Absolute value equations

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.880
Let's do some equations that
deal with absolute values.

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And just as a bit of a review,
when you take the absolute

00:00:06.260 --> 00:00:07.650
value of a number.

00:00:07.650 --> 00:00:10.680
Let's say I take the absolute
value of negative 1.

00:00:10.680 --> 00:00:13.700
What you're really doing is
you're saying, how far is that

00:00:13.700 --> 00:00:16.090
number from 0?

00:00:16.090 --> 00:00:20.620
And in the case of negative 1,
if we draw a number line right

00:00:20.620 --> 00:00:23.310
there-- that's a very badly
drawn number line.

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If we draw a number line
right there, that's 0.

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You have a negative
1 right there.

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Well, it's 1 away from 0.

00:00:30.230 --> 00:00:33.250
So the absolute value
of negative 1 is 1.

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And the absolute value of
1 is also 1 away from 0.

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It's also equal to 1.

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So on some level, absolute value
is the distance from 0.

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But another, I guess simpler way
to think of it, it always

00:00:46.190 --> 00:00:48.600
results in the positive
version of the number.

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The absolute value of negative
7,346 is equal to 7,346.

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So with that in mind, let's try
to solve some equations

00:01:02.240 --> 00:01:05.050
with absolute values in them.

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So let's say I have the equation
the absolute value of

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x minus 5 is equal to 10.

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And one way you can interpret
this, and I want you to think

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about this, this is actually
saying that the distance

00:01:18.840 --> 00:01:23.120
between x and 5 is
equal to 10.

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So how many numbers that are
exactly 10 away from 5?

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And you can already think of the
solution to this equation,

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but I'll show you how to solve
it systematically.

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Now this is going to be true
in two situations.

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Either x minus 5 is equal
to positive 10.

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If this evaluates out to
positive 10, then when you

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take the absolute value
of it, you're going to

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get positive 10.

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Or x minus 5 might evaluate
to negative 10.

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If x minus 5 evaluated to
negative 10, when you take the

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absolute value of it, you
would get 10 again.

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So x minus 5 could also be
equal to negative 10.

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Both of these would satisfy
this equation.

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Now, to solve this one,
add 5 to both

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sides of this equation.

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You get x is equal to 15.

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To solve this one, add 5 to both
sides of this equation.

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x is equal to negative 5.

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So our solution, there's
two x's that

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satisfy this equation.

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x could be 15.

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15 minus 5 is 10, take the
absolute value, you're going

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to get 10, or x could
be negative 5.

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Negative 5 minus 5
is negative 10.

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Take the absolute value,
you get 10.

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And notice, both of these
numbers are exactly 10 away

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from the number 5.

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Let's do another one of these.

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Let's do another one.

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Let's say we have the absolute
value of x plus

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2 is equal to 6.

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So what does that tell us?

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That tells us that either x plus
2, that the thing inside

00:03:03.890 --> 00:03:07.030
the absolute value sign,
is equal to 6.

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Or the thing inside of the
absolute value sign, the x

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plus 2, could also
be negative 6.

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If this whole thing evaluated
to negative 6, you take the

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absolute value, you'd get 6.

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So, or x plus 2 could
equal negative 6.

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And then if you subtract 2
from both sides of this

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equation, you get x could
be equal to 4.

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If you subtract 2 from both
sides of this equation, you

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get x could be equal
to negative 8.

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So these are the two solutions
to the equation.

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And just to kind of have it
gel in your mind, that

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absolute value, you can kind of
view it as a distance, you

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could rewrite this problem as
the absolute value of x minus

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negative 2 is equal to 6.

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And so this is asking me, what
are the x's that are exactly 6

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away from negative 2?

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Remember, up here we said,
what are the x's that are

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exactly 10 away from
positive 5?

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Whatever number you're
subtracting from positive 5,

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these are both 10 away
from positive 5.

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This is asking, what
is exactly 6 away

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from negative 2?

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And it's going to be
4, or negative 8.

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You could try those numbers
out for yourself.

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Let's do another one of these.

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Let's do another one, and
we'll do it in purple.

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Let's say we have the absolute
value of 4x-- I'm going to

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change this problem
up a little bit.

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4x minus 1.

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The absolute value of 4x minus
1, is equal to-- actually,

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I'll just keep it--
is equal to 19.

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So, just like the last few
problems, 4x minus 1 could be

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equal to 19.

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Or 4x minus 1 might evaluate
to negative 19.

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Because then when you take the
absolute value, you're going

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to get 19 again.

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Or 4x minus 1 could be
equal to negative 19.

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Then you just solve these
two equations.

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Add 1 to both sides of this
equation-- we could do them

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simultaneous, even.

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Add 1 to both sides of this,
you get 4x is equal to 20.

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Add 1 to both sides of this
equation, you get 4x is equal

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to negative 18.

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Divide both sides of this by
4, you get x is equal to 5.

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Divide both sides of this by
4, you get x is equal to

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negative 18/4, which is
equal to negative 9/2.

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So both of these x values
satisfy the equation.

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Try it out.

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Negative 9/2 times 4.

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This will become
a negative 18.

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Negative 18 minus 1
is negative 19.

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Take the absolute value,
you get 19.

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You put a 5 here,
4 times 5 is 20.

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Minus 1 is positive 19.

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So you take the absolute
value.

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Once again, you'll get a 19.

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Let's try to graph one of
these, just for fun.

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So let's say I have y is
equal to the absolute

00:06:03.630 --> 00:06:04.990
value of x plus 3.

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So this is a function,
or a graph, with an

00:06:07.840 --> 00:06:09.410
absolute value in it.

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So let's think about
two scenarios.

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There's one scenario where the
thing inside of the absolute

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value is positive.

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So you have the scenario where
x plus 3-- I'll write it over

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here-- x plus 3 is
greater than 0.

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And then you have the scenario
where x plus 3 is less than 0.

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When x plus 3 is greater than 0,
this graph, or this line--

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or I guess we can't call it a
line-- this function, is the

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same thing as y is equal
to x plus 3.

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If this thing over here is
greater than 0, then the

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absolute value sign
is irrelevant.

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So then this thing is the
same thing as y is

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equal to x plus 3.

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But when is x plus
3 greater than 0?

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Well, if you subtract 3 from
both sides, you get x is

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greater than negative 3.

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So when x is greater than
negative 3, this graph is

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going to look just like y
is equal to x plus 3.

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Now, when x plus 3
is less than 0.

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When the situation where this--
the inside of our

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absolute value sign-- is
negative, in that situation

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this equation is going to
be y is equal to the

00:07:22.030 --> 00:07:26.250
negative of x plus 3.

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How can I say that?

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Well, look, if this is going to
be a negative number, if x

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plus 3 is going to be a negative
number-- that's what

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we're assuming here-- if it's
going to be a negative number,

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then when you take the absolute
value of a negative

00:07:38.090 --> 00:07:40.050
number, you're going to
make it positive.

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That's just like multiplying
it by negative 1.

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If you know you're taking the
absolute value of a negative

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number, it's just like
multiplying it by negative 1,

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because you're going to
make it positive.

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And this is going to
be the situation.

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x plus 3 is less than 0.

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If we subtract 3 from both
sides, when x is less than

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negative 3.

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So when x is less than negative
3, the graph will

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look like this.

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When x is greater than negative
3, the graph will

00:08:08.280 --> 00:08:09.600
look like that.

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So let's see what that
would make the

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entire graph look like.

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Let me draw my axes.

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That's my x-axis, that's
my y-axis.

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So just let me multiply this
out, just so we have it in mx

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plus b form.

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So this is equal to negative
x minus 3.

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So let's just figure out
what this graph would

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look like in general.

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Negative x minus 3.

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The y-intercept is negative
3, so 1, 2, 3.

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And negative x means it
slopes downward, has a

00:08:51.060 --> 00:08:52.290
downward slope of 1.

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So it would look like this.

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The x-intercept would be
at x is equal to--.

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So if you say y is equal to 0,
that would happen when x is

00:09:07.740 --> 00:09:08.575
equal to negative 3.

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So it's going to go
through that line,

00:09:10.380 --> 00:09:11.920
that point right there.

00:09:11.920 --> 00:09:14.190
And the graph, if we didn't
have this constraint right

00:09:14.190 --> 00:09:15.600
here, would look something
like this.

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That's if we didn't constrain
it to a certain interval on

00:09:22.760 --> 00:09:23.880
the x-axis.

00:09:23.880 --> 00:09:27.080
Now this graph, what
does it look like?

00:09:27.080 --> 00:09:27.480
Let's see.

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It has its y-intercept
at positive 3.

00:09:31.810 --> 00:09:33.230
Just like that.

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And where's its x-intecept?

00:09:35.260 --> 00:09:37.970
When y is equal to 0,
x is negative 3.

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So it also goes through that
point right there, and it has

00:09:39.760 --> 00:09:40.620
a slope of 1.

00:09:40.620 --> 00:09:43.710
So it would look something
like this.

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That's what this graph
looks like.

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Now, what we figured out is
that this absolute value

00:09:48.100 --> 00:09:52.030
function, it looks like this
purple graph when x is less

00:09:52.030 --> 00:09:53.830
than negative 3.

00:09:53.830 --> 00:09:57.070
So when x is less than negative
3-- that's x is equal

00:09:57.070 --> 00:09:59.593
to negative 3 right there-- when
x is less than negative

00:09:59.593 --> 00:10:03.170
3, it looks like this
purple graph.

00:10:03.170 --> 00:10:04.570
Right there.

00:10:04.570 --> 00:10:07.390
So that's when x is less
than negative 3.

00:10:07.390 --> 00:10:10.830
But when x is greater than
negative 3, it looks like the

00:10:10.830 --> 00:10:12.160
green graph.

00:10:12.160 --> 00:10:14.640
It looks like that.

00:10:14.640 --> 00:10:17.480
So this graph looks like
this strange v.

00:10:17.480 --> 00:10:21.430
When x is greater than negative
3, this is positive.

00:10:21.430 --> 00:10:24.950
So we have the graph of-- we
have a positive slope.

00:10:24.950 --> 00:10:28.270
But then when x is less than
negative 3, we're essentially

00:10:28.270 --> 00:10:30.550
taking the negative of the
function, if you want to view

00:10:30.550 --> 00:10:32.280
it that way, and so we have
this negative slope.

00:10:32.280 --> 00:10:35.060
So you kind of have this
v-shaped function, this

00:10:35.060 --> 00:10:38.250
v-shaped graph, which is
indicative of an absolute

00:10:38.250 --> 00:10:39.950
value function.


Multi-step inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.270
Let's do a few more problems
that bring together the

00:00:03.270 --> 00:00:05.950
concepts that we learned
in the last two videos.

00:00:05.950 --> 00:00:11.140
So let's say we have the
inequality 4x plus 3 is less

00:00:11.140 --> 00:00:12.850
than negative 1.

00:00:12.850 --> 00:00:16.440
So let's find all of the
x's that satisfy this.

00:00:16.440 --> 00:00:18.880
So the first thing I'd like to
do is get rid of this 3.

00:00:18.880 --> 00:00:23.430
So let's subtract 3 from both
sides of this equation.

00:00:23.430 --> 00:00:27.980
So the left-hand side is just
going to end up being 4x.

00:00:27.980 --> 00:00:29.320
These 3's cancel out.

00:00:29.320 --> 00:00:31.060
That just ends up with a zero.

00:00:31.060 --> 00:00:33.200
No reason to change the
inequality just yet.

00:00:33.200 --> 00:00:35.630
We're just adding and
subtracting from both sides,

00:00:35.630 --> 00:00:36.375
in this case, subtracting.

00:00:36.375 --> 00:00:38.780
That doesn't change the
inequality as long as we're

00:00:38.780 --> 00:00:40.460
subtracting the same value.

00:00:40.460 --> 00:00:43.300
We have negative 1 minus 3.

00:00:43.300 --> 00:00:45.680
That is negative 4.

00:00:45.680 --> 00:00:49.610
Negative 1 minus 3
is negative 4.

00:00:49.610 --> 00:00:53.380
And then we'll want to-- let's
see, we can divide both sides

00:00:53.380 --> 00:00:55.375
of this equation by 4.

00:01:00.390 --> 00:01:02.760
Once again, when you multiply
or divide both sides of an

00:01:02.760 --> 00:01:06.190
inequality by a positive number,
it doesn't change the

00:01:06.190 --> 00:01:06.990
inequality.

00:01:06.990 --> 00:01:09.255
So the left-hand
side is just x.

00:01:09.255 --> 00:01:15.070
x is less than negative 4
divided by 4 is negative 1.

00:01:15.070 --> 00:01:16.960
x is less than negative 1.

00:01:16.960 --> 00:01:19.210
Or we could write this
in interval notation.

00:01:19.210 --> 00:01:24.080
All of the x's from negative
infinity to negative 1, but

00:01:24.080 --> 00:01:25.900
not including negative
1, so we put a

00:01:25.900 --> 00:01:27.900
parenthesis right there.

00:01:27.900 --> 00:01:31.980
Let's do a slightly
harder one.

00:01:31.980 --> 00:01:39.500
Let's say we have 5x is greater
than 8x plus 27.

00:01:39.500 --> 00:01:41.620
So let's get all our x's on the
left-hand side, and the

00:01:41.620 --> 00:01:45.360
best way to do that is subtract
8x from both sides.

00:01:45.360 --> 00:01:49.360
So you subtract 8x
from both sides.

00:01:49.360 --> 00:01:53.640
The left-hand side becomes
5x minus 8x.

00:01:53.640 --> 00:01:56.040
That's negative 3x.

00:01:56.040 --> 00:01:57.520
We still have a greater
than sign.

00:01:57.520 --> 00:01:59.560
We're just adding or subtracting
the same

00:01:59.560 --> 00:02:01.040
quantities on both sides.

00:02:01.040 --> 00:02:04.960
These 8x's cancel out and you're
just left with a 27.

00:02:04.960 --> 00:02:08.180
So you have negative 3x
is greater than 27.

00:02:08.180 --> 00:02:11.320
Now, to just turn this into an
x, we want to divide both

00:02:11.320 --> 00:02:12.900
sides by negative 3.

00:02:12.900 --> 00:02:16.450
But remember, when you multiply
or divide both sides

00:02:16.450 --> 00:02:19.140
of an inequality by a negative
number, you swap the

00:02:19.140 --> 00:02:20.550
inequality.

00:02:20.550 --> 00:02:29.140
So if we divide both sides of
this by negative 3, we have to

00:02:29.140 --> 00:02:30.760
swap this inequality.

00:02:30.760 --> 00:02:33.800
It will go from being
a greater than sign

00:02:33.800 --> 00:02:36.410
to a less than sign.

00:02:36.410 --> 00:02:39.090
And just as a bit of a way that
I remember greater than

00:02:39.090 --> 00:02:41.410
is that the left-hand side
just looks bigger.

00:02:41.410 --> 00:02:42.550
This is greater than.

00:02:42.550 --> 00:02:45.800
If you just imagine this height,
that height is greater

00:02:45.800 --> 00:02:48.170
than that height right there,
which is just a point.

00:02:48.170 --> 00:02:49.970
I don't know if that confuses
you or not.

00:02:49.970 --> 00:02:50.830
This is less than.

00:02:50.830 --> 00:02:53.090
This little point
is less than the

00:02:53.090 --> 00:02:55.160
distance of that big opening.

00:02:55.160 --> 00:02:56.490
That's how I remember it.

00:02:56.490 --> 00:03:00.550
But anyway, 3x over
negative 3.

00:03:00.550 --> 00:03:04.280
So now that we divided both
sides by a negative number, by

00:03:04.280 --> 00:03:07.410
negative 3, we swapped the
inequality from greater than

00:03:07.410 --> 00:03:08.870
to less than.

00:03:08.870 --> 00:03:11.490
And the left-hand side, the
negative 3's cancel out.

00:03:11.490 --> 00:03:16.430
You get x is less than
27 over negative 3,

00:03:16.430 --> 00:03:18.820
which is negative 9.

00:03:18.820 --> 00:03:21.600
Or in interval notation, it
would be everything from

00:03:21.600 --> 00:03:26.720
negative infinity to negative
9, not including negative 9.

00:03:26.720 --> 00:03:29.350
If you wanted to do it
as a number line, it

00:03:29.350 --> 00:03:31.530
would look like this.

00:03:31.530 --> 00:03:35.080
This would be negative 9, maybe
this would be negative

00:03:35.080 --> 00:03:37.350
8, maybe this would
be negative 10.

00:03:37.350 --> 00:03:40.770
You would start at negative 9,
not included, because we don't

00:03:40.770 --> 00:03:44.190
have an equal sign here, and
you go everything less than

00:03:44.190 --> 00:03:47.860
that, all the way down, as we
see, to negative infinity.

00:03:47.860 --> 00:03:52.540
Let's do a nice,
hairy problem.

00:03:52.540 --> 00:04:05.930
So let's say we have 8x minus
5 times 4x plus 1 is greater

00:04:05.930 --> 00:04:12.290
than or equal to negative 1
plus 2 times 4x minus 3.

00:04:12.290 --> 00:04:14.330
Now, this might seem very
daunting, but if we just

00:04:14.330 --> 00:04:17.240
simplify it step by step, you'll
see it's no harder than

00:04:17.240 --> 00:04:20.269
any of the other problems
we've tackled.

00:04:20.269 --> 00:04:21.519
So let's just simplify this.

00:04:21.519 --> 00:04:26.990
You get 8x minus-- let's
distribute this negative 5.

00:04:26.990 --> 00:04:29.590
So let me say 8x, and then
distribute the negative 5.

00:04:29.590 --> 00:04:33.560
Negative 5 times 4x
is negative 20x.

00:04:33.560 --> 00:04:35.760
Negative 5-- when I say negative
5, I'm talking about

00:04:35.760 --> 00:04:36.650
this whole thing.

00:04:36.650 --> 00:04:42.510
Negative 5 times 1 is negative
5, and then that's going to be

00:04:42.510 --> 00:04:45.660
greater than or equal
to negative 1 plus

00:04:45.660 --> 00:04:48.210
2 times 4x is 8x.

00:04:48.210 --> 00:04:51.940
2 times negative 3
is negative 6.

00:04:51.940 --> 00:04:56.490
And now we can merge these two
terms. 8x minus 20x is

00:04:56.490 --> 00:05:04.120
negative 12x minus 5 is greater
than or equal to-- we

00:05:04.120 --> 00:05:10.040
can merge these constant terms.
Negative 1 minus 6,

00:05:10.040 --> 00:05:16.070
that's negative 7, and then we
have this plus 8x left over.

00:05:16.070 --> 00:05:18.620
Now, I like to get all my x
terms on the left-hand side,

00:05:18.620 --> 00:05:22.255
so let's subtract 8x from both
sides of this equation.

00:05:30.670 --> 00:05:32.700
I'm subtracting 8x.

00:05:32.700 --> 00:05:34.900
This left-hand side, negative
12 minus 8,

00:05:34.900 --> 00:05:36.900
that's negative 20.

00:05:36.900 --> 00:05:41.270
Negative 20x minus 5.

00:05:41.270 --> 00:05:42.710
Once again, no reason
to change the

00:05:42.710 --> 00:05:44.380
inequality just yet.

00:05:44.380 --> 00:05:46.910
All we're doing is simplifying
the sides, or adding and

00:05:46.910 --> 00:05:48.740
subtracting from them.

00:05:48.740 --> 00:05:53.220
The right-hand side becomes--
this thing cancels out, 8x

00:05:53.220 --> 00:05:54.600
minus 8x, that's 0.

00:05:54.600 --> 00:05:57.690
So you're just left
with a negative 7.

00:05:57.690 --> 00:05:59.690
And now I want to get rid
of this negative 5.

00:05:59.690 --> 00:06:01.910
So let's add 5 to both sides
of this equation.

00:06:06.860 --> 00:06:12.540
The left-hand side, you're just
left with a negative 20x.

00:06:12.540 --> 00:06:14.600
These 5's cancel out.

00:06:14.600 --> 00:06:17.300
No reason to change the
inequality just yet.

00:06:17.300 --> 00:06:22.330
Negative 7 plus 5, that's
negative 2.

00:06:22.330 --> 00:06:23.710
Now, we're at an interesting
point.

00:06:23.710 --> 00:06:26.810
We have negative 20x is greater
than or equal to

00:06:26.810 --> 00:06:28.460
negative 2.

00:06:28.460 --> 00:06:31.000
If this was an equation, or
really any type of an

00:06:31.000 --> 00:06:33.740
inequality, we want to divide
both sides by negative 20.

00:06:33.740 --> 00:06:36.520
But we have to remember, when
you multiply or divide both

00:06:36.520 --> 00:06:39.370
sides of an inequality by a
negative number, you have to

00:06:39.370 --> 00:06:40.810
swap the inequality.

00:06:40.810 --> 00:06:42.250
So let's remember that.

00:06:42.250 --> 00:06:46.761
So if we divide this side by
negative 20 and we divide this

00:06:46.761 --> 00:06:50.900
side by negative 20, all I did
is took both of these sides

00:06:50.900 --> 00:06:54.350
divided by negative 20, we have
to swap the inequality.

00:06:54.350 --> 00:06:58.560
The greater than or equal to has
to become a less than or

00:06:58.560 --> 00:06:59.990
equal sign.

00:06:59.990 --> 00:07:03.380
And, of course, these cancel
out, and you get x is less

00:07:03.380 --> 00:07:05.550
than or equal to-- the
negatives cancel

00:07:05.550 --> 00:07:10.590
out-- 2/20 is 1/10.

00:07:10.590 --> 00:07:13.490
If we were writing it in
interval notation, the upper

00:07:13.490 --> 00:07:15.440
bound would be 1/10.

00:07:15.440 --> 00:07:18.670
Notice, we're including it,
because we have an equal sign,

00:07:18.670 --> 00:07:22.220
less than or equal, so we're
including 1/10, and we're

00:07:22.220 --> 00:07:24.210
going to go all the way down
to negative infinity,

00:07:24.210 --> 00:07:27.260
everything less than
or equal to 1/10.

00:07:27.260 --> 00:07:29.600
This is just another way
of writing that.

00:07:29.600 --> 00:07:33.670
And just for fun, let's
draw the number line.

00:07:33.670 --> 00:07:35.380
Let's draw the number
line right here.

00:07:35.380 --> 00:07:38.520
This is maybe 0, that is 1.

00:07:38.520 --> 00:07:42.610
1/10 might be over here.

00:07:42.610 --> 00:07:45.090
Everything less than
or equal to 1/10.

00:07:45.090 --> 00:07:51.620
So we're going to include the
1/10 and everything less than

00:07:51.620 --> 00:07:54.010
that is included in
the solution set.

00:07:54.010 --> 00:07:56.900
And you could try out any
value less than 1/10 and

00:07:56.900 --> 00:08:00.400
verify that it will satisfy
this inequality.


Multiplying and dividing with inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:03.150
In this video, I want to tackle
some inequalities that

00:00:03.150 --> 00:00:06.290
involve multiplying and dividing
by positive and

00:00:06.290 --> 00:00:08.070
negative numbers, and you'll
see that it's a little bit

00:00:08.070 --> 00:00:11.530
more tricky than just the adding
and subtracting numbers

00:00:11.530 --> 00:00:12.540
that we saw in the last video.

00:00:12.540 --> 00:00:14.960
I also want to introduce you
to some other types of

00:00:14.960 --> 00:00:18.905
notations for describing the
solution set of an inequality.

00:00:18.905 --> 00:00:21.390
So let's do a couple
of examples.

00:00:21.390 --> 00:00:28.510
So let's say I had negative
0.5x is less

00:00:28.510 --> 00:00:31.115
than or equal to 7.5.

00:00:33.700 --> 00:00:36.810
Now, if this was an equality,
your natural impulse is to

00:00:36.810 --> 00:00:39.430
say, hey, let's divide both
sides by the coefficient on

00:00:39.430 --> 00:00:42.670
the x term, and that is a
completely legitimate thing to

00:00:42.670 --> 00:00:46.680
do: divide both sides
by negative 0.5.

00:00:46.680 --> 00:00:48.850
The important thing you need to
realize, though, when you

00:00:48.850 --> 00:00:52.160
do it with an inequality is
that when you multiply or

00:00:52.160 --> 00:00:55.920
divide both sides of the
equation by a negative number,

00:00:55.920 --> 00:00:57.776
you swap the inequality.

00:01:00.410 --> 00:01:01.360
Think of it this way.

00:01:01.360 --> 00:01:03.210
I'll do a simple example here.

00:01:03.210 --> 00:01:07.900
If I were to tell you that 1
is less than 2, I think you

00:01:07.900 --> 00:01:08.600
would agree with that.

00:01:08.600 --> 00:01:10.970
1 is definitely less than 2.

00:01:10.970 --> 00:01:13.470
Now, what happens if I multiply
both sides of this by

00:01:13.470 --> 00:01:14.700
negative 1?

00:01:14.700 --> 00:01:17.930
Negative 1 versus negative 2?

00:01:17.930 --> 00:01:20.700
Well, all of a sudden, negative
2 is more negative

00:01:20.700 --> 00:01:21.730
than negative 1.

00:01:21.730 --> 00:01:25.510
So here, negative 2 is actually
less than negative 1.

00:01:25.510 --> 00:01:27.800
Now, this isn't a proof, but I
think it'll give you comfort

00:01:27.800 --> 00:01:29.680
on why you're swapping
the sign.

00:01:29.680 --> 00:01:33.120
If something is larger, when you
take the negative of both

00:01:33.120 --> 00:01:36.820
of it, it'll be more negative,
or vice versa.

00:01:36.820 --> 00:01:40.100
So that's why, if we're going to
multiply both sides of this

00:01:40.100 --> 00:01:43.080
equation or divide both sides of
the equation by a negative

00:01:43.080 --> 00:01:45.940
number, we need to
swap the sign.

00:01:45.940 --> 00:01:48.610
So let's multiply both sides
of this equation.

00:01:48.610 --> 00:01:54.950
Dividing by 0.5 is the same
thing as multiplying by 2.

00:01:54.950 --> 00:01:57.650
Our whole goal here is to have
a 1 coefficient there.

00:01:57.650 --> 00:01:59.400
So let's multiply both
sides of this

00:01:59.400 --> 00:02:02.830
equation by negative 2.

00:02:02.830 --> 00:02:08.320
So we have negative 2
times negative 0.5.

00:02:08.320 --> 00:02:10.770
And you might say, hey, how
did Sal get this 2 here?

00:02:10.770 --> 00:02:13.310
My brain is just thinking what
can I multiply negative

00:02:13.310 --> 00:02:15.380
0.5 by to get 1?

00:02:15.380 --> 00:02:18.230
And negative 0.5 is the same
thing as negative 1/2.

00:02:18.230 --> 00:02:20.510
The inverse of that
is negative 2.

00:02:20.510 --> 00:02:22.270
So I'm multiplying negative
2 times both

00:02:22.270 --> 00:02:24.430
sides of this equation.

00:02:24.430 --> 00:02:27.660
And I have the 7.5 on
the other side.

00:02:27.660 --> 00:02:32.110
I'm going to multiply that
by negative 2 as well.

00:02:32.110 --> 00:02:34.520
And remember, when you multiply
or divide both sides

00:02:34.520 --> 00:02:37.330
of an inequality by a negative,
you swap the

00:02:37.330 --> 00:02:37.690
inequality.

00:02:37.690 --> 00:02:38.820
You had less than or equal?

00:02:38.820 --> 00:02:42.790
Now it'll be greater
than or equal.

00:02:42.790 --> 00:02:44.960
So the left-hand side,
negative 2 times

00:02:44.960 --> 00:02:47.070
negative 0.5 is just 1.

00:02:47.070 --> 00:02:52.450
You get x is greater than or
equal to 7.5 times negative 2.

00:02:52.450 --> 00:02:56.670
That's negative 15, which
is our solution set.

00:02:56.670 --> 00:03:01.190
All x's larger than negative 15
will satisfy this equation.

00:03:01.190 --> 00:03:03.200
I challenge you to try it.

00:03:03.200 --> 00:03:04.610
For example, 0 will work.

00:03:04.610 --> 00:03:07.460
0 is greater than negative 15.

00:03:07.460 --> 00:03:10.710
But try something like--
try negative 16.

00:03:10.710 --> 00:03:13.070
Negative 16 will not work.

00:03:13.070 --> 00:03:17.820
Negative 16 times negative
0.5 is 8, which is

00:03:17.820 --> 00:03:19.790
not less than 7.5.

00:03:19.790 --> 00:03:23.470
So the solution set is all of
the x's-- let me draw a number

00:03:23.470 --> 00:03:27.840
line here-- greater
than negative 15.

00:03:27.840 --> 00:03:30.610
So that is negative 15 there,
maybe that's negative 16,

00:03:30.610 --> 00:03:32.040
that's negative 14.

00:03:32.040 --> 00:03:41.570
Greater than or equal to
negative 15 is the solution.

00:03:41.570 --> 00:03:44.830
Now, you might also see solution
sets to inequalities

00:03:44.830 --> 00:03:46.610
written in interval notation.

00:03:46.610 --> 00:03:48.640
And interval notation, it
just takes a little

00:03:48.640 --> 00:03:50.010
getting used to.

00:03:50.010 --> 00:03:53.280
We want to include negative 15,
so our lower bound to our

00:03:53.280 --> 00:03:55.780
interval is negative 15.

00:03:55.780 --> 00:03:58.190
And putting in this bracket here
means that we're going to

00:03:58.190 --> 00:03:59.530
include negative 15.

00:03:59.530 --> 00:04:01.880
The set includes the
bottom boundary.

00:04:01.880 --> 00:04:03.780
It includes negative 15.

00:04:03.780 --> 00:04:05.963
And we're going to go all
the way to infinity.

00:04:08.500 --> 00:04:13.110
And we put a parentheses here.

00:04:13.110 --> 00:04:15.290
Parentheses normally means that
you're not including the

00:04:15.290 --> 00:04:16.100
upper bound.

00:04:16.100 --> 00:04:18.779
You also do it for infinity,
because infinity really isn't

00:04:18.779 --> 00:04:20.850
a normal number, so to speak.

00:04:20.850 --> 00:04:22.940
You can't just say, oh,
I'm at infinity.

00:04:22.940 --> 00:04:24.540
You're never at infinity.

00:04:24.540 --> 00:04:26.040
So that's why you put
that parentheses.

00:04:26.040 --> 00:04:28.070
But the parentheses tends to
mean that you don't include

00:04:28.070 --> 00:04:31.040
that boundary, but you also
use it with infinity.

00:04:31.040 --> 00:04:35.680
So this and this are the
exact same thing.

00:04:35.680 --> 00:04:38.940
Sometimes you might also see
set notations, where the

00:04:38.940 --> 00:04:48.630
solution of that, they might
say x is a real number such

00:04:48.630 --> 00:04:51.650
that-- that little line, that
vertical line thing, just

00:04:51.650 --> 00:04:58.070
means such that-- x is greater
than or equal to negative 15.

00:04:58.070 --> 00:05:01.260
These curly brackets mean the
set of all real numbers, or

00:05:01.260 --> 00:05:04.320
the set of all numbers, where x
is a real number, such that

00:05:04.320 --> 00:05:06.430
x is greater than or equal
to negative 15.

00:05:06.430 --> 00:05:11.390
All of this, this, and this
are all equivalent.

00:05:11.390 --> 00:05:16.610
Let's keep that in mind and do
a couple of more examples.

00:05:16.610 --> 00:05:25.400
So let's say we had 75x is
greater than or equal to 125.

00:05:25.400 --> 00:05:31.030
So here we can just divide
both sides by 75.

00:05:31.030 --> 00:05:34.020
And since 75 is a positive
number, you don't have to

00:05:34.020 --> 00:05:35.620
change the inequality.

00:05:35.620 --> 00:05:41.070
So you get x is greater than
or equal to 125/75.

00:05:41.070 --> 00:05:43.240
And if you divide the numerator
and denominator by

00:05:43.240 --> 00:05:46.440
25, this is 5/3.

00:05:46.440 --> 00:05:50.950
So x is greater than
or equal to 5/3.

00:05:50.950 --> 00:05:56.220
Or we could write the solution
set being from including 5/3

00:05:56.220 --> 00:05:58.380
to infinity.

00:05:58.380 --> 00:06:00.230
And once again, if you were
to graph it on a number

00:06:00.230 --> 00:06:04.630
line, 5/3 is what?

00:06:04.630 --> 00:06:06.650
That's 1 and 2/3.

00:06:06.650 --> 00:06:12.940
So you have 0, 1, 2, and
1 and 2/3 will be

00:06:12.940 --> 00:06:13.660
right around there.

00:06:13.660 --> 00:06:15.200
We're going to include it.

00:06:15.200 --> 00:06:17.320
That right there is 5/3.

00:06:17.320 --> 00:06:20.960
And everything greater than or
equal to that will be included

00:06:20.960 --> 00:06:22.970
in our solution set.

00:06:22.970 --> 00:06:24.850
Let's do another one.

00:06:24.850 --> 00:06:33.170
Let's say we have x over
negative 3 is greater than

00:06:33.170 --> 00:06:36.900
negative 10/9.

00:06:36.900 --> 00:06:39.720
So we want to just isolate the
x on the left-hand side.

00:06:39.720 --> 00:06:45.990
So let's multiply both sides
by negative 3, right?

00:06:45.990 --> 00:06:49.770
The coefficient, you could
imagine, is negative 1/3, so

00:06:49.770 --> 00:06:51.360
we want to multiply by the
inverse, which should be

00:06:51.360 --> 00:06:52.700
negative 3.

00:06:52.700 --> 00:06:55.590
So if you multiply both sides
by negative 3, you get

00:06:55.590 --> 00:06:59.940
negative 3 times-- this you
could rewrite it as negative

00:06:59.940 --> 00:07:06.000
1/3x, and on this side, you
have negative 10/9 times

00:07:06.000 --> 00:07:07.220
negative 3.

00:07:07.220 --> 00:07:10.350
And the inequality will
switch, because we are

00:07:10.350 --> 00:07:13.330
multiplying or dividing
by a negative number.

00:07:13.330 --> 00:07:14.980
So the inequality will switch.

00:07:14.980 --> 00:07:18.940
It'll go from greater
than to less than.

00:07:18.940 --> 00:07:22.630
So the left-hand side of the
equation just becomes an x.

00:07:22.630 --> 00:07:23.540
That was the whole point.

00:07:23.540 --> 00:07:24.810
That cancels out with that.

00:07:24.810 --> 00:07:27.870
The negatives cancel out.
x is less than.

00:07:27.870 --> 00:07:30.480
And then you have a negative
times a negative.

00:07:30.480 --> 00:07:32.260
That will make it a positive.

00:07:32.260 --> 00:07:34.010
Then if you divide the numerator
and the denominator

00:07:34.010 --> 00:07:41.430
by 3, you get a 1 and a 3,
so x is less than 10/3.

00:07:41.430 --> 00:07:43.630
So if we were to write this
in interval notation, the

00:07:43.630 --> 00:07:48.900
solution set will-- the upper
bound will be 10/3 and it

00:07:48.900 --> 00:07:50.290
won't include 10/3.

00:07:50.290 --> 00:07:52.270
This isn't less than or equal
to, so we're going to put a

00:07:52.270 --> 00:07:53.870
parentheses here.

00:07:53.870 --> 00:07:55.390
Notice, here it included 5/3.

00:07:55.390 --> 00:07:56.180
We put a bracket.

00:07:56.180 --> 00:07:57.760
Here, we're not including
10/3.

00:07:57.760 --> 00:07:59.240
We put a parentheses.

00:07:59.240 --> 00:08:04.560
It'll go from 10/3, all the way
down to negative infinity.

00:08:04.560 --> 00:08:07.850
Everything less than 10/3
is in our solution set.

00:08:07.850 --> 00:08:09.640
And let's draw that.

00:08:09.640 --> 00:08:12.170
Let's draw the solution set.

00:08:12.170 --> 00:08:17.560
So 10/3, so we might
have 0, 1, 2, 3, 4.

00:08:17.560 --> 00:08:21.340
10/3 is 3 and 1/3, so it might
sit-- let me do it in a

00:08:21.340 --> 00:08:22.840
different color.

00:08:22.840 --> 00:08:23.440
It might be over here.

00:08:23.440 --> 00:08:24.410
We're not going to
include that.

00:08:24.410 --> 00:08:25.690
It's less than 10/3.

00:08:25.690 --> 00:08:28.120
10/3 is not in the
solution set.

00:08:28.120 --> 00:08:32.150
That is 10/3 right there, and
everything less than that, but

00:08:32.150 --> 00:08:39.169
not including 10/3, is
in our solution set.

00:08:39.169 --> 00:08:41.720
Let's do one more.

00:08:45.350 --> 00:08:54.430
Say we have x over negative
15 is less than 8.

00:08:54.430 --> 00:08:56.820
So once again, let's multiply
both sides of this equation by

00:08:56.820 --> 00:08:58.520
negative 15.

00:08:58.520 --> 00:09:03.700
So negative 15 times
x over negative 15.

00:09:03.700 --> 00:09:06.880
Then you have an 8 times
a negative 15.

00:09:06.880 --> 00:09:09.640
And when you multiply both sides
of an inequality by a

00:09:09.640 --> 00:09:12.300
negative number or divide both
sides by a negative number,

00:09:12.300 --> 00:09:13.730
you swap the inequality.

00:09:13.730 --> 00:09:17.960
It's less than, you change
it to greater than.

00:09:17.960 --> 00:09:21.260
And now, this left-hand side
just becomes an x, because

00:09:21.260 --> 00:09:22.955
these guys cancel out.

00:09:22.955 --> 00:09:28.860
x is greater than 8 times 15
is 80 plus 40 is 120, so

00:09:28.860 --> 00:09:31.010
negative 120.

00:09:31.010 --> 00:09:31.640
Is that right?

00:09:31.640 --> 00:09:33.270
80 plus 40.

00:09:33.270 --> 00:09:34.810
Yep, negative 120.

00:09:34.810 --> 00:09:38.330
Or we could write the solution
set as starting at negative

00:09:38.330 --> 00:09:41.090
120-- but we're not including
negative 120.

00:09:41.090 --> 00:09:43.450
We don't have an equal
sign here-- going all

00:09:43.450 --> 00:09:46.690
the way up to infinity.

00:09:46.690 --> 00:09:49.280
And if we were to graph
it, let me draw

00:09:49.280 --> 00:09:50.290
the number line here.

00:09:50.290 --> 00:09:52.280
I'll do a real quick one.

00:09:52.280 --> 00:09:54.720
Let's say that that
is negative 120.

00:09:54.720 --> 00:09:56.170
Maybe zero is sitting up here.

00:09:56.170 --> 00:09:58.240
This would be negative 121.

00:09:58.240 --> 00:10:00.110
This would be negative 119.

00:10:00.110 --> 00:10:04.670
We are not going to include
negative 120, because we don't

00:10:04.670 --> 00:10:06.470
have an equal sign there, but
it's going to be everything

00:10:06.470 --> 00:10:09.280
greater than negative 120.

00:10:09.280 --> 00:10:11.970
All of these things that I'm
shading in green would satisfy

00:10:11.970 --> 00:10:12.660
the inequality.

00:10:12.660 --> 00:10:13.510
And you can even try it out.

00:10:13.510 --> 00:10:14.860
Does zero work?

00:10:14.860 --> 00:10:15.980
0/15?

00:10:15.980 --> 00:10:16.820
Yeah, that's zero.

00:10:16.820 --> 00:10:18.250
That's definitely less than 8.

00:10:18.250 --> 00:10:20.020
I mean, that doesn't prove it to
you, but you could try any

00:10:20.020 --> 00:10:22.750
of these numbers and
they should work.

00:10:22.750 --> 00:10:24.470
Anyway, hopefully, you
found that helpful.

00:10:24.470 --> 00:10:26.820
I'll see you in the
next video.


Inequalities using addition and subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:03.670
what i want to do in this video is a handful of fairly

00:00:03.670 --> 00:00:05.470
simple inequality videos.

00:00:05.470 --> 00:00:07.710
But the real value of it, I
think, will be just to get you

00:00:07.710 --> 00:00:10.750
warmed up in the notation
of inequality.

00:00:10.750 --> 00:00:12.390
So, let's just start with one.

00:00:12.390 --> 00:00:19.590
we have x minus 5 is less than 35.

00:00:19.590 --> 00:00:22.480
So let's see if we can find
all of the x's that will

00:00:22.480 --> 00:00:23.940
satisfy this equation.

00:00:23.940 --> 00:00:26.370
And that's one of the
distinctions of an inequality.

00:00:26.370 --> 00:00:30.000
In an equation, you typically
have one solution, or at least

00:00:30.000 --> 00:00:31.220
the ones we've solved so far.

00:00:31.220 --> 00:00:33.150
In the future, we'll see
equations where they have more

00:00:33.150 --> 00:00:34.250
than one solution.

00:00:34.250 --> 00:00:36.150
But in the ones we've solved
so far, you solved for a

00:00:36.150 --> 00:00:37.310
particular x.

00:00:37.310 --> 00:00:40.350
In the inequalities, there's a
whole set of x's that will

00:00:40.350 --> 00:00:41.720
satisfy this inequality.

00:00:41.720 --> 00:00:43.930
So they're saying, what are
all the x's, that when you

00:00:43.930 --> 00:00:48.120
subtract 5 from them, it's
going to be less than 35?

00:00:48.120 --> 00:00:49.170
And we can already
think about it.

00:00:49.170 --> 00:00:50.690
I mean 0 minus 5.

00:00:50.690 --> 00:00:52.290
That's less than 35.

00:00:52.290 --> 00:00:54.310
Minus 100 minus 5.

00:00:54.310 --> 00:00:55.750
That's less than 35.

00:00:55.750 --> 00:00:57.160
5 minus 5.

00:00:57.160 --> 00:00:58.300
That's less than 35.

00:00:58.300 --> 00:01:01.750
So there's clearly a lot of x's
that will satisfy that.

00:01:01.750 --> 00:01:04.610
But what we want to do is come
up with a solution that

00:01:04.610 --> 00:01:07.060
essentially encompasses
all of the x's.

00:01:07.060 --> 00:01:09.640
So the way we do that is
essentially the same way that

00:01:09.640 --> 00:01:10.960
we solved any equations.

00:01:10.960 --> 00:01:13.610
We want to get just the x terms,
in this case, on the

00:01:13.610 --> 00:01:14.720
left-hand side.

00:01:14.720 --> 00:01:17.280
So I want to get rid of this
negative 5, and I can do that

00:01:17.280 --> 00:01:20.400
by adding 5 to both sides
of this equation.

00:01:20.400 --> 00:01:24.350
So I can add 5 to both sides
of this equation.

00:01:24.350 --> 00:01:25.930
That won't change
the inequality.

00:01:25.930 --> 00:01:28.710
It won't change the
less than sign.

00:01:28.710 --> 00:01:31.470
If something is less than
something else, something plus

00:01:31.470 --> 00:01:35.720
5 is still going to be less than
the other thing plus 5.

00:01:35.720 --> 00:01:38.580
So on the left-hand side,
we just have an x.

00:01:38.580 --> 00:01:41.500
This negative 5 and this
positive 5 cancel out.

00:01:41.500 --> 00:01:45.730
x is less than 35 plus
5, which is 40.

00:01:45.730 --> 00:01:47.190
And that's our solution.

00:01:47.190 --> 00:01:50.010
And to just visualize the
set of all numbers that

00:01:50.010 --> 00:01:52.340
represents, let me draw
a number line here.

00:01:56.760 --> 00:02:01.280
And I'll do it around--
let's say that's 40,

00:02:01.280 --> 00:02:05.250
this is 40, 41, 42.

00:02:05.250 --> 00:02:06.730
And then we could go below 40.

00:02:06.730 --> 00:02:09.389
39, 38.

00:02:09.389 --> 00:02:12.240
You can just keep
going below 40.

00:02:12.240 --> 00:02:14.700
It just keeps going on
in both directions.

00:02:14.700 --> 00:02:18.610
And any x that is less than
40 will satisfy this.

00:02:18.610 --> 00:02:22.950
So it can't be equal to 40,
because if x is equal to 40,

00:02:22.950 --> 00:02:25.010
40 minus 5 is 35.

00:02:25.010 --> 00:02:27.130
It's not less than 35.

00:02:27.130 --> 00:02:28.820
So x has to be less than 40.

00:02:28.820 --> 00:02:32.590
And to show this on the number
line, we do a circle around 40

00:02:32.590 --> 00:02:34.620
to show that we're
not including 40.

00:02:34.620 --> 00:02:37.020
But then we can shade in
everything below 40.

00:02:37.020 --> 00:02:41.010
So everything that's just
exactly below 40 is included

00:02:41.010 --> 00:02:41.770
in our solution set.

00:02:41.770 --> 00:02:45.240
So everything I've shaded in
yellow is included in our

00:02:45.240 --> 00:02:46.040
solution set.

00:02:46.040 --> 00:02:52.530
So 39, 39.999999, repeating,
which is about as close as you

00:02:52.530 --> 00:02:55.240
can get to 40 as possible,
that's in our solution set.

00:02:55.240 --> 00:02:56.370
But 40 is not.

00:02:56.370 --> 00:02:58.630
That's why we put that open
circle around it.

00:02:58.630 --> 00:03:00.850
Let's do another one.

00:03:00.850 --> 00:03:03.095
Let me do it in another
color as well.

00:03:03.095 --> 00:03:06.860
So let's say we have x-- let
me do it over on this top

00:03:06.860 --> 00:03:08.360
right corner.

00:03:08.360 --> 00:03:16.320
Say we have x plus 15 is greater
than or equal to

00:03:16.320 --> 00:03:17.570
negative 60.

00:03:17.570 --> 00:03:20.220
Notice, now we have greater
than or equal.

00:03:20.220 --> 00:03:22.110
So let's solve this the
same way we solved

00:03:22.110 --> 00:03:22.830
that one over there.

00:03:22.830 --> 00:03:25.360
We can subtract 15
from both sides.

00:03:25.360 --> 00:03:26.800
And I like to switch
up my notation.

00:03:26.800 --> 00:03:29.680
Here I added the 5, kind
of, on the same line.

00:03:29.680 --> 00:03:32.460
You could also do your adding
or subtracting below

00:03:32.460 --> 00:03:33.480
the line, like this.

00:03:33.480 --> 00:03:37.130
So if I subtract 15 from both
sides, so I do a minus 15

00:03:37.130 --> 00:03:40.170
there, and I do a
minus 15 there.

00:03:40.170 --> 00:03:44.450
Then the left-hand side
just becomes an x.

00:03:44.450 --> 00:03:46.110
Because obviously you
have 15 minus 15.

00:03:46.110 --> 00:03:47.420
That just cancels out.

00:03:47.420 --> 00:03:51.850
And you get x is greater than or
equal to negative 60 minus

00:03:51.850 --> 00:03:56.000
15 is negative 75.

00:03:56.000 --> 00:03:59.050
If something is greater than or
equal to something else, if

00:03:59.050 --> 00:04:01.830
I take 15 away from this and
from that, the greater than or

00:04:01.830 --> 00:04:03.760
equal sign will still apply.

00:04:03.760 --> 00:04:06.320
So our solution is x is greater
than or equal to

00:04:06.320 --> 00:04:07.950
negative 75.

00:04:07.950 --> 00:04:09.995
Let's graph it on
the number line.

00:04:09.995 --> 00:04:13.060
So let me draw a number
line here.

00:04:13.060 --> 00:04:18.310
I'll have-- let's say that's
negative 75, that's negative

00:04:18.310 --> 00:04:24.310
74, that's negative 73,
that's negative 76.

00:04:24.310 --> 00:04:25.450
And so on and so forth.

00:04:25.450 --> 00:04:26.870
I could keep plotting things.

00:04:26.870 --> 00:04:31.280
Now, x has to be greater than
or equal to negative 75.

00:04:31.280 --> 00:04:34.590
So x can be equal
to negative 75.

00:04:34.590 --> 00:04:38.470
So we can include the point,
because we have this greater

00:04:38.470 --> 00:04:39.840
than or equal sign.

00:04:39.840 --> 00:04:41.540
Notice we're not making it
hollow like we did there,

00:04:41.540 --> 00:04:44.090
we're making it filled in
because it can equal negative

00:04:44.090 --> 00:04:46.870
75, or it needs to
be greater than.

00:04:46.870 --> 00:04:48.015
So greater than or equal.

00:04:48.015 --> 00:04:52.340
We'll shade in everything above
negative 75 as well.

00:04:52.340 --> 00:04:55.160
So in orange is the
solution set.

00:04:55.160 --> 00:04:58.200
And this obviously, we could
keep going to the right. x

00:04:58.200 --> 00:04:59.970
could be a million, it
could be a billion,

00:04:59.970 --> 00:05:00.830
it could be a googol.

00:05:00.830 --> 00:05:03.220
It can be an arbitrarily large
number as long as it's greater

00:05:03.220 --> 00:05:06.090
than or equal to negative 75.

00:05:06.090 --> 00:05:08.080
Let's do a couple more.

00:05:08.080 --> 00:05:17.940
Let's do x minus 2 is less
than or equal to 1.

00:05:17.940 --> 00:05:19.650
Once again, we want to
get just our x on

00:05:19.650 --> 00:05:20.440
the left-hand side.

00:05:20.440 --> 00:05:21.830
Get rid of this negative 2.

00:05:21.830 --> 00:05:24.400
Let's add 2 to both sides
of this equation.

00:05:24.400 --> 00:05:27.410
Plus 2.

00:05:27.410 --> 00:05:29.680
The left-hand side just
becomes an x.

00:05:29.680 --> 00:05:31.400
You have a less than
or equal sign.

00:05:31.400 --> 00:05:34.130
That won't change by adding or
subtracting the same thing to

00:05:34.130 --> 00:05:36.140
both sides of the inequality.

00:05:36.140 --> 00:05:37.740
And then 1 plus 2 is 3.

00:05:37.740 --> 00:05:40.010
So x needs to be less
than or equal to 3.

00:05:40.010 --> 00:05:42.960
Any x that is less than
or equal to 3 will

00:05:42.960 --> 00:05:45.120
satisfy this equation.

00:05:45.120 --> 00:05:46.640
So let's plot it.

00:05:46.640 --> 00:05:49.180
And I'd try out any x that's
less than or equal to 3 and

00:05:49.180 --> 00:05:52.160
verify for yourself that it
does indeed satisfy this

00:05:52.160 --> 00:05:52.740
inequality.

00:05:52.740 --> 00:05:53.760
I shouldn't call
it an equation.

00:05:53.760 --> 00:05:55.790
This inequality.

00:05:55.790 --> 00:05:59.520
So let me graph the
solution set.

00:05:59.520 --> 00:06:05.140
So let's say this is
0, 1, 2, 3, 4.

00:06:05.140 --> 00:06:07.830
That's negative 1, negative 2.

00:06:07.830 --> 00:06:10.020
So x has to be less than
or equal to 3.

00:06:10.020 --> 00:06:13.480
It can be equal to 3, so
we fill in the dot,

00:06:13.480 --> 00:06:14.920
or less than 3.

00:06:14.920 --> 00:06:18.730
So the solution set for
over here is in pink.

00:06:18.730 --> 00:06:20.286
Anything less than
or equal to 3.

00:06:20.286 --> 00:06:21.640
And verify it for yourself.

00:06:21.640 --> 00:06:25.670
If x is equal to 3, you get 3
minus 2, which is equal to 1.

00:06:25.670 --> 00:06:29.170
And that is valid because it
could be less than or equal.

00:06:29.170 --> 00:06:34.950
If you do 2.999999 minus 2,
you get 0.999999, which is

00:06:34.950 --> 00:06:35.920
less than 1.

00:06:35.920 --> 00:06:38.500
And you can keep trying for any
of these numbers in this

00:06:38.500 --> 00:06:40.280
pink solution set here.

00:06:40.280 --> 00:06:43.370
Let's do one more.

00:06:43.370 --> 00:06:49.830
Let's say we have x minus 32
is less than or equal to 0.

00:06:49.830 --> 00:06:51.160
Same drill as before.

00:06:51.160 --> 00:06:53.275
Let's add 32 to both sides
of this equation.

00:06:57.360 --> 00:07:00.820
The left-hand side just becomes
x, and then the

00:07:00.820 --> 00:07:04.650
right-hand side is less
than or equal to 32.

00:07:04.650 --> 00:07:05.800
Pretty straightforward.

00:07:05.800 --> 00:07:09.110
Same drill when we graph
this equation.

00:07:09.110 --> 00:07:11.250
We draw the number line.

00:07:11.250 --> 00:07:16.200
If this is 32, this
is 33, this is 31.

00:07:16.200 --> 00:07:20.140
I could keep adding things
above and below 32.

00:07:20.140 --> 00:07:23.000
For the solution set is
everything less than or equal,

00:07:23.000 --> 00:07:26.410
so we can-- it could be equal
to 32, or less than.

00:07:26.410 --> 00:07:30.580
So we fill in everything
below that.

00:07:30.580 --> 00:07:32.790
Remember, the reason why we're
filling in this solid, the

00:07:32.790 --> 00:07:36.960
reason why 32 is an acceptable
solution to this original

00:07:36.960 --> 00:07:41.090
inequality, is because of this
less than or equal sign.

00:07:41.090 --> 00:07:43.040
Over here, you didn't have less
than or equal, and that's

00:07:43.040 --> 00:07:45.890
why 40 wasn't part of
the solution set.


Using a Linear Model

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.160
Let's do another example where
we have some data.

00:00:03.160 --> 00:00:06.880
We want to fit a line to that
data, and maybe use that to

00:00:06.880 --> 00:00:10.870
estimate some of the
data that we didn't

00:00:10.870 --> 00:00:12.440
collect, so to speak.

00:00:12.440 --> 00:00:15.010
So the first thing they ask us,
let's look at the chart

00:00:15.010 --> 00:00:15.490
first of all.

00:00:15.490 --> 00:00:16.480
This is birth year.

00:00:16.480 --> 00:00:17.680
They started in 1930.

00:00:17.680 --> 00:00:19.720
They go all the way up
to the year 2000 in

00:00:19.720 --> 00:00:21.720
increments of 10 years.

00:00:21.720 --> 00:00:23.990
They tell us the life
expectancy in years.

00:00:23.990 --> 00:00:25.470
So a good thing for us.

00:00:25.470 --> 00:00:27.660
I think this is the
United States.

00:00:27.660 --> 00:00:30.930
The life expectancy has been
steadily increasing.

00:00:30.930 --> 00:00:33.560
The first thing they ask us to
do is to make a scatter plot

00:00:33.560 --> 00:00:34.540
of the data.

00:00:34.540 --> 00:00:36.510
I did this on Excel
ahead of time.

00:00:36.510 --> 00:00:37.900
This is what I got.

00:00:37.900 --> 00:00:43.940
As you can see in 1930
it was 59.7.

00:00:43.940 --> 00:00:46.970
In 1940 it was 62.9.

00:00:46.970 --> 00:00:49.340
These blue dots are
just plotting each

00:00:49.340 --> 00:00:51.600
of these data points.

00:00:51.600 --> 00:00:53.270
1930 is 57.9.

00:00:53.270 --> 00:00:55.230
It's that data point
right there.

00:00:55.230 --> 00:00:58.770
In 1940 it was 62.9.

00:00:58.770 --> 00:01:00.410
It's that data point
right there.

00:01:00.410 --> 00:01:02.820
So I just went ahead and
plotted all of them.

00:01:02.820 --> 00:01:09.060
Then I let Excel figure out
a line of best fit.

00:01:09.060 --> 00:01:10.970
I didn't go into the math
of how to do this.

00:01:10.970 --> 00:01:12.610
We'll do that in
future videos.

00:01:12.610 --> 00:01:15.310
It's out of the scope of
introductory algebra.

00:01:15.310 --> 00:01:20.480
But Excel figured out that the
line of best fit, the line

00:01:20.480 --> 00:01:23.290
that is least far away from
all of these points, that

00:01:23.290 --> 00:01:26.930
minimizes the distance from all
of these points, is y is

00:01:26.930 --> 00:01:35.150
equal to 0.2395x minus 400.99.

00:01:35.150 --> 00:01:37.170
I'm not going to go into what
r squared is just yet.

00:01:37.170 --> 00:01:41.680
But it's a measure of how good
this line is actually fitting.

00:01:41.680 --> 00:01:43.300
So let's use that.

00:01:43.300 --> 00:01:44.300
Let's use this table.

00:01:44.300 --> 00:01:46.480
Let's use our line of best
fit to answer the

00:01:46.480 --> 00:01:48.760
rest of their questions.

00:01:48.760 --> 00:01:50.730
So the second question is, use
the line of best fit to

00:01:50.730 --> 00:01:55.210
estimate the life expectancy
of a person born in 1955.

00:01:55.210 --> 00:01:56.130
So we have two options.

00:01:56.130 --> 00:01:57.170
We could just eyeball it.

00:01:57.170 --> 00:01:59.610
1955 is right there.

00:01:59.610 --> 00:02:03.590
So if you go up from 1955 you're
going to hit the line

00:02:03.590 --> 00:02:04.880
of best fit.

00:02:04.880 --> 00:02:06.360
Let me do it in different color,
because that's the same

00:02:06.360 --> 00:02:08.220
color as the dots.

00:02:08.220 --> 00:02:11.590
In 1955 you're going
to hit the line of

00:02:11.590 --> 00:02:14.340
best fit right there.

00:02:14.340 --> 00:02:16.990
If I were to just eyeball
it, it looks

00:02:16.990 --> 00:02:18.810
like it's right there.

00:02:18.810 --> 00:02:21.070
This is 1, 2, 3, 4, 5.

00:02:21.070 --> 00:02:22.600
So these are increments of 2.

00:02:22.600 --> 00:02:25.690
So it looks like
about 77 years.

00:02:25.690 --> 00:02:33.320
So 1955, it looks roughly equal
to 77-- no, no sorry--

00:02:33.320 --> 00:02:39.550
67 years-- let me be clear, it
sounded suspicious-- a 67 year

00:02:39.550 --> 00:02:40.400
life expectancy.

00:02:40.400 --> 00:02:43.330
Now we can use the equation of
this line of best fit to come

00:02:43.330 --> 00:02:44.820
up with the exact value.

00:02:44.820 --> 00:02:47.940
When x is equal to
1955, what is y?

00:02:47.940 --> 00:02:49.930
What is the life expectancy
in years?

00:02:49.930 --> 00:02:52.280
Let's get our calculator
out for this.

00:02:52.280 --> 00:02:55.240
Actually let me use a graphing
calculator for

00:02:55.240 --> 00:02:56.195
this one right now.

00:02:56.195 --> 00:02:58.950
So let me clear this.

00:02:58.950 --> 00:03:01.740
Let me clear everything.

00:03:01.740 --> 00:03:04.950
Then let's just punch
in 1955 for x.

00:03:04.950 --> 00:03:15.900
So we have 0.2395 times 1955,
that's our x-value, minus

00:03:15.900 --> 00:03:24.040
400.99 is equal to 67.23.

00:03:24.040 --> 00:03:26.900
So our line of best fit
gives us the exact

00:03:26.900 --> 00:03:30.750
answer of 67.23 years.

00:03:30.750 --> 00:03:32.990
So our eyeballing it wasn't
that far off.

00:03:32.990 --> 00:03:36.260
So that's what our line of best
fit would predict, if we

00:03:36.260 --> 00:03:39.010
assume that a linear model
is a good model to

00:03:39.010 --> 00:03:40.500
describe this data.

00:03:40.500 --> 00:03:42.070
All we know right now are
linear models, so

00:03:42.070 --> 00:03:43.000
that's what we'll use.

00:03:43.000 --> 00:03:44.180
It actually looks pretty good.

00:03:44.180 --> 00:03:47.280
It seems like it fits the data
pretty, pretty good.

00:03:47.280 --> 00:03:48.970
Let's do the next part.

00:03:48.970 --> 00:03:52.410
Use linear interpolation to
estimate the life expectancy

00:03:52.410 --> 00:03:54.890
of a person born in 1955.

00:03:54.890 --> 00:03:59.430
Now linear interpolation means
let's take the data points

00:03:59.430 --> 00:04:01.010
around 1955.

00:04:01.010 --> 00:04:03.780
So you have this data
point from 1950.

00:04:03.780 --> 00:04:06.390
You have this data
point from 1960.

00:04:06.390 --> 00:04:10.120
Let's draw a line between those
two data points and

00:04:10.120 --> 00:04:14.900
assume that the 1955 number
would be right in between on

00:04:14.900 --> 00:04:18.310
the line there.

00:04:18.310 --> 00:04:20.510
Now we could figure out the
equation of that line.

00:04:20.510 --> 00:04:21.640
We have two points.

00:04:21.640 --> 00:04:26.000
We have the 1950 data and
we have the 1960 data.

00:04:26.000 --> 00:04:27.500
We could figure out the
slope of the line.

00:04:27.500 --> 00:04:29.240
We could figure out
its y-intercept.

00:04:29.240 --> 00:04:31.980
Then we can substitute
1955 in there.

00:04:31.980 --> 00:04:35.260
But this is a line, and
1955 is right in

00:04:35.260 --> 00:04:37.130
between these two points.

00:04:37.130 --> 00:04:38.740
It's 5 from either one.

00:04:38.740 --> 00:04:42.400
So on this point right here,
that's on that linear

00:04:42.400 --> 00:04:44.630
interpolation, on that orange,
will just be halfway

00:04:44.630 --> 00:04:45.550
between these two.

00:04:45.550 --> 00:04:47.290
So we can even average those.

00:04:47.290 --> 00:04:48.960
So let's do that.

00:04:48.960 --> 00:04:58.780
So if we just average those,
68.2 plus 69.7 is equal, and

00:04:58.780 --> 00:05:00.000
then we want to divide it.

00:05:00.000 --> 00:05:01.900
That answer means our
previous answer.

00:05:01.900 --> 00:05:08.040
So that number divided by
2 is equal to 68.95.

00:05:08.040 --> 00:05:15.640
So part 2, through linear
interpolation we get 68.95.

00:05:15.640 --> 00:05:16.670
Is that what I got?

00:05:16.670 --> 00:05:20.580
68.95, so that is this
point right here.

00:05:20.580 --> 00:05:29.530
So linear interpolation is
giving us a slightly higher

00:05:29.530 --> 00:05:32.770
estimate than our line of best
fit across all of the data.

00:05:32.770 --> 00:05:35.210
You can even eyeball it if you
look at it right there.

00:05:35.210 --> 00:05:36.990
Let's do the next one.

00:05:36.990 --> 00:05:39.720
Use a line of best fit to
estimate the life expectancy

00:05:39.720 --> 00:05:42.370
of a person born in 1976.

00:05:42.370 --> 00:05:44.740
Once again, we can use the
equation of the line of best

00:05:44.740 --> 00:05:47.820
fit, plug in 1976 here and
see what it gets us.

00:05:47.820 --> 00:05:48.640
We can eyeball it.

00:05:48.640 --> 00:05:49.840
1975 is there.

00:05:49.840 --> 00:05:51.790
1976 is there.

00:05:51.790 --> 00:05:54.440
It'll be right around there.

00:05:54.440 --> 00:05:56.130
Let's actually use the
actual equation.

00:05:56.130 --> 00:06:01.300
So in 1976, this is going to be
the line of best fit, the

00:06:01.300 --> 00:06:05.320
line of best fit is going to
give us, if we actually figure

00:06:05.320 --> 00:06:13.080
out what this line value gives
us, so we get 0.2395 times

00:06:13.080 --> 00:06:21.560
1976 minus 400.99.

00:06:21.560 --> 00:06:28.340
So 72.26 years, so it gives
us 72.26 years.

00:06:28.340 --> 00:06:31.580
So it equals 72.26.

00:06:31.580 --> 00:06:33.370
Let me make this clear here.

00:06:33.370 --> 00:06:39.280
These columns, this one right
here, is line of best fit.

00:06:39.280 --> 00:06:40.755
This one right here
is interpolation.

00:06:43.450 --> 00:06:46.450
Now they want to use linear
interpolation to estimate the

00:06:46.450 --> 00:06:49.500
life expectancy of a person
born in 1976.

00:06:49.500 --> 00:06:53.000
So what they want us to do is
draw a line between these two

00:06:53.000 --> 00:06:57.030
points, and then use that line,
not the whole line of

00:06:57.030 --> 00:07:00.890
best fit, to figure out what
1976 might have been.

00:07:00.890 --> 00:07:05.310
Now before, 1955 was right in
between those two values.

00:07:05.310 --> 00:07:08.180
1976 is not right in between
those values.

00:07:08.180 --> 00:07:11.140
So let's actually figure out
the equation of this line

00:07:11.140 --> 00:07:21.370
using the points, these two
points, 1970 comma 70.8, and

00:07:21.370 --> 00:07:24.280
1980 comma 73.7.

00:07:24.280 --> 00:07:27.070
Let's figure out the slope
of that line, of

00:07:27.070 --> 00:07:29.170
this line right here.

00:07:29.170 --> 00:07:34.060
So the slope is the change
in y over change in x.

00:07:34.060 --> 00:07:42.660
It's going to be 73.7
minus 70.8.

00:07:42.660 --> 00:07:45.980
That's how much we change in
the y direction, divided by

00:07:45.980 --> 00:07:53.640
how much we change in the x
direction, 1980 minus 1970.

00:07:53.640 --> 00:08:02.380
So this is going to be equal
to 3.7 minus 0.8 is 2.9.

00:08:02.380 --> 00:08:05.515
So it's 2.9 over 10.

00:08:05.515 --> 00:08:08.420
So it's equal to 0.29.

00:08:08.420 --> 00:08:14.030
We changed 2.9 years in life
expectancy every 10 years over

00:08:14.030 --> 00:08:19.040
that time period right there,
or 0.29 per year.

00:08:19.040 --> 00:08:20.770
Now what's the equation
of that line?

00:08:20.770 --> 00:08:29.780
Well we know that the point 1970
and 70.8 is on that line.

00:08:29.780 --> 00:08:34.880
So if we say the equation is y
is equal to mx plus b, we know

00:08:34.880 --> 00:08:46.970
that y is equal to 70.8 when--
we know m is 0.29-- when x is

00:08:46.970 --> 00:08:50.800
equal to, or when the year
is equal to 1970,

00:08:50.800 --> 00:08:54.830
times 1970 plus b.

00:08:54.830 --> 00:08:58.240
Now we can calculate
our y-intercept.

00:08:58.240 --> 00:08:59.385
So what is our y-intercept?

00:08:59.385 --> 00:09:00.950
We'll get our calculator out.

00:09:00.950 --> 00:09:03.550
We have to subtract this
from both sides.

00:09:03.550 --> 00:09:13.060
So b is equal to 70.8 minus
0.29 times 1970.

00:09:13.060 --> 00:09:16.420
Let's get our calculator out.

00:09:16.420 --> 00:09:17.780
There we go.

00:09:17.780 --> 00:09:29.460
All right, 70.8 minus
0.29 times 1970 is

00:09:29.460 --> 00:09:33.070
equal to minus 500.5.

00:09:33.070 --> 00:09:37.620
So this is equal to
negative 500.5.

00:09:37.620 --> 00:09:40.140
So the equation of our
interpolation line-- remember

00:09:40.140 --> 00:09:41.920
we're just finding the equation
of this line between

00:09:41.920 --> 00:09:48.150
1970 and 1980-- is, that
equation is y is equal to

00:09:48.150 --> 00:09:57.360
0.29x minus 500.5.

00:09:57.360 --> 00:10:00.615
Now if we want to figure out
what that line, if we want to

00:10:00.615 --> 00:10:05.060
interpolate 1976, we just have
to put 1976 in there for x.

00:10:05.060 --> 00:10:07.460
So what is life expectancy
in 1976?

00:10:07.460 --> 00:10:09.230
Get our calculator out again.

00:10:09.230 --> 00:10:23.120
0.29 times 1976 minus 500.5
is equal to 72.54.

00:10:23.120 --> 00:10:27.200
So we get 72.54 is
what happens?

00:10:27.200 --> 00:10:30.720
It's the number we get when
we do linear interpolation

00:10:30.720 --> 00:10:32.930
between those two data points.

00:10:32.930 --> 00:10:35.210
All we did is we figured out
the equation of the line

00:10:35.210 --> 00:10:37.080
between those two points.

00:10:37.080 --> 00:10:41.370
Then we plugged in the number
1976 to get 72.54.

00:10:41.370 --> 00:10:44.490
That's a slightly, slightly
higher number than what the

00:10:44.490 --> 00:10:45.920
line of best fit gave us.

00:10:45.920 --> 00:10:49.880
But it's pretty, pretty close
in either scenario.

00:10:49.880 --> 00:10:51.890
Now let's do this last part.

00:10:51.890 --> 00:10:55.340
Use the line of best fit to
estimate the life expectancy

00:10:55.340 --> 00:10:57.960
of a person born 2012.

00:10:57.960 --> 00:11:00.540
This is what's cool about
the line of best fit.

00:11:00.540 --> 00:11:02.250
We don't have 2012 here.

00:11:02.250 --> 00:11:03.470
2012 goes off the table.

00:11:03.470 --> 00:11:07.830
But we could just keep
continuing on with this line

00:11:07.830 --> 00:11:11.400
right here-- now I'm using the
wrong color-- keep continuing

00:11:11.400 --> 00:11:15.320
with this line right here and
see what does this line equal

00:11:15.320 --> 00:11:17.540
when the year is
equal to 2012?

00:11:17.540 --> 00:11:21.370
So all we have to do is put 2012
in for x and we'll get

00:11:21.370 --> 00:11:23.640
what the line of best
fits tells us.

00:11:23.640 --> 00:11:26.840
We'll get the calculator
out again.

00:11:26.840 --> 00:11:36.780
So we have 0.2395
times 2012 minus

00:11:36.780 --> 00:11:46.430
400.99 is equal to 80.88.

00:11:46.430 --> 00:11:51.090
So this last part, in 2012 if
you were to use a line of best

00:11:51.090 --> 00:11:53.940
fit, the average life expectancy
in the U.S. will be

00:11:53.940 --> 00:11:56.240
80.88 years.

00:11:56.240 --> 00:11:58.620
Anyway, hopefully you found
that to be interesting.

00:11:58.620 --> 00:12:00.550
Sometimes dealing with data
can be a little bit hairy.

00:12:00.550 --> 00:12:02.730
These aren't nice,
clean numbers.

00:12:02.730 --> 00:12:04.680
But it gives you a little sense
of why slope, and y-

00:12:04.680 --> 00:12:07.750
intercept, and interpolation,
and fitting a line, is

00:12:07.750 --> 00:12:10.670
actually very important to
analyzing data and coming to

00:12:10.670 --> 00:12:13.350
conclusions from that data.


Fitting a line to data

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.190
In this video I want to give
you an example of what it

00:00:03.190 --> 00:00:06.170
means to fit data to a line.

00:00:06.170 --> 00:00:08.730
Instead of doing my traditional
video using my

00:00:08.730 --> 00:00:11.840
little pen tablet, I'm going to
do it straight on Excel so

00:00:11.840 --> 00:00:14.250
you could see how to do this for
yourself, so if you have

00:00:14.250 --> 00:00:18.090
Excel or some other type of
a spreadsheet program.

00:00:18.090 --> 00:00:19.380
We're not going to go
into the math of it.

00:00:19.380 --> 00:00:21.540
I really just want you to get
the conceptual understanding

00:00:21.540 --> 00:00:26.840
of what it means to fit data
with line, or do a linear

00:00:26.840 --> 00:00:27.880
regression.

00:00:27.880 --> 00:00:29.550
So here, let's just
read the problem.

00:00:29.550 --> 00:00:33.550
The following table shows the
median California income--

00:00:33.550 --> 00:00:37.190
remember median is the middle,
the middle California income

00:00:37.190 --> 00:00:40.850
--from 1995 to 2002
as reported by

00:00:40.850 --> 00:00:42.680
the U.S. Census Bureau.

00:00:42.680 --> 00:00:45.320
Draw a scatter plot and
find the equation.

00:00:45.320 --> 00:00:47.540
What would you expect the median
annual income of a

00:00:47.540 --> 00:00:50.370
California family to be
in the year 2010?

00:00:50.370 --> 00:00:52.970
What are the meanings of the
slope and the y-intercept of

00:00:52.970 --> 00:00:53.580
this problem?

00:00:53.580 --> 00:00:56.050
So the first thing you'd want
to do-- I just copied and

00:00:56.050 --> 00:00:59.060
pasted this image --we have to
get the data in a form that

00:00:59.060 --> 00:01:00.920
the spreadsheet can
understand it.

00:01:00.920 --> 00:01:03.830
So let's make some
tables here.

00:01:03.830 --> 00:01:08.800
Let's say years since 1995.

00:01:08.800 --> 00:01:10.710
Let's make that one column.

00:01:10.710 --> 00:01:13.670
Let me make this a
little bit wider.

00:01:13.670 --> 00:01:16.670
Then let me put median income.

00:01:16.670 --> 00:01:20.100
This is the median income in
California for a family.

00:01:20.100 --> 00:01:23.560
So we start off with 1 year,
or 0 years since

00:01:23.560 --> 00:01:28.100
1995, 0, 1, 2, 3, 4.

00:01:28.100 --> 00:01:32.710
Actually if you want, it'll
figure out the trend if you

00:01:32.710 --> 00:01:33.590
just keep going down.

00:01:33.590 --> 00:01:36.650
It'll figure out you're just
incrementing by 1.

00:01:36.650 --> 00:01:38.340
Then the income, I'll
just copy in these

00:01:38.340 --> 00:01:39.300
numbers right there.

00:01:39.300 --> 00:01:54.710
So that's $53,807, $55,217,
$55,209, $55,415 $63,100,

00:01:54.710 --> 00:02:08.680
$63,206, $63,761, and then
we have $65,766.

00:02:08.680 --> 00:02:10.520
So I don't need these
over here.

00:02:10.520 --> 00:02:13.380
So I'm going to get
rid of them.

00:02:13.380 --> 00:02:14.810
I can clear them.

00:02:14.810 --> 00:02:16.180
So let me make sure I
have enough entries.

00:02:16.180 --> 00:02:24.040
This is 1, 2, 3, 4, 5, 6, 7, 8,
and I have 1, 2, 3, 4, 5,

00:02:24.040 --> 00:02:25.570
6, 7, 8 entries.

00:02:25.570 --> 00:02:29.930
I want to make sure I got my
data right. $53,807, $55,217,

00:02:29.930 --> 00:02:34.510
$55,209, 415, 100,
206, 761, 766.

00:02:34.510 --> 00:02:36.040
OK, there we go.

00:02:36.040 --> 00:02:38.880
Now you're going to find that
in Excel this is incredibly

00:02:38.880 --> 00:02:41.080
easy if you know what
to click on.

00:02:41.080 --> 00:02:44.430
One, plot this data, create a
scatter plot, and then even

00:02:44.430 --> 00:02:46.220
better, create a regression
of that data.

00:02:46.220 --> 00:02:48.970
So all you have to do is
you select the data.

00:02:48.970 --> 00:02:51.480
Then you go to insert,
and I'm going to

00:02:51.480 --> 00:02:53.170
insert a scatter plot.

00:02:53.170 --> 00:02:54.390
Then you can pick the different

00:02:54.390 --> 00:02:56.160
types of scatter plots.

00:02:56.160 --> 00:02:58.820
I just want to plot the data.

00:02:58.820 --> 00:02:59.670
There you go.

00:02:59.670 --> 00:03:02.470
It plotted the data for me.

00:03:02.470 --> 00:03:03.050
There you go.

00:03:03.050 --> 00:03:07.320
If you go by this is the actual
income, and this is by

00:03:07.320 --> 00:03:08.870
year since 1995.

00:03:08.870 --> 00:03:10.570
So this is 1995.

00:03:10.570 --> 00:03:13.710
It was $53,807.

00:03:13.710 --> 00:03:17.370
In 1996 it's $55,217.

00:03:17.370 --> 00:03:19.250
So it plotted all the data.

00:03:19.250 --> 00:03:22.110
Now what I want to
do is fit a line.

00:03:22.110 --> 00:03:24.000
So this isn't exactly a line.

00:03:24.000 --> 00:03:27.580
But let's see, if we assume
that a line can model this

00:03:27.580 --> 00:03:31.500
data well, I'm going to get
Excel to fit a line for me.

00:03:31.500 --> 00:03:34.580
So what I can do is I have all
of these options up here for

00:03:34.580 --> 00:03:37.720
different ways to fit
a line, all of

00:03:37.720 --> 00:03:38.710
these different options.

00:03:38.710 --> 00:03:39.780
I'm going to pick
this one here.

00:03:39.780 --> 00:03:40.940
You might not be
able to see it.

00:03:40.940 --> 00:03:42.460
It looks like it has a
line between dots.

00:03:42.460 --> 00:03:45.450
It also has fx which tells me
going to tell me the equation

00:03:45.450 --> 00:03:46.470
of the line.

00:03:46.470 --> 00:03:49.520
So if I click on that,
there you go.

00:03:49.520 --> 00:03:52.570
It not only fit, it replotted
that same data

00:03:52.570 --> 00:03:54.530
on a different graph.

00:03:54.530 --> 00:03:55.830
Let me make it a little
bit bigger.

00:03:58.430 --> 00:03:59.930
No, I don't want to that.

00:03:59.930 --> 00:04:01.060
Let me make it a little
bit bigger.

00:04:01.060 --> 00:04:04.430
We can cover up the data now,
just because I think we know

00:04:04.430 --> 00:04:06.410
what's going on.

00:04:06.410 --> 00:04:09.460
So let me cover it up
right like that.

00:04:09.460 --> 00:04:12.620
So not only did it plot the
various data points, it

00:04:12.620 --> 00:04:16.029
actually fit a line to that
data and it gave me the

00:04:16.029 --> 00:04:17.355
equation of that line.

00:04:21.279 --> 00:04:22.975
Let me see if I can make this
a little bit bigger.

00:04:27.330 --> 00:04:30.910
I'll move it out of the way so
you can read it at least.

00:04:30.910 --> 00:04:33.400
So it tells me right here, that
the equation for this

00:04:33.400 --> 00:04:40.400
line is y is equal to 1,882.3x
plus 52,847.

00:04:40.400 --> 00:04:43.250
So if you remember what we
know about slope and

00:04:43.250 --> 00:04:50.660
y-intercept, the y-intercept
is 52,847, which is, if you

00:04:50.660 --> 00:04:55.000
use this line as your measure,
where this line intersects at

00:04:55.000 --> 00:04:57.470
year 0, or in 1995.

00:04:57.470 --> 00:05:01.830
So if you use this line as a
model, in 1995 the line would

00:05:01.830 --> 00:05:05.580
say that you're going
to make $52,847.

00:05:05.580 --> 00:05:07.280
The actual data was a little
bit off of that.

00:05:07.280 --> 00:05:09.970
It was a little bit
higher, $53,807.

00:05:09.970 --> 00:05:11.250
So it was a little bit higher.

00:05:11.250 --> 00:05:13.850
But we're trying to get a line
that gets as close as possible

00:05:13.850 --> 00:05:14.980
to all of this data.

00:05:14.980 --> 00:05:17.530
It's actually trying to minimize
the distance, the

00:05:17.530 --> 00:05:20.470
square of the distance, between
each of these points

00:05:20.470 --> 00:05:20.970
in the line.

00:05:20.970 --> 00:05:22.440
We won't go into
the math there.

00:05:22.440 --> 00:05:23.830
But it gave us this
nice equation.

00:05:23.830 --> 00:05:26.810
Now we can use this nice
equation to predict things.

00:05:26.810 --> 00:05:30.290
If we say that this is a good a
model for the data-- let me

00:05:30.290 --> 00:05:31.400
bring this down a little
bit --let's try

00:05:31.400 --> 00:05:32.400
to answer our question.

00:05:32.400 --> 00:05:35.210
So we drew a scatter plot--
really Excel did it for us.

00:05:35.210 --> 00:05:38.000
We found the equation
right there.

00:05:38.000 --> 00:05:40.060
They say, what would you expect
the median annual

00:05:40.060 --> 00:05:44.490
income of a California family
to be in the year 2010?

00:05:44.490 --> 00:05:50.010
So here, we can just use the
equation they gave us.

00:05:50.010 --> 00:05:51.770
This right here, was 2002.

00:05:51.770 --> 00:05:52.850
So I could write
down the year.

00:05:52.850 --> 00:05:57.240
This was the year of 2002.

00:05:57.240 --> 00:06:02.490
So the year 2010 is
8 more years.

00:06:02.490 --> 00:06:04.350
Let me make a little
column here.

00:06:08.390 --> 00:06:13.540
So this is the year,
1995, 1996.

00:06:13.540 --> 00:06:15.710
Then Excel will be able to
figure out if I select those,

00:06:15.710 --> 00:06:18.950
and I go to this little bottom
right square and I scroll

00:06:18.950 --> 00:06:23.920
down, Excel will actually figure
out that I want to

00:06:23.920 --> 00:06:26.980
increment by 1 year
every time.

00:06:26.980 --> 00:06:31.670
If I say years since 1995, once
again I can just continue

00:06:31.670 --> 00:06:33.550
this trend right here.

00:06:33.550 --> 00:06:36.600
So 2010 would be 15 years.

00:06:36.600 --> 00:06:38.660
So we can just apply
this equation.

00:06:38.660 --> 00:06:41.760
We could say it's going to be
equal to, according to this

00:06:41.760 --> 00:06:44.570
line-- I'm just going to type it
in, hopefully you can read

00:06:44.570 --> 00:06:51.000
what I'm saying --1,882.3
times x.

00:06:51.000 --> 00:06:55.500
x here is the year since 1995.

00:06:55.500 --> 00:06:58.430
I could just select this
cell, or I could type

00:06:58.430 --> 00:06:59.320
in the number 15.

00:06:59.320 --> 00:07:03.590
That means times this
cell, times 15.

00:07:03.590 --> 00:07:11.390
Then plus 52,847, plus
that right there.

00:07:11.390 --> 00:07:17.600
Click enter and it predicts
$81,081.50.

00:07:17.600 --> 00:07:21.610
So if you just continue this
line for another 8 or so

00:07:21.610 --> 00:07:24.550
years, it predicts that the
median income in California

00:07:24.550 --> 00:07:27.730
for a family will be $81,000.

00:07:27.730 --> 00:07:30.270
Anyway, hopefully you found
that interesting.

00:07:30.270 --> 00:07:33.340
Spreadsheets are very useful
tools for manipulating data.

00:07:33.340 --> 00:07:36.520
It'll give you a sense of why
linear models are interesting,

00:07:36.520 --> 00:07:39.130
why lines are interesting, and
how you can actually use these

00:07:39.130 --> 00:07:42.600
tools to interpret data and
maybe even extrapolate some

00:07:42.600 --> 00:07:43.370
type of a prediction.

00:07:43.370 --> 00:07:45.450
This right here, is an
extrapolation using this

00:07:45.450 --> 00:07:47.260
linear regression.


Predicting with Linear Models

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.130
Let's do a couple of problems
where we estimate or predict

00:00:04.130 --> 00:00:07.390
answers to questions using
linear models.

00:00:10.270 --> 00:00:12.030
It sounds very fancy.

00:00:12.030 --> 00:00:16.550
But what a linear model is is
just using a line to describe

00:00:16.550 --> 00:00:19.000
some trend in data.

00:00:19.000 --> 00:00:21.490
With our linear models we
will do a little bit of

00:00:21.490 --> 00:00:27.390
interpolation and a little
bit of extrapolation.

00:00:27.390 --> 00:00:29.280
You might have already heard
these words before in your

00:00:29.280 --> 00:00:30.530
everyday life.

00:00:34.690 --> 00:00:38.830
Interpolation means trying to
estimate what happened between

00:00:38.830 --> 00:00:39.620
two data points.

00:00:39.620 --> 00:00:41.700
We're going to do a couple
of examples of

00:00:41.700 --> 00:00:42.890
that in this video.

00:00:42.890 --> 00:00:46.680
Extrapolation means we'll see
what the last few data points

00:00:46.680 --> 00:00:49.950
were, and see what that trend
would look like, and then keep

00:00:49.950 --> 00:00:53.720
on continuing that trend and see
what might happen if that

00:00:53.720 --> 00:00:54.950
trend were to continue.

00:00:54.950 --> 00:00:57.380
I'll show you some
examples of that.

00:00:57.380 --> 00:00:59.990
So let's do some example
questions here.

00:00:59.990 --> 00:01:02.050
So I have this chart here.

00:01:02.050 --> 00:01:04.650
Median age, remember median
is the middle.

00:01:04.650 --> 00:01:14.310
So if I have the numbers 3, 7,
and 9, the median is the 7.

00:01:14.310 --> 00:01:17.390
It's the middle age.

00:01:17.390 --> 00:01:21.900
So this is the median age of
males and females at first

00:01:21.900 --> 00:01:23.750
marriage by year.

00:01:23.750 --> 00:01:26.650
So if we just look at this data
right here, if we look at

00:01:26.650 --> 00:01:31.200
1900, the average man was
getting married, not the

00:01:31.200 --> 00:01:32.730
average, the median male.

00:01:32.730 --> 00:01:36.080
So the middle man was getting
married at looks like at

00:01:36.080 --> 00:01:40.930
around the age a little over
25, almost 26, while the

00:01:40.930 --> 00:01:47.030
median woman was getting married
looks at around 22 or

00:01:47.030 --> 00:01:48.790
23 years old.

00:01:48.790 --> 00:01:51.160
As we went through the century,
those ages got lower,

00:01:51.160 --> 00:01:53.830
and lower, and lower, until
we got to about 1960.

00:01:53.830 --> 00:01:55.410
This problem is interesting
irrespective

00:01:55.410 --> 00:01:57.010
of the actual problem.

00:01:57.010 --> 00:02:00.330
It's just interesting to see
this trend that at the

00:02:00.330 --> 00:02:03.970
beginning of the century, people
got married reasonably

00:02:03.970 --> 00:02:06.020
similar to when they're getting
married right now.

00:02:06.020 --> 00:02:07.950
But there was a minimum point.

00:02:07.950 --> 00:02:11.070
Around the 1960s, people got
married at a younger age.

00:02:11.070 --> 00:02:12.870
This was true for males
and females.

00:02:12.870 --> 00:02:17.420
You can also see back here the
age difference between the

00:02:17.420 --> 00:02:19.400
median male and the
median female.

00:02:19.400 --> 00:02:22.370
That's just gotten smaller
and smaller over time.

00:02:22.370 --> 00:02:23.580
Now it's really small.

00:02:23.580 --> 00:02:24.910
So these are just
all interesting

00:02:24.910 --> 00:02:25.820
things from this chart.

00:02:25.820 --> 00:02:28.720
This is the actual data
from that chart.

00:02:28.720 --> 00:02:30.160
So actually I didn't even have
to estimate it here.

00:02:30.160 --> 00:02:36.620
In 1900, the median age of
males getting married was

00:02:36.620 --> 00:02:41.570
25.9, that's that right there,
and the median age of women

00:02:41.570 --> 00:02:44.520
getting married was 21.9,
that's that over there.

00:02:44.520 --> 00:02:47.060
So this is just a scatter
plot of this data.

00:02:47.060 --> 00:02:50.220
We've just plotted each of these
points over here, these

00:02:50.220 --> 00:02:52.660
in blue, these in red.

00:02:52.660 --> 00:02:55.230
Let's do the questions.

00:02:55.230 --> 00:03:00.020
Use the data from example 1--
this is example 1 up here-- to

00:03:00.020 --> 00:03:05.850
estimate the age at marriage
for females in 1946.

00:03:05.850 --> 00:03:10.750
Fit a line by hand to the
data before 1970.

00:03:10.750 --> 00:03:12.660
So what they want to do, is they
want me to look at all of

00:03:12.660 --> 00:03:14.370
the data before 1970.

00:03:14.370 --> 00:03:18.200
So let's see, this is
1970 right here.

00:03:18.200 --> 00:03:20.460
That's 1970 right there.

00:03:20.460 --> 00:03:26.420
They want to fit a line by hand
to the data before 1970.

00:03:26.420 --> 00:03:29.650
What they mean is-- and I'm just
going to eyeball -- but

00:03:29.650 --> 00:03:34.830
they want me to draw a line that
gets as close to all of

00:03:34.830 --> 00:03:38.390
this data before 1970
as possible.

00:03:38.390 --> 00:03:40.200
So it can't go through
all of these points.

00:03:40.200 --> 00:03:43.680
Because all these points don't
sit exactly on one line.

00:03:43.680 --> 00:03:47.710
But let me try to draw a line,
or a linear model, for the

00:03:47.710 --> 00:03:53.040
data from, what is this, 1890
all the way to 1970.

00:03:53.040 --> 00:03:57.370
So I'm going to try my best to
draw a line here that gets

00:03:57.370 --> 00:03:59.030
close to all of these points.

00:03:59.030 --> 00:04:01.090
It won't be able to go
through all of them.

00:04:01.090 --> 00:04:04.780
So all the way to 1970, maybe
a line would look

00:04:04.780 --> 00:04:07.310
something like that.

00:04:07.310 --> 00:04:09.296
Let me draw a better
one than that.

00:04:09.296 --> 00:04:11.600
I'll draw it in red because
we're dealing

00:04:11.600 --> 00:04:12.810
with the red data.

00:04:12.810 --> 00:04:15.620
So I want to go all
the way to 1970.

00:04:15.620 --> 00:04:18.130
It would have been nice to
have a line drawing tool.

00:04:18.130 --> 00:04:20.329
Maybe the line might look
something like that.

00:04:20.329 --> 00:04:23.880
There are actual mathematical
ways to figure out the best

00:04:23.880 --> 00:04:25.040
fit line on this.

00:04:25.040 --> 00:04:26.530
We won't go into
that right now.

00:04:26.530 --> 00:04:28.100
That's called linear
regression.

00:04:28.100 --> 00:04:29.130
But I just eyeballed it.

00:04:29.130 --> 00:04:31.930
That line looks like a pretty
good fit for all of that data.

00:04:31.930 --> 00:04:34.810
None of the data is too far
away from that line.

00:04:34.810 --> 00:04:37.960
So that's what they mean by fit
a line by hand to the data

00:04:37.960 --> 00:04:39.330
before 1970.

00:04:39.330 --> 00:04:42.540
This is the data before 1970.

00:04:42.540 --> 00:04:46.080
If we assume that this is a
good linear model for that

00:04:46.080 --> 00:04:53.340
data, we can use it to estimate
the age at marriage

00:04:53.340 --> 00:04:57.340
for females in 1946.

00:04:57.340 --> 00:04:59.380
Let's see, this is 1950.

00:04:59.380 --> 00:05:02.200
This would be 1945.

00:05:02.200 --> 00:05:04.060
1945 would be right over here.

00:05:04.060 --> 00:05:07.740
So 1946 would be right
over there.

00:05:07.740 --> 00:05:11.840
So if we wanted to estimate
that, if I were to figure out

00:05:11.840 --> 00:05:15.380
where that hits the vertical
axis, it seems to hit the

00:05:15.380 --> 00:05:22.695
vertical axis at around-- I
don't know, my best to guess

00:05:22.695 --> 00:05:26.015
is, I should have zoomed
in more-- it hits

00:05:26.015 --> 00:05:27.780
it right about there.

00:05:27.780 --> 00:05:31.260
This is 22.5.

00:05:31.260 --> 00:05:34.740
So this looks like maybe 20.5.

00:05:34.740 --> 00:05:38.520
Because this would be
maybe 20.5 years.

00:05:38.520 --> 00:05:41.300
So I'll call it twenty
20.5 years.

00:05:41.300 --> 00:05:43.550
Obviously I can't do it exact,
but you get the idea.

00:05:43.550 --> 00:05:44.800
I am interpolating.

00:05:47.090 --> 00:05:49.510
Here I'm using the
model to do.

00:05:49.510 --> 00:05:51.620
I won't use necessarily the
word interpolation yet.

00:05:51.620 --> 00:05:55.530
We're going to do a much more
direct form of interpolation

00:05:55.530 --> 00:05:56.700
later on in this video.

00:05:56.700 --> 00:06:00.040
But here I fit a line, and then
I just use that line as a

00:06:00.040 --> 00:06:06.425
model to estimate what was the
median age of women in 1946.

00:06:06.425 --> 00:06:08.090
Let's do part 2.

00:06:08.090 --> 00:06:10.890
Use the data from example 1 to
estimate the age of marriage

00:06:10.890 --> 00:06:13.280
for females in 1984.

00:06:13.280 --> 00:06:17.220
Fit a line by hand to the data
from 1970 on in order to

00:06:17.220 --> 00:06:18.700
estimate this accurately.

00:06:18.700 --> 00:06:21.050
So now they want us to fit
a line from the data on.

00:06:21.050 --> 00:06:22.220
So let me make a line.

00:06:22.220 --> 00:06:26.750
So fit a line by hand from
the data 1970 on.

00:06:26.750 --> 00:06:29.340
So let me draw a line that gets
close to all of these

00:06:29.340 --> 00:06:31.680
points, that gets close
to describing

00:06:31.680 --> 00:06:32.470
all of these points.

00:06:32.470 --> 00:06:35.520
Maybe it looks like something
like that.

00:06:35.520 --> 00:06:38.080
Not all of them sit exactly on
the line, but they're all

00:06:38.080 --> 00:06:39.520
pretty close.

00:06:39.520 --> 00:06:42.110
If we assume that this is a
good a linear model-- I

00:06:42.110 --> 00:06:44.970
haven't done this
mathematically, I'm eyeballing

00:06:44.970 --> 00:06:47.660
it-- but let's see what
it would be in 1984.

00:06:47.660 --> 00:06:50.410
This is 1990.

00:06:50.410 --> 00:06:55.540
So 1984 is going to be
right around there.

00:06:58.370 --> 00:07:02.340
Actually it looks amazingly
similar to the median age of

00:07:02.340 --> 00:07:04.340
men in 1970.

00:07:04.340 --> 00:07:06.170
So we can actually get
the exact number.

00:07:06.170 --> 00:07:08.020
It's about 23.2.

00:07:08.020 --> 00:07:14.550
So this looks to be
about 23.2 years.

00:07:14.550 --> 00:07:18.890
Part 3, use the data from
example 1 to estimate the age

00:07:18.890 --> 00:07:21.870
at marriage for males in 1995.

00:07:21.870 --> 00:07:23.160
Use linear interpolation.

00:07:23.160 --> 00:07:25.750
Now we're going to do more
direct form of interpolation.

00:07:25.750 --> 00:07:27.270
We're not going to use a linear
model or we're not

00:07:27.270 --> 00:07:28.220
going to fit a line.

00:07:28.220 --> 00:07:33.600
We're going to use linear
interpolation between 1990 and

00:07:33.600 --> 00:07:35.180
2000 data points.

00:07:35.180 --> 00:07:36.270
We're looking at men now.

00:07:36.270 --> 00:07:37.530
So I'll do it in blue.

00:07:37.530 --> 00:07:43.140
Between 1990, that is 1990
and that is 2000 for men.

00:07:43.140 --> 00:07:46.810
So if I were to draw just a line
there, we could assume

00:07:46.810 --> 00:07:48.650
that the trend would have looked
something like that

00:07:48.650 --> 00:07:51.030
between 1990 and 2000.

00:07:51.030 --> 00:07:54.330
They want us to interpolate
what happened in 1995.

00:07:54.330 --> 00:07:57.280
So 1995 is that point
right there.

00:07:57.280 --> 00:07:59.320
So this is pure interpolation.

00:07:59.320 --> 00:08:01.940
We're using the data point
in 1990 and the

00:08:01.940 --> 00:08:03.225
data point in 2000.

00:08:03.225 --> 00:08:07.340
We're drawing a line between
them and assuming that 1995

00:08:07.340 --> 00:08:08.970
would have been right
in between them.

00:08:08.970 --> 00:08:12.130
We're interpolating between
the 1990 data

00:08:12.130 --> 00:08:13.910
and the 2000 data.

00:08:13.910 --> 00:08:19.150
So if you were to try to
estimate where that is, it

00:08:19.150 --> 00:08:22.710
looks like it's a little
bit over 26.

00:08:22.710 --> 00:08:25.550
Actually if we say that it's
right in between these two

00:08:25.550 --> 00:08:31.900
values right there, it'll
be 26.45 years.

00:08:31.900 --> 00:08:35.330
If it's exactly in between,
1995 is exactly

00:08:35.330 --> 00:08:36.600
5 years from each.

00:08:36.600 --> 00:08:40.080
So 26.45 looks pretty good.

00:08:40.080 --> 00:08:44.010
So let's do another example So
here's a percentage of women

00:08:44.010 --> 00:08:45.160
smokers by year.

00:08:45.160 --> 00:08:47.580
I have to say these charts
are interesting in and of

00:08:47.580 --> 00:08:48.510
themselves.

00:08:48.510 --> 00:08:52.420
You'll notice in 1990 there were
a good number of women

00:08:52.420 --> 00:08:54.660
who were pregnant who smoked.

00:08:54.660 --> 00:09:00.270
Almost 18% or 19% of pregnant
women smoked in 1990.

00:09:00.270 --> 00:09:01.050
Now that's down.

00:09:01.050 --> 00:09:05.070
Well this is 2004 so hopefully
this trend is continuing to go

00:09:05.070 --> 00:09:06.710
down eventually to 0%.

00:09:06.710 --> 00:09:08.090
But it went down to 10%.

00:09:08.090 --> 00:09:11.060
So there still are a percentage
of pregnant women

00:09:11.060 --> 00:09:14.870
who unfortunately smoke.

00:09:14.870 --> 00:09:17.260
Let's do these problems. Use
the data from example 2--

00:09:17.260 --> 00:09:19.870
that's this right here-- to
estimate the percentage of

00:09:19.870 --> 00:09:22.130
pregnant smokers in 1997.

00:09:22.130 --> 00:09:23.790
Use linear interpolation
between the

00:09:23.790 --> 00:09:28.650
1996 and 2000 data.

00:09:28.650 --> 00:09:30.850
We're going to use linear
interpolation.

00:09:30.850 --> 00:09:34.220
So let me draw a line.

00:09:34.220 --> 00:09:36.820
You could always do this much
more exact if you have better

00:09:36.820 --> 00:09:37.950
tools or a spreadsheet.

00:09:37.950 --> 00:09:47.480
But I'm going to draw a line,
and I want to get 1997.

00:09:47.480 --> 00:09:48.530
This is 1997.

00:09:48.530 --> 00:09:51.350
It's going to be right
over there.

00:09:51.350 --> 00:09:53.220
If we use linear interpolation,
we are

00:09:53.220 --> 00:09:54.620
interpolating between
that data point

00:09:54.620 --> 00:09:55.500
and that data point.

00:09:55.500 --> 00:09:56.355
It's a linear interpolation.

00:09:56.355 --> 00:09:59.530
We assume It's a linear
trend between the two.

00:09:59.530 --> 00:10:04.700
So this looks like it's-- let me
go all the way over here--

00:10:04.700 --> 00:10:06.850
looks a little bit more
than 12, maybe

00:10:06.850 --> 00:10:11.810
13% is my best estimate.

00:10:11.810 --> 00:10:13.260
Now part 5.

00:10:13.260 --> 00:10:16.060
Use the data from example 2--
same example right here-- to

00:10:16.060 --> 00:10:21.030
estimate the percentage of
pregnant smokers in 2006.

00:10:21.030 --> 00:10:22.450
There isn't any 2006.

00:10:22.450 --> 00:10:24.290
This chart ends at 2004.

00:10:24.290 --> 00:10:28.440
So we have to use linear
extrapolation with the final

00:10:28.440 --> 00:10:29.190
two data points.

00:10:29.190 --> 00:10:30.580
So this here you're going to
see the difference between

00:10:30.580 --> 00:10:32.520
interpolation and
extrapolation.

00:10:32.520 --> 00:10:35.050
Interpolation, you are
estimating between two data

00:10:35.050 --> 00:10:36.290
points you already knew.

00:10:36.290 --> 00:10:40.300
Extrapolation, we're going to
take the final two data points

00:10:40.300 --> 00:10:41.790
and we're going to continue
that line.

00:10:41.790 --> 00:10:43.610
We're going to extrapolate
that line.

00:10:43.610 --> 00:10:45.990
So if you were to just continue
this line between the

00:10:45.990 --> 00:10:47.400
final two data points,
it might look

00:10:47.400 --> 00:10:48.830
something like that.

00:10:48.830 --> 00:10:52.300
So if you had 2006 out here--
these are going up by two, so

00:10:52.300 --> 00:10:59.830
2004, 2006-- we extrapolate
this line.

00:10:59.830 --> 00:11:03.020
The percentage of women smoking
might be, well it

00:11:03.020 --> 00:11:07.160
looks a little bit under 10%.

00:11:07.160 --> 00:11:09.980
So maybe 9.5%.

00:11:09.980 --> 00:11:11.520
Once again, it's just
an estimate.

00:11:11.520 --> 00:11:13.290
But we're just extrapolating
the trend.

00:11:13.290 --> 00:11:15.000
We're assuming the
trend of the last

00:11:15.000 --> 00:11:16.660
two data points continues.

00:11:16.660 --> 00:11:20.370
Then using that, it would
be around 9.5% in 2006.

00:11:20.370 --> 00:11:24.300
Now maybe, that's not definitely
the only prediction

00:11:24.300 --> 00:11:24.730
you could make.

00:11:24.730 --> 00:11:28.390
You could extrapolate based on
all of the data and get a line

00:11:28.390 --> 00:11:31.010
that looks something-- let me
try to do it a little bit

00:11:31.010 --> 00:11:35.060
better than that-- that looks
something like that.

00:11:35.060 --> 00:11:37.620
Then this line would predict
a lower smoking rate.

00:11:37.620 --> 00:11:40.080
So it depends on what data you
use, what you're using to do

00:11:40.080 --> 00:11:42.760
your linear model with, and
how you're extrapolating.

00:11:42.760 --> 00:11:45.580
But anyway, hopefully you
found this useful.