is C.

00:10:05.490 --> 00:10:06.520
Oh I'm all out of time.

00:10:06.520 --> 00:10:08.560
I'll see in the next video.


CA Algebra I: Quadratic roots

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:02.580
We're on problem 58.

00:00:02.580 --> 00:00:06.040
The graph of the equation y is
equal to x squared minus 3x

00:00:06.040 --> 00:00:07.300
minus 4 is shown below.

00:00:07.300 --> 00:00:09.270
Fair enough.

00:00:09.270 --> 00:00:14.380
For what value or values
of x is y equal to 0?

00:00:14.380 --> 00:00:16.350
So they're essentially
saying is, when does

00:00:16.350 --> 00:00:18.470
this here equal 0?

00:00:18.470 --> 00:00:20.030
They want to know when
does y equal 0?

00:00:20.030 --> 00:00:22.590
So what values of x
does that happen?

00:00:22.590 --> 00:00:25.180
And we could factor this and
solve for the roots, but they

00:00:25.180 --> 00:00:26.970
drew us the graph, so let's
just inspect it.

00:00:26.970 --> 00:00:28.270
So when does y equal 0?

00:00:28.270 --> 00:00:31.180
So let me draw the line
of y is equal to 0.

00:00:31.180 --> 00:00:32.590
So that's right here.

00:00:32.590 --> 00:00:34.130
Let me draw it as a line.

00:00:34.130 --> 00:00:35.160
y equals 0.

00:00:35.160 --> 00:00:36.970
That's y equals 0 right there.

00:00:36.970 --> 00:00:41.240
So what values of x
makes y equal 0?

00:00:41.240 --> 00:00:43.970
If I can see this properly,
it's when x is equal to

00:00:43.970 --> 00:00:47.330
negative 1 and when
x is equal to 4.

00:00:47.330 --> 00:00:50.715
So x is equal to negative
1 or 4.

00:00:50.715 --> 00:00:53.480
And if we substitute either of
these values into this right

00:00:53.480 --> 00:00:57.240
here, we should get
y is equal to 0.

00:00:57.240 --> 00:00:57.850
And let's see.

00:00:57.850 --> 00:01:00.850
The choices, do they have
negative 1 and 4?

00:01:00.850 --> 00:01:02.180
Yep, sure enough.

00:01:02.180 --> 00:01:05.150
Negative 1 and 4 right there.

00:01:05.150 --> 00:01:12.450
Next question, 59.

00:01:12.450 --> 00:01:16.720
Let me copy and paste it.

00:01:16.720 --> 00:01:20.190
OK, I've copied it.

00:01:20.190 --> 00:01:22.450
I'll paste it below.

00:01:22.450 --> 00:01:24.300
I'll do it right
on top of this.

00:01:24.300 --> 00:01:24.730
There you go.

00:01:24.730 --> 00:01:25.280
59.

00:01:25.280 --> 00:01:28.560
Let me erase this stuff
right here.

00:01:28.560 --> 00:01:31.960
This is unrelated
to this problem.

00:01:31.960 --> 00:01:35.970
So they are asking-- let me get
the pen right-- which best

00:01:35.970 --> 00:01:39.060
represents the graph of
y is equal to minus x

00:01:39.060 --> 00:01:41.470
squared plus 3?

00:01:41.470 --> 00:01:44.020
So here, just to get an
intuition of what parabolas

00:01:44.020 --> 00:01:46.205
look like, because these are all
parabolas, or the graph of

00:01:46.205 --> 00:01:48.010
a quadratic equation.

00:01:48.010 --> 00:01:52.230
So if I had the graph of y is
equal to x squared, what does

00:01:52.230 --> 00:01:54.010
that look like?

00:01:54.010 --> 00:01:57.380
Let me just draw a quick and
dirty x- and y-axis.

00:01:57.380 --> 00:02:00.510
And I think you're familiar
with what that looks like.

00:02:00.510 --> 00:02:02.750
So if I were to just draw y is
equal to x squared, that looks

00:02:02.750 --> 00:02:03.855
something like this.

00:02:03.855 --> 00:02:07.750
It looks something like this.

00:02:07.750 --> 00:02:09.210
It will look something
like that.

00:02:09.210 --> 00:02:10.930
I think you're familiar
with it.

00:02:10.930 --> 00:02:12.270
Because you're taking x squared,
you always get

00:02:12.270 --> 00:02:13.160
positive values.

00:02:13.160 --> 00:02:14.930
Even if you have a negative
number squared that still

00:02:14.930 --> 00:02:16.870
becomes a positive.

00:02:16.870 --> 00:02:22.740
And it's symmetric around the
line x is equal to 0.

00:02:22.740 --> 00:02:24.180
That's the graph of y
equals x squared.

00:02:24.180 --> 00:02:24.910
Now let me ask you a question.

00:02:24.910 --> 00:02:28.600
What is the graph of y is equal
to minus x squared?

00:02:28.600 --> 00:02:29.370
Let me do that.

00:02:29.370 --> 00:02:32.680
y is equal to minus x squared.

00:02:32.680 --> 00:02:35.430
So it's essentially the same
thing as this graph, but it's

00:02:35.430 --> 00:02:39.150
going to be the negative
of whatever you get.

00:02:39.150 --> 00:02:41.860
So here, x squared is always
going to be positive.

00:02:41.860 --> 00:02:43.510
You square any real number
and you're going to

00:02:43.510 --> 00:02:44.670
get a positive number.

00:02:44.670 --> 00:02:47.370
Here, you square any real
number, this part right here

00:02:47.370 --> 00:02:49.070
becomes positive.

00:02:49.070 --> 00:02:50.430
But then you take the
negative of it.

00:02:50.430 --> 00:02:52.470
So this is always going to
be a negative number.

00:02:52.470 --> 00:02:57.520
So x equals 0 is still going to
be there, but regardless of

00:02:57.520 --> 00:02:59.667
whether you go in the positive
x direction or the negative x

00:02:59.667 --> 00:03:01.550
direction, this is going
to be positive.

00:03:01.550 --> 00:03:02.460
But then you put the
negative sign, it's

00:03:02.460 --> 00:03:03.330
going to become negative.

00:03:03.330 --> 00:03:05.806
So the graph is going
to look like this.

00:03:05.806 --> 00:03:08.580
The graph is going to
look like that.

00:03:08.580 --> 00:03:11.650
I didn't draw that well, let
me give that another shot.

00:03:11.650 --> 00:03:15.100
The graph is going to
look like that.

00:03:15.100 --> 00:03:18.270
It's essentially like the mirror
image of this one if

00:03:18.270 --> 00:03:21.210
you were to reflect
it on the x-axis.

00:03:21.210 --> 00:03:22.820
This is y equals minus
x squared.

00:03:22.820 --> 00:03:24.150
So now you're pointing
it down.

00:03:24.150 --> 00:03:26.440
The u goes-- opens
up downward.

00:03:26.440 --> 00:03:28.990
And hopefully that makes
a little bit of sense.

00:03:28.990 --> 00:03:31.200
And now, what happens if
you do plus or minus 3?

00:03:31.200 --> 00:03:33.650
So what is y is equal
to x squared plus 3?

00:03:37.510 --> 00:03:39.820
Not minus x squared plus 3,
but just x squared plus 3.

00:03:39.820 --> 00:03:43.410
So if you start with x squared,
now every y value for

00:03:43.410 --> 00:03:45.450
every given x is just going
to be 3 higher.

00:03:45.450 --> 00:03:48.390
So it's just going to shift
the graph up by 3.

00:03:48.390 --> 00:03:50.910
It's going to look like that.

00:03:50.910 --> 00:03:54.190
So if you go from x squared to
x squared plus 3, you're just

00:03:54.190 --> 00:03:55.690
shifting up by 3.

00:03:55.690 --> 00:03:58.890
Similar, if you go from minus
x squared to minus x squared

00:03:58.890 --> 00:04:01.200
plus 3, which is what they gave
us in the problem, you're

00:04:01.200 --> 00:04:03.375
just going to shift
the graph up by 3.

00:04:03.375 --> 00:04:05.550
So I'll do that in
this brown color.

00:04:05.550 --> 00:04:07.500
So it's just going to take this
graph, which is minus x

00:04:07.500 --> 00:04:09.590
squared and you're going
to shift it up by 3.

00:04:09.590 --> 00:04:11.840
So it's going to look
something like this.

00:04:11.840 --> 00:04:14.200
It's going to look something
like that.

00:04:14.200 --> 00:04:16.610
So let's see, out of all the
choices they gave us, it

00:04:16.610 --> 00:04:19.740
should be opening downward and
it should have its y-intercept

00:04:19.740 --> 00:04:23.520
at y is equal to 3.

00:04:23.520 --> 00:04:24.950
If you put x is equal to
0, y is equal to 3.

00:04:24.950 --> 00:04:25.750
So let's see.

00:04:25.750 --> 00:04:28.290
It's opening downwards.

00:04:28.290 --> 00:04:30.650
So these two are the only two
that are opening downwards.

00:04:30.650 --> 00:04:32.620
And the y-intercept should
be at 3 because we

00:04:32.620 --> 00:04:33.780
shifted it up by 3.

00:04:33.780 --> 00:04:36.900
So this is the choice B.

00:04:36.900 --> 00:04:39.470
Problem 60.

00:04:39.470 --> 00:04:44.400
Which quadratic funtion, when
graphed, has x-intercepts of 4

00:04:44.400 --> 00:04:46.180
and minus 3?

00:04:46.180 --> 00:04:48.860
So x-intercepts of 4 and minus
3 means that when you

00:04:48.860 --> 00:04:50.880
substitute x of either of
these values into the

00:04:50.880 --> 00:04:54.450
equation, you get
y is equal to 0.

00:04:54.450 --> 00:04:56.630
Because when y is equal to 0,
you're at the x-intercepts.

00:04:56.630 --> 00:04:58.370
This is when y equals to 0.

00:04:58.370 --> 00:04:59.620
So that's what they mean
by x-intercepts.

00:05:02.550 --> 00:05:05.240
So how do we set up an equation
where if I put in one

00:05:05.240 --> 00:05:08.170
of these numbers I'm
going to get 0?

00:05:08.170 --> 00:05:16.090
Well, if I make it the product
of x minus the first root and

00:05:16.090 --> 00:05:17.790
x minus the second root.

00:05:17.790 --> 00:05:22.040
So x minus minus
3 is x plus 3.

00:05:22.040 --> 00:05:22.720
So think about it.

00:05:22.720 --> 00:05:26.790
If you put 4 here for x, you
get 4 minus 4, which is 0.

00:05:26.790 --> 00:05:27.940
Times 4 plus 3 is 7.

00:05:27.940 --> 00:05:30.080
So 0 times 7 is 0.

00:05:30.080 --> 00:05:31.350
So that works.

00:05:31.350 --> 00:05:32.880
And then, for minus 3.

00:05:32.880 --> 00:05:34.780
Minus 3 minus 4 is minus 7.

00:05:34.780 --> 00:05:36.960
But then minus 3
plus 3 is a 0.

00:05:36.960 --> 00:05:38.880
So either of these, when you
substitute it into this

00:05:38.880 --> 00:05:40.820
expression, you get 0.

00:05:40.820 --> 00:05:42.185
Let's see, which
choice is that?

00:05:42.185 --> 00:05:49.210
x minus 4 times x plus 3.

00:05:49.210 --> 00:05:57.590
Have the x-intercepts
of 4 and minus 3.

00:05:57.590 --> 00:05:58.350
Right.

00:05:58.350 --> 00:06:00.610
This should be right.

00:06:00.610 --> 00:06:01.920
They're being tricky
right here.

00:06:01.920 --> 00:06:03.620
So x plus 3 is there.

00:06:03.620 --> 00:06:05.330
I see that in a couple
of them, right?

00:06:05.330 --> 00:06:07.990
But I don't see the x
minus 4 anywhere.

00:06:07.990 --> 00:06:11.910
But that's because we can
multiply this by any constant.

00:06:11.910 --> 00:06:14.350
Because 0 times some number
times some constant is still

00:06:14.350 --> 00:06:15.440
going to be 0.

00:06:15.440 --> 00:06:20.270
So if you look at this one
right here, 2x minus 8.

00:06:20.270 --> 00:06:21.030
We could factor out a 2.

00:06:21.030 --> 00:06:26.460
That's the same thing as
2 times x minus 4.

00:06:26.460 --> 00:06:30.720
So choice B is x plus 3,
times x minus 4, times

00:06:30.720 --> 00:06:32.100
some constant 2.

00:06:32.100 --> 00:06:34.740
So choice B is our answer.

00:06:34.740 --> 00:06:39.560
If this is equal to 0 when x is
equal to 4 minus 3, this--

00:06:39.560 --> 00:06:45.220
any constant, x minus 4 times x
plus 3--I that's still going

00:06:45.220 --> 00:06:46.260
to be equal to 0.

00:06:46.260 --> 00:06:47.915
Because when x is equal
to 4, this is going

00:06:47.915 --> 00:06:48.740
to be equal to 0.

00:06:48.740 --> 00:06:50.450
So 0 times anything times
anything else

00:06:50.450 --> 00:06:51.190
is going to be 0.

00:06:51.190 --> 00:06:53.070
Same thing with x is
equal to minus 3.

00:06:53.070 --> 00:06:54.710
So they just put a 2 here.

00:06:54.710 --> 00:06:55.600
That's a good problem.

00:06:55.600 --> 00:06:57.910
It made you realize that you
could put a constant in there

00:06:57.910 --> 00:06:59.110
and it's a little tricky.

00:06:59.110 --> 00:07:00.360
Next problem.

00:07:05.790 --> 00:07:16.130
OK, so they want to know-- let
me copy and paste it-- how

00:07:16.130 --> 00:07:19.650
many times does the graph of y
equals 2x squared minus 2x

00:07:19.650 --> 00:07:23.170
plus 3 intersect the x-axis?

00:07:23.170 --> 00:07:25.450
So the easiest thing-- because
maybe it doesn't intersect the

00:07:25.450 --> 00:07:26.420
x-axis at all.

00:07:26.420 --> 00:07:28.450
Maybe if you use a quadratic
equation

00:07:28.450 --> 00:07:29.650
there are no real solutions.

00:07:29.650 --> 00:07:31.280
So let's just apply
the quadratic.

00:07:31.280 --> 00:07:36.670
So the roots or the times that--
I guess the x values

00:07:36.670 --> 00:07:42.650
that solve this equation, 2x
squared minus 2x plus 3 is

00:07:42.650 --> 00:07:44.030
equal to 0.

00:07:44.030 --> 00:07:45.540
And these are the x
values where you

00:07:45.540 --> 00:07:46.730
intersect the x-axis.

00:07:46.730 --> 00:07:47.500
And why do I say that?

00:07:47.500 --> 00:07:50.820
Because the x-axis is the
line y is equal to 0.

00:07:50.820 --> 00:07:53.200
So I set y equal to
0 and I get this.

00:07:53.200 --> 00:07:55.530
And we know from the quadratic
equation the solution to this

00:07:55.530 --> 00:07:58.680
is negative b-- let me do
this in another color.

00:07:58.680 --> 00:08:00.210
So negative b.

00:08:00.210 --> 00:08:02.500
So minus minus 2 is 2.

00:08:02.500 --> 00:08:06.370
Plus or minus the square
root of b squared.

00:08:06.370 --> 00:08:09.100
Minus 2 squared is 4.

00:08:09.100 --> 00:08:16.320
Minus 4 times a, which is 2.

00:08:16.320 --> 00:08:18.220
Times c, times 3.

00:08:18.220 --> 00:08:19.860
All of that over 2a.

00:08:19.860 --> 00:08:22.670
2 times 2, which is 4.

00:08:22.670 --> 00:08:23.840
Now they don't want
us to figure out

00:08:23.840 --> 00:08:24.500
the roots or anything.

00:08:24.500 --> 00:08:25.420
They just want to know,
how many times does it

00:08:25.420 --> 00:08:26.570
intersect the axis?

00:08:26.570 --> 00:08:27.750
So let's think about this.

00:08:27.750 --> 00:08:29.140
What happens under this
radical sign?

00:08:29.140 --> 00:08:31.350
We have 4 times 2
times 3 is 24.

00:08:31.350 --> 00:08:37.970
So this becomes 2 plus or
minus 4 minus 24 over 4.

00:08:37.970 --> 00:08:40.960
This is minus 20.

00:08:40.960 --> 00:08:44.010
So you end up with minus 20
under the radical sign.

00:08:44.010 --> 00:08:46.220
And we know if we're dealing
with real numbers, if we want

00:08:46.220 --> 00:08:50.430
real solutions, you can't take
the square root of minus 20.

00:08:50.430 --> 00:08:53.040
So this actually has no
solutions or, another way to

00:08:53.040 --> 00:08:56.700
put it is, there is no x values
where y is equal to 0.

00:08:56.700 --> 00:08:58.720
Or another way to put it
is, this never does

00:08:58.720 --> 00:09:00.820
intersect the x-axis.

00:09:00.820 --> 00:09:01.680
So it's A.

00:09:01.680 --> 00:09:04.270
none.

00:09:04.270 --> 00:09:06.970
What gave that away was the fact
that when you apply the

00:09:06.970 --> 00:09:09.750
quadratic equation, you get a
negative number under the

00:09:09.750 --> 00:09:10.410
radical sign.

00:09:10.410 --> 00:09:11.570
So if we're dealing
with real numbers,

00:09:11.570 --> 00:09:14.340
there's no answer there.

00:09:14.340 --> 00:09:15.590
Next question.

00:09:19.740 --> 00:09:27.700
62.

00:09:27.700 --> 00:09:30.260
An object that is projected
straight down-- oh, this is

00:09:30.260 --> 00:09:32.070
good, this is projectile
motion-- is projected straight

00:09:32.070 --> 00:09:35.650
downward with initial velocity
v feet per second, Travels a

00:09:35.650 --> 00:09:40.400
distance of s v times t plus
16t squared, where t equals

00:09:40.400 --> 00:09:41.580
time in seconds.

00:09:41.580 --> 00:09:45.520
If Ramon is standing on a
balcony 84 feet above the

00:09:45.520 --> 00:09:48.170
ground and throws a penny
straight down with an initial

00:09:48.170 --> 00:09:51.760
velocity of 10 feet per second,
in how many seconds

00:09:51.760 --> 00:09:54.330
will it reach the ground?

00:09:54.330 --> 00:09:58.840
OK, so he's 84 feet
above the ground.

00:09:58.840 --> 00:10:00.450
Let's draw a diagram.

00:10:00.450 --> 00:10:01.800
He's 84 feet.

00:10:01.800 --> 00:10:05.180
This is 84 feet above
the ground.

00:10:05.180 --> 00:10:07.580
It says, how many seconds will
it reach the ground?

00:10:07.580 --> 00:10:10.500
So we essentially want to know
how many seconds will it take

00:10:10.500 --> 00:10:13.080
it to travel? s is
distance, right?

00:10:13.080 --> 00:10:14.560
So s is equal to 84 feet.

00:10:14.560 --> 00:10:17.690
It has to go down 84 feet.

00:10:17.690 --> 00:10:19.490
Let's see if we can
figure this out.

00:10:19.490 --> 00:10:22.240
Now this is something that might
be a little bit-- so how

00:10:22.240 --> 00:10:24.620
long does it take it to go 84
feet, I guess is the best way

00:10:24.620 --> 00:10:25.540
to think about it.

00:10:25.540 --> 00:10:31.490
So we say 84 is equal to
velocity times-- your initial

00:10:31.490 --> 00:10:32.870
velocity times time.

00:10:32.870 --> 00:10:37.090
And your initial velocity
is 10 feet per second.

00:10:37.090 --> 00:10:38.710
So it's 10 feet per second.

00:10:38.710 --> 00:10:40.150
Everything is in feet I think.

00:10:40.150 --> 00:10:42.010
Right, everything is--
v feet per second.

00:10:42.010 --> 00:10:45.110
Initial velocity of
v feet per second.

00:10:45.110 --> 00:10:47.100
So 10 feet per second
times time.

00:10:47.100 --> 00:10:48.870
I just substituted what
they gave us.

00:10:48.870 --> 00:10:50.670
10 for v.

00:10:50.670 --> 00:10:53.920
Plus 16t squared.

00:10:53.920 --> 00:10:56.590
And now I just solve
this quadratic.

00:10:56.590 --> 00:10:58.070
That is a t right there.

00:10:58.070 --> 00:11:00.450
So let me put everything on the
same-- let me subtract 84

00:11:00.450 --> 00:11:02.080
from both sides and I'll
rearrange a little bit.

00:11:02.080 --> 00:11:11.700
So you get 16t squared plus 10t
minus 84 is equal to 0.

00:11:11.700 --> 00:11:13.720
Well I swapped the sides.

00:11:13.720 --> 00:11:14.940
I put these on the left.

00:11:14.940 --> 00:11:16.730
Well, let me just show
you what I did.

00:11:16.730 --> 00:11:17.710
I swapped the sides.

00:11:17.710 --> 00:11:23.180
So I made this 16t squared
plus 10t equals 84.

00:11:23.180 --> 00:11:24.110
I just swapped them.

00:11:24.110 --> 00:11:26.910
And then I subtracted 84 from
both sides to get this.

00:11:26.910 --> 00:11:29.650
And now we just have to
solve when t equals 0.

00:11:29.650 --> 00:11:32.190
So I guess the first thing we
could do is we could simplify

00:11:32.190 --> 00:11:32.840
this a little bit.

00:11:32.840 --> 00:11:35.080
Everything here is
divisible by 2.

00:11:35.080 --> 00:11:39.886
So this is 8t squared plus-- I'm
just dividing both sides

00:11:39.886 --> 00:11:41.150
of this equation by 2.

00:11:41.150 --> 00:11:43.930
Plus 5t minus what?

00:11:43.930 --> 00:11:48.670
Minus 42 is equal to 0.

00:11:48.670 --> 00:11:51.200
And then we can use the
quadratic equation.

00:11:51.200 --> 00:11:55.200
So what are the solutions? t
is equal to negatives b.

00:11:55.200 --> 00:12:02.980
So minus 5 plus or minus the
square root of b squared, so

00:12:02.980 --> 00:12:07.880
25, minus 4 times a.

00:12:07.880 --> 00:12:11.360
times 8, times c.
c is minus 42.

00:12:11.360 --> 00:12:13.720
So instead of times minus
42, let's put a plus

00:12:13.720 --> 00:12:16.540
here and do plus 42.

00:12:16.540 --> 00:12:18.610
Just a negative times a negative
is a positive.

00:12:18.610 --> 00:12:22.540
All of that, over 2 times a.

00:12:22.540 --> 00:12:24.652
2a is 16.

00:12:24.652 --> 00:12:26.150
So let's see where
that gets me.

00:12:26.150 --> 00:12:32.030
So I t is equal to minus 5 plus
or minus the square root.

00:12:32.030 --> 00:12:32.550
What is this?

00:12:32.550 --> 00:12:35.330
25 plus-- let's see.

00:12:35.330 --> 00:12:39.440
4 times 8 times 42.

00:12:39.440 --> 00:12:41.580
That's 32 times 42.

00:12:41.580 --> 00:12:46.840
32 times 42.

00:12:46.840 --> 00:12:49.590
2 times 32 is 64.

00:12:49.590 --> 00:12:50.340
Put a 0.

00:12:50.340 --> 00:12:52.250
4 times 2 is 8.

00:12:52.250 --> 00:12:56.970
4 times 3 is 12.

00:12:56.970 --> 00:13:05.810
You end up with 4,
14, 3 and 1.

00:13:05.810 --> 00:13:06.880
So this is 1,344.

00:13:06.880 --> 00:13:08.530
And we're going to
add this 25 here.

00:13:08.530 --> 00:13:09.200
So let me see.

00:13:09.200 --> 00:13:11.620
Plus 1,344.

00:13:11.620 --> 00:13:13.800
All of that over 16.

00:13:13.800 --> 00:13:13.970
Let's see.

00:13:13.970 --> 00:13:15.640
1,344 plus 25.

00:13:15.640 --> 00:13:20.280
So it's minus 5 plus or
minus the square root

00:13:20.280 --> 00:13:24.370
of-- what is this?

00:13:24.370 --> 00:13:29.150
1,369 over 16.

00:13:29.150 --> 00:13:30.230
And actually, I don't
know what the square

00:13:30.230 --> 00:13:31.430
root of 1369 is.

00:13:31.430 --> 00:13:34.040
Let me get the calculator.

00:13:34.040 --> 00:13:37.040
Let me open it up.

00:13:37.040 --> 00:13:38.060
Give me one second.

00:13:38.060 --> 00:13:42.540
Accessories, calculator.

00:13:42.540 --> 00:13:43.510
All right.

00:13:43.510 --> 00:13:48.630
So 1,369.

00:13:48.630 --> 00:13:49.410
37.

00:13:49.410 --> 00:13:50.390
Look at that.

00:13:50.390 --> 00:13:53.440
OK, so it's minus 5
plus or minus 37.

00:13:53.440 --> 00:13:55.440
That's the square
root of 1,369.

00:13:55.440 --> 00:14:01.600
So minus 5 plus or minus 37 over
16 is equal to the time.

00:14:01.600 --> 00:14:03.020
Now we don't have to worry about
the minus because that's

00:14:03.020 --> 00:14:03.950
going to give us a
negative number.

00:14:03.950 --> 00:14:06.110
Minus 5 minus 37 over 16.

00:14:06.110 --> 00:14:08.780
We don't want a negative time,
we want a positive time.

00:14:08.780 --> 00:14:10.080
So let's just do the positive.

00:14:10.080 --> 00:14:13.240
So minus 5 plus 37.

00:14:13.240 --> 00:14:13.450
Let's see.

00:14:13.450 --> 00:14:16.830
Minus 5 plus 37 over 16.

00:14:16.830 --> 00:14:21.340
So that's 32/16, which
equals to 2 seconds.

00:14:21.340 --> 00:14:23.290
And that's choice A.

00:14:23.290 --> 00:14:25.590
Anyway, see you in
the next video.


CA Algebra I: Rational Expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.170
We're on problem 66.

00:00:03.170 --> 00:00:10.870
And it says what is x squared
minus 4x plus 4, divided by x

00:00:10.870 --> 00:00:16.070
squared minus 3x plus 2, reduced
to lowest terms?

00:00:16.070 --> 00:00:18.280
So they probably want us
to factor each of these

00:00:18.280 --> 00:00:20.610
quadratics and see if any of
these terms cancel out.

00:00:20.610 --> 00:00:22.060
So let's try to do that.

00:00:22.060 --> 00:00:25.550
So the numerator, this seems
pretty easy to factor.

00:00:25.550 --> 00:00:28.100
What two numbers when I
multiply them equal 4?

00:00:28.100 --> 00:00:29.910
And when I add them
equal minus 4?

00:00:29.910 --> 00:00:31.010
Well it's minus 2, right?

00:00:31.010 --> 00:00:33.140
Minus 2 and minus
2 is minus 4.

00:00:33.140 --> 00:00:34.930
Minus 2 squared is plus 4.

00:00:34.930 --> 00:00:41.380
So this is x minus 2
times x minus 2.

00:00:41.380 --> 00:00:42.820
And you could test it if
you don't believe it.

00:00:42.820 --> 00:00:44.390
Multiply that out.

00:00:44.390 --> 00:00:47.580
Divided by, let's see,
what two numbers?

00:00:47.580 --> 00:00:49.250
This looks factorable.

00:00:49.250 --> 00:00:51.490
They both have to be the same
sign because when you multiply

00:00:51.490 --> 00:00:53.640
them you get a positive.

00:00:53.640 --> 00:00:55.530
And they're both going to be
negative, because when you add

00:00:55.530 --> 00:00:56.630
them you get a negative 3.

00:00:56.630 --> 00:00:58.910
So let's see, minus
2 and minus 1.

00:00:58.910 --> 00:01:01.390
Minus 2 times minus
1 is positive 2.

00:01:01.390 --> 00:01:03.740
Minus 2 plus minus
1 is minus 3.

00:01:03.740 --> 00:01:09.140
So x minus 2, times x minus 1.

00:01:09.140 --> 00:01:11.810
And if we assume that x is never
equal to 2, because that

00:01:11.810 --> 00:01:15.250
would make this expression
undefined, we cancel that out.

00:01:15.250 --> 00:01:17.490
You'll learn later that would
cause a hole in the graph,

00:01:17.490 --> 00:01:19.240
because the function
is undefined there.

00:01:19.240 --> 00:01:23.660
And you're left with minus
2 over minus 1.

00:01:23.660 --> 00:01:26.225
And that is choice A.

00:01:28.820 --> 00:01:31.880
Problem 67.

00:01:31.880 --> 00:01:32.740
This is good practice.

00:01:32.740 --> 00:01:33.820
They give a bunch of it.

00:01:33.820 --> 00:01:42.250
They say what is-- I'll just
write it-- 12a cubed minus 20a

00:01:42.250 --> 00:01:49.550
squared over 16a squared
plus 8a.

00:01:49.550 --> 00:01:53.300
Reduce to lowest terms. So let's
just try to factor out

00:01:53.300 --> 00:01:55.270
things on the top and the bottom
and see what happens.

00:01:55.270 --> 00:01:57.880
So in the top, in the
numerator-- let me switch

00:01:57.880 --> 00:02:02.980
colors-- both terms are
divisible by 4 and a squared.

00:02:02.980 --> 00:02:04.670
So let's fact out
a 4a squared.

00:02:04.670 --> 00:02:07.274
So we get 4a squared.

00:02:09.990 --> 00:02:12.490
12 divided by 4 is a 3.

00:02:12.490 --> 00:02:15.210
And a cubed divided by
a squared is an a.

00:02:15.210 --> 00:02:19.240
So 12a cubed divided by
4a squared is 3a.

00:02:19.240 --> 00:02:22.620
Minus 20-- I could say
plus minus 20--

00:02:22.620 --> 00:02:23.750
but you get the idea.

00:02:23.750 --> 00:02:26.440
20 divided by 4 is 5.

00:02:26.440 --> 00:02:29.500
And a squared divided by
a squared is just a.

00:02:29.500 --> 00:02:31.660
And if you don't believe
this, multiply it out.

00:02:31.660 --> 00:02:34.850
4a squared times 3a
is 12a cubed.

00:02:34.850 --> 00:02:38.290
And 4a squared times minus
5 is minus 20a squared.

00:02:38.290 --> 00:02:39.950
So it works out.

00:02:39.950 --> 00:02:40.690
You do the denominator.

00:02:40.690 --> 00:02:43.660
Let's see, both of these are
divisible by 8a, so let's

00:02:43.660 --> 00:02:46.690
factor that out.

00:02:46.690 --> 00:02:48.990
16 divided by 8 is 2.

00:02:48.990 --> 00:02:52.950
a squared divided by a is a.

00:02:52.950 --> 00:02:55.260
So 16a squared divided
by 8s is 2a.

00:02:55.260 --> 00:02:57.540
And if you go the other
way, 8a times 2a

00:02:57.540 --> 00:02:58.490
squared is 16a squared.

00:02:58.490 --> 00:02:59.960
So it all works out.

00:02:59.960 --> 00:03:02.110
Plus 1.

00:03:02.110 --> 00:03:04.150
8a times 1 is 8a.

00:03:04.150 --> 00:03:06.750
So let's see what we
could do here.

00:03:06.750 --> 00:03:08.310
This is becomes a 1.

00:03:08.310 --> 00:03:09.820
This becomes a 2.

00:03:09.820 --> 00:03:15.090
And a squared divided by a, this
becomes a 1 and this just

00:03:15.090 --> 00:03:16.340
becomes just an a.

00:03:16.340 --> 00:03:23.330
And we're left with a times
3a minus 5, over 2

00:03:23.330 --> 00:03:26.290
times 2a plus 1.

00:03:26.290 --> 00:03:26.920
And let's see.

00:03:26.920 --> 00:03:30.520
That is choice D.

00:03:30.520 --> 00:03:32.870
I thought maybe they'd want us
to re-multiply this out again.

00:03:32.870 --> 00:03:35.350
But that is choice D.

00:03:35.350 --> 00:03:44.240
Problem 68.

00:03:44.240 --> 00:03:46.280
Oh this is a good one.

00:03:46.280 --> 00:03:47.160
I'll just write it.

00:03:47.160 --> 00:03:48.490
They want us to multiply
something.

00:03:48.490 --> 00:04:00.570
So they say 7z squared plus
7z-- all of that--

00:04:00.570 --> 00:04:04.650
over 4z plus 8.

00:04:04.650 --> 00:04:11.840
Times z squared minus 4-- all of
that-- over z to the third

00:04:11.840 --> 00:04:16.649
plus 2z squared plus z equals.

00:04:16.649 --> 00:04:18.250
So you must be like oh my god,
I have to multiply all of

00:04:18.250 --> 00:04:19.430
these things and I have
to divide them.

00:04:19.430 --> 00:04:21.640
But the best thing, I'm
guessing, is to just factor

00:04:21.640 --> 00:04:24.070
these out and all sorts of
things will start canceling

00:04:24.070 --> 00:04:24.670
out with each other.

00:04:24.670 --> 00:04:26.880
And it will turn into a
pretty simple problem.

00:04:26.880 --> 00:04:29.280
Let's see, both of these terms
are divisible by 7z.

00:04:29.280 --> 00:04:30.960
So le'ts factor that out.

00:04:30.960 --> 00:04:37.110
So that top part becomes, 7z
squared divided by 7z, you

00:04:37.110 --> 00:04:38.930
just have a z left.

00:04:38.930 --> 00:04:41.520
If multiply these, you
get 7z squared.

00:04:41.520 --> 00:04:43.950
Plus 1.

00:04:43.950 --> 00:04:46.200
If you multiply this out, you
get 7z squared plus 7z.

00:04:50.450 --> 00:04:52.220
When you multiply fractions,
it's just the numerator times

00:04:52.220 --> 00:04:55.300
the numerator, over the
denominator times the

00:04:55.300 --> 00:04:56.570
denominator.

00:04:56.570 --> 00:04:58.840
So this is times
the numerator.

00:04:58.840 --> 00:05:02.020
Z squared minus 4, that's a
squared minus b squared.

00:05:02.020 --> 00:05:07.970
So that's z plus 2, a plus b,
times z minus 2, a minus b.

00:05:07.970 --> 00:05:10.630
That's just the pattern when I
say all those a's and b's.

00:05:10.630 --> 00:05:12.860
So that's z plus 2 times z minus
2, hopefully you can

00:05:12.860 --> 00:05:14.660
recognize that at this point.

00:05:14.660 --> 00:05:17.930
And then all of that over--
let's see, we can definitely

00:05:17.930 --> 00:05:24.470
factor out a 4 here, so that's
4 times z plus 2.

00:05:24.470 --> 00:05:30.390
8 divided by 4 is 2 times-- so
we can definitely factor out a

00:05:30.390 --> 00:05:39.020
z here, so we get z times z
squared plus 2z plus 1.

00:05:39.020 --> 00:05:40.200
I think we're almost done.

00:05:40.200 --> 00:05:41.580
Now we have to factor this.

00:05:41.580 --> 00:05:42.840
Let me just rewrite
everything.

00:05:42.840 --> 00:05:50.520
So this is equal to 7z times
z plus 1, times z plus 2,

00:05:50.520 --> 00:05:53.140
times z minus 6.

00:05:53.140 --> 00:06:02.400
All of that over 4 times
z plus 2, times z.

00:06:02.400 --> 00:06:03.130
And what's this?

00:06:03.130 --> 00:06:05.650
This is z plus 1 squared.

00:06:05.650 --> 00:06:07.720
Z plus 1 times z plus
1, 1 times 1 is 1,

00:06:07.720 --> 00:06:09.790
and 1 plus 1 is 2.

00:06:09.790 --> 00:06:14.540
So times z plus 1,
times z plus 1.

00:06:14.540 --> 00:06:16.100
And now is fun part.

00:06:16.100 --> 00:06:18.350
This is a 1 here, that's
parentheses.

00:06:18.350 --> 00:06:19.900
Now we can start
canceling out.

00:06:19.900 --> 00:06:21.860
And we assume that the
denominator would never equal

00:06:21.860 --> 00:06:23.120
0 and all that.

00:06:23.120 --> 00:06:25.400
Let's see, this z plus
2 cancels out

00:06:25.400 --> 00:06:27.370
with this z plus 2.

00:06:27.370 --> 00:06:31.030
This z plus 1 cancels out with
one of these z plus 1's.

00:06:31.030 --> 00:06:33.410
I'll do the one that's
written messier.

00:06:33.410 --> 00:06:37.310
And let's see, this z cancels
out with this z.

00:06:37.310 --> 00:06:38.920
And what are we left with?

00:06:38.920 --> 00:06:46.020
Everything simplified to 7
times z minus 6 over 4

00:06:46.020 --> 00:06:48.210
times z plus 1.

00:06:59.500 --> 00:07:00.740
I wrote a z minus b here.

00:07:00.740 --> 00:07:03.320
It's z plus 2 times z minus 2.

00:07:03.320 --> 00:07:05.090
All that pattern matching,
I made a mistake.

00:07:05.090 --> 00:07:08.620
Z squared minus 4 is z plus
2 times z minus 2.

00:07:08.620 --> 00:07:10.690
Not z minus b, and I though
that was a 6.

00:07:10.690 --> 00:07:13.790
So this is z minus 2.

00:07:13.790 --> 00:07:15.920
So this is z minus 2.

00:07:15.920 --> 00:07:19.550
And so that is choice A.

00:07:19.550 --> 00:07:22.640
Sorry about that error.

00:07:22.640 --> 00:07:26.760
Brain malfunctions
all the time.

00:07:26.760 --> 00:07:29.220
All right, now they want
us to do it again.

00:07:29.220 --> 00:07:36.790
They want us to find the product
of x plus 5, over 3x

00:07:36.790 --> 00:07:44.940
plus 2, times 2x minus
3, over x minus 5.

00:07:44.940 --> 00:07:46.780
Frankly, there's not a lot of
simplification we can do, we

00:07:46.780 --> 00:07:47.740
just have to multiply it out.

00:07:47.740 --> 00:07:53.765
So this is going to be equal to
x plus 5 times 2x minus 3.

00:07:53.765 --> 00:07:56.240
All of that over 3x--

00:07:56.240 --> 00:07:59.240
I'm just multiplying the
numerator and then multiplying

00:07:59.240 --> 00:08:03.880
the denominators-- 3x plus
2 times x minus 5.

00:08:03.880 --> 00:08:08.040
And now we just multiply
both binomials, x

00:08:08.040 --> 00:08:11.470
times 2x, 2x squared.

00:08:11.470 --> 00:08:14.780
X times minus 3, minus 3x.

00:08:14.780 --> 00:08:18.820
5 times 2x, plus 10x.

00:08:18.820 --> 00:08:22.730
5 times minus 3, minus 15.

00:08:22.730 --> 00:08:23.290
Fair enough.

00:08:23.290 --> 00:08:24.750
Now you do the denominator.

00:08:24.750 --> 00:08:28.950
3x times x is 3x squared.

00:08:28.950 --> 00:08:33.990
3x times minus 5, minus 15x.

00:08:33.990 --> 00:08:37.520
2 times x, plus 2x.

00:08:37.520 --> 00:08:41.460
2 times minus 5, minus 10.

00:08:41.460 --> 00:08:43.000
And now let's see if
we can simplify.

00:08:43.000 --> 00:08:46.690
We have the numerator is
equal to 2x squared

00:08:46.690 --> 00:08:48.410
minus 3x plus 10x.

00:08:48.410 --> 00:08:53.150
So that's plus 7x minus 15.

00:08:53.150 --> 00:08:56.360
All of that over 3x squared.

00:08:56.360 --> 00:09:00.060
And minus 15x plus 2x.

00:09:00.060 --> 00:09:05.890
That's minus 13x minus 10.

00:09:05.890 --> 00:09:07.790
And that is choice D.

00:09:10.550 --> 00:09:13.260
Next problem.

00:09:13.260 --> 00:09:14.290
Problem 70.

00:09:14.290 --> 00:09:17.110
Boy, they they want us
to keep this up.

00:09:17.110 --> 00:09:18.480
This is good practice.

00:09:18.480 --> 00:09:30.540
So they write, x squared plus
8x plus 16, over x plus 3,

00:09:30.540 --> 00:09:41.482
divided by 2x plus 8, over
x squared minus 9.

00:09:41.482 --> 00:09:44.190
So the first thing you do, when
you divide by a fraction,

00:09:44.190 --> 00:09:46.200
it's the same thing is
multiplying by its inverse.

00:09:46.200 --> 00:09:55.210
So this is equal to x squared
plus 8x plus 16, over x plus 3

00:09:55.210 --> 00:09:59.650
times the inverse of this,
x squared minus 9,

00:09:59.650 --> 00:10:02.280
over 2x plus 8.

00:10:02.280 --> 00:10:02.850
Fair enough.

00:10:02.850 --> 00:10:04.870
Now let's see if we can simplify
these a little bit.

00:10:04.870 --> 00:10:06.810
I'll do that in yellow.

00:10:06.810 --> 00:10:10.980
So this is, 4 plus 4 is
8, 4 times 4 is 16.

00:10:10.980 --> 00:10:17.190
So this we can rewrite as
x plus 4 times x plus 4.

00:10:20.550 --> 00:10:23.680
x squared minus 9, that's a
squared minus b squared.

00:10:23.680 --> 00:10:29.720
So this we can rewrite as x
plus 3 times x minus 3.

00:10:29.720 --> 00:10:31.070
It's going with the pattern.

00:10:31.070 --> 00:10:33.880
We can factor out a 2 here, so
we can rewrite this as 2

00:10:33.880 --> 00:10:37.240
times x plus 4.

00:10:37.240 --> 00:10:38.510
We have an x plus 3 there.

00:10:38.510 --> 00:10:40.440
And of course, when we multiply
fractions, we're just

00:10:40.440 --> 00:10:42.770
multiplying all the numerators
over all the denominators.

00:10:42.770 --> 00:10:46.480
So it's almost like you
make this one line.

00:10:46.480 --> 00:10:49.270
So the numerator is x plus 4
times x plus 4 times x plus 3

00:10:49.270 --> 00:10:50.520
times x minus 3.

00:10:50.520 --> 00:10:54.580
All of that over x plus 3
times 2 times x plus 4.

00:10:54.580 --> 00:10:55.790
So now let's do some
cancellation.

00:10:55.790 --> 00:10:56.730
This is the fun part.

00:10:56.730 --> 00:10:59.030
So we have an x plus 4 and and
x plus 4, cancel them out.

00:11:01.710 --> 00:11:04.535
We have an x plus 3 and an x
plus 3, cancel them out.

00:11:07.070 --> 00:11:10.220
And what are we left with?

00:11:10.220 --> 00:11:18.020
We are left with an x plus
4 times an x minus 3.

00:11:18.020 --> 00:11:20.390
All of that over 2.

00:11:20.390 --> 00:11:24.320
And that is choice C.

00:11:24.320 --> 00:11:26.930
And I will see you in
the next video.


CA Algebra I: Rational Expressions 1

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.280
We are on problem 63.

00:00:03.280 --> 00:00:07.140
The height of a triangle
is 4 inches greater

00:00:07.140 --> 00:00:08.580
than twice its base.

00:00:08.580 --> 00:00:10.215
Let me this triangle
in question.

00:00:12.900 --> 00:00:17.300
That's the triangle, that's its
height, that's its base.

00:00:20.400 --> 00:00:21.530
Let's say this is the base.

00:00:21.530 --> 00:00:22.970
Let's call that b.

00:00:22.970 --> 00:00:28.030
So then the height, that's that,
is 4 inches greater than

00:00:28.030 --> 00:00:29.070
twice its base.

00:00:29.070 --> 00:00:33.450
So 4 inches more than
2 times the base.

00:00:33.450 --> 00:00:34.440
Fair enough.

00:00:34.440 --> 00:00:36.980
The area of the triangle
is 168 square inches.

00:00:36.980 --> 00:00:38.920
What is the base of
the triangle?

00:00:38.920 --> 00:00:39.900
So what's the area
of a triangle?

00:00:39.900 --> 00:00:43.640
It's the base times the
height times 1/2.

00:00:43.640 --> 00:00:49.080
So area is equal to 1/2
base times height.

00:00:49.080 --> 00:00:51.660
Well they told us that the
height is four more than twice

00:00:51.660 --> 00:00:53.710
the base, so we can just
substitute that in.

00:00:53.710 --> 00:00:57.385
And then we'll have a just
in terms of the base.

00:00:57.385 --> 00:00:58.510
Let me keep that there.

00:00:58.510 --> 00:01:02.920
So the area is equal to 1/2
the base times the height,

00:01:02.920 --> 00:01:08.210
which they told us is 4 more
than twice the base.

00:01:08.210 --> 00:01:12.710
And now if we distribute
the 1/2 b.

00:01:12.710 --> 00:01:16.240
1/2 b times 4 is equal to 2b.

00:01:16.240 --> 00:01:21.500
And then 1/2 b times 2b,
that's plus b squared.

00:01:21.500 --> 00:01:23.890
1/2 time 2 is 1 and then
b times b is b squared.

00:01:23.890 --> 00:01:24.590
Fair enough.

00:01:24.590 --> 00:01:27.780
And they tell us that
the area is 168.

00:01:27.780 --> 00:01:33.160
OK, so we have area is 168, and
that's equal to 2 times

00:01:33.160 --> 00:01:36.110
the base plus the
base sqaured.

00:01:36.110 --> 00:01:38.110
Let's swap the sides just
to make it look

00:01:38.110 --> 00:01:38.980
a little bit familiar.

00:01:38.980 --> 00:01:43.220
B squared plus 2 times the
base is equal to 168.

00:01:43.220 --> 00:01:45.650
We could subtract 168
from both sides.

00:01:45.650 --> 00:01:54.850
You get b squared plus 2b
minus 168 is equal to 0.

00:01:54.850 --> 00:01:56.730
And 168 is a bit of
a strange number.

00:01:56.730 --> 00:02:00.060
I don't know exactly which
numbers go into 168.

00:02:00.060 --> 00:02:07.770
Let's see, 4 isn't
going to work.

00:02:13.060 --> 00:02:14.770
Does 12 go into it?

00:02:14.770 --> 00:02:16.970
Yeah, 12 goes into 168,
and I think it

00:02:16.970 --> 00:02:19.140
goes into it 14 times.

00:02:19.140 --> 00:02:20.200
So that works.

00:02:20.200 --> 00:02:22.820
So you could use the quadratic
equation to solve this.

00:02:22.820 --> 00:02:26.490
But I think that 12
times 14 is 168.

00:02:26.490 --> 00:02:28.190
And they have to be of different
signs, since this is

00:02:28.190 --> 00:02:29.610
a negative.

00:02:29.610 --> 00:02:31.290
Only different signs
when multiplied get

00:02:31.290 --> 00:02:32.220
you a negative number.

00:02:32.220 --> 00:02:33.790
And they're 2 apart.

00:02:33.790 --> 00:02:34.510
So that works out.

00:02:34.510 --> 00:02:42.840
So you get b plus 14 times
b minus 12 is equal to 0.

00:02:42.840 --> 00:02:45.610
Let me just confirm and
not waste your time.

00:02:45.610 --> 00:02:48.600
14 times 12, I'm pretty
sure I'm right.

00:02:48.600 --> 00:02:50.570
Yeah, 168.

00:02:50.570 --> 00:02:51.630
I just factored that out.

00:02:51.630 --> 00:02:54.260
So b could be equal
to minus 14.

00:02:54.260 --> 00:02:56.360
Right, that would make that 0.

00:02:56.360 --> 00:02:59.470
Or b could be equal to 12,
which would make that 0.

00:02:59.470 --> 00:03:01.390
Obviously if you can make one
of these terms 0, you're

00:03:01.390 --> 00:03:03.190
making this whole thing 0.

00:03:03.190 --> 00:03:06.200
And the base, since we're
dealing with distances or

00:03:06.200 --> 00:03:07.960
lengths, we can't have
a negative length.

00:03:07.960 --> 00:03:11.250
So the base has to
be 12 inches.

00:03:11.250 --> 00:03:12.805
Which is choice C in
that question.

00:03:16.140 --> 00:03:17.840
Next question.

00:03:17.840 --> 00:03:23.550
Problem 64.

00:03:23.550 --> 00:03:24.800
All right.

00:03:26.960 --> 00:03:29.920
This is a good one.

00:03:29.920 --> 00:03:33.620
What is that thing reduced
to lowest terms?

00:03:33.620 --> 00:03:35.750
So essentially what they want
us to do is factor stuff out

00:03:35.750 --> 00:03:39.390
and factor it out and write it
as a product of factors and

00:03:39.390 --> 00:03:40.410
then cancel out the ones.

00:03:40.410 --> 00:03:41.320
So let's see what we can do.

00:03:41.320 --> 00:03:44.030
Can we factor that
top expression?

00:03:44.030 --> 00:03:44.300
Let's see.

00:03:44.300 --> 00:03:46.340
This is looks like a perfect
square, right?

00:03:46.340 --> 00:03:51.430
Because if you have minus 2y
squared is 4y squared.

00:03:51.430 --> 00:03:54.020
And minus 2y and minus
2y is minus 4y.

00:03:54.020 --> 00:04:01.150
So the top expression is
x minus 2y squared.

00:04:01.150 --> 00:04:02.060
And I'll just write it twice.

00:04:02.060 --> 00:04:04.360
Because I'm guessing that one
of them are going to have to

00:04:04.360 --> 00:04:07.050
be factored out.

00:04:07.050 --> 00:04:12.520
And once, the intuition is that
if I had this, x squared

00:04:12.520 --> 00:04:17.380
minus 4x plus 4, you say OK,
what two numbers when you

00:04:17.380 --> 00:04:19.204
multiply them equal four
and when you add

00:04:19.204 --> 00:04:20.170
them equal minus 4?

00:04:20.170 --> 00:04:22.250
You say, minus 2 and you'd so
that's equal to x minus

00:04:22.250 --> 00:04:25.720
2 and x minus 2.

00:04:25.720 --> 00:04:27.500
The only difference between
that and that is

00:04:27.500 --> 00:04:29.560
they put y's here.

00:04:29.560 --> 00:04:31.540
Y and a y squared.

00:04:31.540 --> 00:04:37.690
OK, so you could visualize this
thing as x squared minus

00:04:37.690 --> 00:04:45.810
4yx plus 4y squared.

00:04:45.810 --> 00:04:52.000
Or you could visualize this as x
squared minus two times 2y x

00:04:52.000 --> 00:04:56.630
plus 2y squared.

00:04:56.630 --> 00:04:57.450
This isn't 4y squared.

00:04:57.450 --> 00:04:59.410
This is 4 y squared.

00:04:59.410 --> 00:05:00.600
Forget the parentheses.

00:05:00.600 --> 00:05:03.540
But I hope you can see that this
thing up here is the same

00:05:03.540 --> 00:05:05.940
thing is this thing
right here.

00:05:05.940 --> 00:05:07.910
And that's where I got the
intuition that that's going to

00:05:07.910 --> 00:05:10.900
be x minus 2y times
x minus 2y.

00:05:10.900 --> 00:05:12.970
You say, OK what two numbers
when you add

00:05:12.970 --> 00:05:14.740
them are 2 times 2y?

00:05:14.740 --> 00:05:16.410
Well, obviously 2y.

00:05:16.410 --> 00:05:19.130
And what numbers when you
squared them are 2y squared?

00:05:19.130 --> 00:05:21.120
Well 2y times 2y
is 2y squared.

00:05:21.120 --> 00:05:23.660
So that's where you get the
intuition that it's x minus 2y

00:05:23.660 --> 00:05:25.490
times x minus 2y.

00:05:25.490 --> 00:05:27.130
That's the numerator, let's see
what we can do with the

00:05:27.130 --> 00:05:29.910
denominator.

00:05:29.910 --> 00:05:34.230
Well immediately, both factors
are divisible by 3 and both

00:05:34.230 --> 00:05:36.040
terms are divisible by y.

00:05:36.040 --> 00:05:36.980
So let's factor those out.

00:05:36.980 --> 00:05:39.781
So this is equal to 3y.

00:05:39.781 --> 00:05:42.640
Now if you divide this term
by 3y, all you're

00:05:42.640 --> 00:05:43.880
left with is an x.

00:05:43.880 --> 00:05:46.780
3xy divided by 3y, you
just have an x.

00:05:46.780 --> 00:05:53.120
And then if you divide minus
6y squared by 3y, minus six

00:05:53.120 --> 00:05:58.470
divided by 3 is minus 2, y
squared divided by y is y.

00:05:58.470 --> 00:06:02.180
Let's see, if we assume that x
does not equal 2y, because

00:06:02.180 --> 00:06:05.940
then this would be undefined,
we can cancel these two out.

00:06:05.940 --> 00:06:11.680
And you're just left with
x minus 2y over 3y.

00:06:11.680 --> 00:06:15.020
And that is choice B.

00:06:15.020 --> 00:06:19.710
Problem 65.

00:06:19.710 --> 00:06:20.960
They want us to do it again.

00:06:23.910 --> 00:06:25.290
OK, let's see.

00:06:29.550 --> 00:06:33.110
So this top expression,
everything is divisible by 3,

00:06:33.110 --> 00:06:35.070
so let's rewrite it.

00:06:35.070 --> 00:06:36.950
Let's factor out a 3 there.

00:06:36.950 --> 00:06:42.620
So it'd be 3 times to 2x
squared, plus 7x-- or 21

00:06:42.620 --> 00:06:46.540
divided by 3-- Plus 3.

00:06:46.540 --> 00:06:48.480
Let's see, what's
the bottom one?

00:06:48.480 --> 00:06:49.520
Well this fits a pattern.

00:06:49.520 --> 00:06:53.190
A squared minus b
squared, right?

00:06:53.190 --> 00:06:54.130
This is something squared.

00:06:54.130 --> 00:06:56.030
4x squared is 2x squared.

00:06:56.030 --> 00:06:57.980
And then 1 is obviously
1 squared.

00:06:57.980 --> 00:07:03.950
So we can write this as a
plus b times a minus b.

00:07:03.950 --> 00:07:10.550
And that would be 2x plus
1, times 2x minus 1.

00:07:13.800 --> 00:07:16.800
2x squared is 4x squared,
and 1 squared is 1.

00:07:16.800 --> 00:07:18.960
So let's see, we factored
that top part.

00:07:18.960 --> 00:07:20.860
And I'm guessing it's going to
cancel out with something in

00:07:20.860 --> 00:07:21.450
the denominator.

00:07:21.450 --> 00:07:23.890
So let's see what we can get.

00:07:23.890 --> 00:07:26.251
Let me do it on the side here.

00:07:26.251 --> 00:07:33.620
So if I have to factor 2x
squared plus 7x plus 3, it's

00:07:33.620 --> 00:07:40.815
going to be 2x plus something
times x plus something.

00:07:43.460 --> 00:07:46.610
Because 2x times the
x is 2x squared.

00:07:46.610 --> 00:07:51.070
And the b times the a is
going to be equal to 3.

00:07:51.070 --> 00:07:53.960
So I'm guessing one of them is
going to be a 3 and one of

00:07:53.960 --> 00:07:57.200
them is going to be 1.

00:07:57.200 --> 00:07:58.610
And I could already guess, if
we could do it a little bit

00:07:58.610 --> 00:08:00.440
more systematically than
how it's going, so

00:08:00.440 --> 00:08:01.210
let's multiply it out.

00:08:01.210 --> 00:08:05.690
2x times x, 2x squared.

00:08:05.690 --> 00:08:08.840
2x times a, plus 2ax.

00:08:08.840 --> 00:08:11.120
B times x, plus bx.

00:08:11.120 --> 00:08:13.520
B times a, plus ba.

00:08:13.520 --> 00:08:17.360
So you get 2x squared, then
merge these terms, add the

00:08:17.360 --> 00:08:18.040
coefficients.

00:08:18.040 --> 00:08:26.660
Plus 2a plus bx, that's those
two terms. Plus ba is equal to

00:08:26.660 --> 00:08:28.610
this thing up here.

00:08:28.610 --> 00:08:31.090
I'll write it right below it so
we can compare terms. It's

00:08:31.090 --> 00:08:37.600
equal to 2x squared
plus 7x plus 3.

00:08:37.600 --> 00:08:39.600
So what could a and b equal?

00:08:39.600 --> 00:08:45.900
Let's see, 2 times a plus b is
equal to-- so if a is 3 and

00:08:45.900 --> 00:08:50.630
this is 1, 2 times 3 is
6 plus 1 is 7, works.

00:08:50.630 --> 00:08:53.980
And then 3 times 1 is 7.

00:08:53.980 --> 00:08:57.746
So b is 1 and a is 3.

00:09:02.160 --> 00:09:04.690
And you could multiply that
out if you're not

00:09:04.690 --> 00:09:05.430
comfortable with it.

00:09:05.430 --> 00:09:09.190
But anyway, we could rewrite
this top thing as 3 times--

00:09:09.190 --> 00:09:11.690
and we did the factoring here,
we figured out that b and a

00:09:11.690 --> 00:09:18.990
are 2x plus 1, times x plus 3.

00:09:18.990 --> 00:09:24.620
All of that over 2x plus
1, times 2x minus 1.

00:09:24.620 --> 00:09:26.160
And we know we're on the right
track because we've got

00:09:26.160 --> 00:09:27.010
something to cancel out.

00:09:27.010 --> 00:09:29.860
And sure enough, we
cancel those out.

00:09:29.860 --> 00:09:31.800
Assuming that this
never equals 0.

00:09:31.800 --> 00:09:34.600
This x does not equal minus 1/2,
because that would make

00:09:34.600 --> 00:09:35.630
that equal 0.

00:09:35.630 --> 00:09:43.510
And we get 3 times x plus
3 over 2x minus 1.

00:09:43.510 --> 00:09:48.500
And that is choice B.

00:09:48.500 --> 00:09:50.180
Let me see the next problem,
see if we have time

00:09:50.180 --> 00:09:52.620
to do it right now.

00:09:52.620 --> 00:09:53.510
Oh, it's another one of these.

00:09:53.510 --> 00:09:56.170
I'll wait for the next
video, see you soon.