Solving systems of equations by elimination

WEBVTT
Kind: captions
Language: en

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Let's explore a few
more methods for

00:00:02.210 --> 00:00:04.410
solving systems of equations.

00:00:04.410 --> 00:00:12.370
Let's say I have the equation,
3x plus 4y is equal to 2.5.

00:00:12.370 --> 00:00:24.080
And I have another equation, 5x
minus 4y is equal to 25.5.

00:00:24.080 --> 00:00:28.150
And we want to find an x and y
value that satisfies both of

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these equations.

00:00:29.220 --> 00:00:31.260
If you think of it graphically,
this would be the

00:00:31.260 --> 00:00:34.850
intersection of the lines that
represent the solution sets to

00:00:34.850 --> 00:00:37.080
both of these equations.

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So how can we proceed?

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We saw in substitution,
we like to

00:00:40.520 --> 00:00:42.350
eliminate one of the variables.

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We did it through substitution
last time.

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But is there anything we can add
or subtract-- let's focus

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on this yellow, on this top
equation right here-- is there

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anything that we can add
or subtract to both

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sides of this equation?

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Remember, any time you deal with
an equation you have to

00:00:56.290 --> 00:00:59.240
add or subtract the same
thing to both sides.

00:00:59.240 --> 00:01:01.590
But is there anything that we
could add or subtract to both

00:01:01.590 --> 00:01:03.410
sides of this equation
that might

00:01:03.410 --> 00:01:05.069
eliminate one of the variables?

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And then we would have one
equation in one variable, and

00:01:08.170 --> 00:01:09.610
we can solve for it.

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And it's probably not obvious,
even though it's sitting right

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in front of your face.

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Well, what if we
just added this

00:01:17.620 --> 00:01:19.190
equation to that equation?

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What I mean by that is, what if
we were to add 5x minus 4y

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to the left-hand side, and add
25.5 to the right-hand side?

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So if I were to literally add
this to the left-hand side,

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and add that to the
right-hand side.

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And you're probably saying, Sal,
hold on, how can you just

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add two equations like that?

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And remember, when you're doing
any equation, if I have

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any equation of the form-- well,
really, any equation--

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Ax plus By is equal to C, if I
want to do something to this

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equation, I just have to add the
same thing to both sides

00:01:53.460 --> 00:01:54.470
of the equation.

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So I could, for example, I could
add D to both sides of

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the equation.

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Because D is equal to
D, so I won't be

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changing the equation.

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You would get Ax plus By, plus
D is equal to C plus D.

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And we've seen that multiple,
multiple times.

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Anything you do to one side of
the equation, you have to do

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to the other side.

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But you're saying, hey, Sal,
wait, on the left-hand side,

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you're adding 5x minus
4y to the equation.

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On the right-hand side, you're
adding 25.5 to the equation.

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Aren't you adding two different
things to both sides

00:02:27.976 --> 00:02:28.840
of the equation?

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And my answer would be no.

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We know that 5x minus
4y is 25.5.

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This quantity and this quantity
are the same.

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They're both 25.5.

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This second equation is telling
me that explicitly.

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So I can add this to
the left-hand side.

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I'm essentially adding
25.5 to it.

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And I could add 25.5 to
the right-hand side.

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So let's do that.

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If we were to add the
left-hand side,

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3x plus 5x is 8x.

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And then what is 4y minus 4y?

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And this was the whole point.

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When I looked at these two
equations, I said, oh, I have

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a 4y, I have a negative 4y.

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If you just add these two
together, they are going to

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cancel out.

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They're going to be plus 0y.

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Or that whole term is just
going to go away.

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And that's going to be equal
to 2.5 plus 25.5 is 28.

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So you divide both sides.

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So you get 8x is equal to 28.

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And you divide both sides by 8,
and we get x is equal to 28

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over 8, or you divide the
numerator and the

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denominator by 4.

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That's equal to 7 over 2.

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That's our x value.

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Now we want to solve
for our y value.

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And we could substitute
this back into either

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of these two equations.

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Let's use the top one.

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You could do it with the
bottom one as well.

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So we know that 3 times x, 3
times 7 over 2-- I'm just

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substituting the x value we
figured out into this top

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equation-- 3 times 7 over 2,
plus 4y is equal to 2.5.

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Let me just write that as 5/2.

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We're going to stay in
the fraction world.

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So this is going to be 21 over
2 plus 4y is equal to 5/2.

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Subtract 21 over 2
from both sides.

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So minus 21 over 2,
minus 21 over 2.

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The left-hand side-- you're just
left with a 4y, because

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these two guys cancel out--
is equal to-- this is 5

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minus 21 over 2.

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That's negative 16 over 2.

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So that's negative 16 over 2,
which is the same thing--

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well, I'll write it out
as negative 16 over 2.

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Or we could write that-- let's
continue up here-- 4y-- I'm

00:05:02.030 --> 00:05:06.340
just continuing this train of
thought up here-- 4y is equal

00:05:06.340 --> 00:05:08.400
to negative 8.

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Divide both sides by
4, and you get y is

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equal to negative 2.

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So the solution to this equation
is x is equal to 7/2,

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y is equal to negative 2.

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This would be the coordinate
of their intersection.

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And you could try it out
on both of these

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equations right here.

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So let's verify that it also

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satisfies this bottom equation.

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5 times 7/2 is 35 over 2 minus
4 times negative 2, so minus

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negative 8.

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That's equivalent to-- let's
see, this is 17.5 plus 8.

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And that indeed does
equal 25.5.

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So this satisfies
both equations.

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Now let's see if we can use
our newly found skills to

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tackle a word problem, our
newly found skills in

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elimination.

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So here it says, Nadia and Peter
visit the candy store.

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Nadia buys 3 candy bars and 4
Fruit Roll-Ups for $2.84.

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Peter also buys 3 candy bars,
but can only afford 1

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additional Fruit Roll-Up.

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His purchase costs $1.79.

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What is the cost of each candy
bar and each Fruit Roll-Up?

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So let's define some
variables.

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Let's just use x and y.

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Let's let x equal cost of candy
bar-- I was going to do

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a c and a f for Fruit Roll-Up,
but I'll just stick with x and

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y-- cost of candy bar.

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And let y equal the cost
of a Fruit Roll-Up.

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All right.

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So what does this first
statement tell us?

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Nadia buys 3 candy bars, so the
cost of 3 candy bars is

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going to be 3x.

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And 4 Fruit Roll-Ups.

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Plus 4 times y, the cost
of a Fruit Roll-Up.

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This is how much Nadia spends.

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3 candy bars, 4 Fruit
Roll-Ups.

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And it's going to cost $2.84.

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That's what this first
statement tells us.

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It translates into
that equation.

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The second statement.

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Peter also buys 3 candy bars,
but could only afford 1

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additional Fruit Roll-Up.

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So plus 1 additional
Fruit Roll-Up.

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His purchase cost is
equal to $1.79.

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What is the cost of each candy
bar and each Fruit Roll-Up?

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And we're going to solve
this using elimination.

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You could solve this using any
of the techniques we've seen

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so far-- substitution,
elimination, even graphing,

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although it's kind of
hard to eyeball

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things with the graphing.

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So how can we do this?

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Remember, with elimination,
you're going to add-- let's

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focus on this top equation
right here.

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Is there something we could
add to both sides of this

00:08:08.380 --> 00:08:09.910
equation that'll help us

00:08:09.910 --> 00:08:12.170
eliminate one of the variables?

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Or let me put it this way, is
there something we could add

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or subtract to both sides of
this equation that will help

00:08:18.100 --> 00:08:20.550
us eliminate one of
the variables?

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Well, like in the problem we
did a little bit earlier in

00:08:22.950 --> 00:08:27.860
the video, what if we were to
subtract this equation, or

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what if we were to subtract 3x
plus y from 3x plus 4y on the

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left-hand side, and subtract
$1.79 from

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the right-hand side?

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And remember, by doing that, I
would be subtracting the same

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thing from both sides
of the equation.

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This is $1.79.

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How do I know?

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Because it says this
is equal to $1.79.

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So if we did that we would be
subtracting the same thing

00:08:51.810 --> 00:08:53.760
from both sides of
the equation.

00:08:53.760 --> 00:08:57.320
So let's subtract 3x
plus y from the

00:08:57.320 --> 00:08:58.620
left-hand side of the equation.

00:09:01.720 --> 00:09:03.550
And let me just do this
over on the right.

00:09:03.550 --> 00:09:08.550
If I subtract 3x plus y, that is
the same thing as negative

00:09:08.550 --> 00:09:12.680
3x minus y, if you just
distribute the negative sign.

00:09:12.680 --> 00:09:13.750
So let's subtract it.

00:09:13.750 --> 00:09:20.820
So you get negative 3x minus
y-- maybe I should make it

00:09:20.820 --> 00:09:23.750
very clear this is not a plus
sign; you could imagine I'm

00:09:23.750 --> 00:09:27.560
multiplying the second equation
by negative 1-- is

00:09:27.560 --> 00:09:30.460
equal to negative $1.79.

00:09:30.460 --> 00:09:32.000
I'm just taking the
second equation.

00:09:32.000 --> 00:09:34.090
You could imagine I'm
multiplying it by negative 1,

00:09:34.090 --> 00:09:37.440
and now I'm going to add the
left-hand side to the

00:09:37.440 --> 00:09:39.310
left-hand side of this equation,
and the right-hand

00:09:39.310 --> 00:09:42.160
side to the right-hand side
of that equation.

00:09:42.160 --> 00:09:45.450
And what do we get?

00:09:45.450 --> 00:09:49.010
When you add 3x plus 4y,
minus 3x, minus y,

00:09:49.010 --> 00:09:50.540
the 3x's cancel out.

00:09:50.540 --> 00:09:52.730
3x minus 3x is 0x.

00:09:52.730 --> 00:09:54.080
I won't even write it down.

00:09:54.080 --> 00:09:58.650
You get 4x minus-- sorry,
4y minus y.

00:09:58.650 --> 00:10:00.630
That is 3y.

00:10:00.630 --> 00:10:06.940
And that is going to be equal
to $2.84 minus $1.79.

00:10:06.940 --> 00:10:07.930
What is that?

00:10:07.930 --> 00:10:12.980
That's $1.05.

00:10:12.980 --> 00:10:15.100
So 3y is equal to $1.05.

00:10:15.100 --> 00:10:16.903
Divide both sides by 3.

00:10:20.360 --> 00:10:24.700
y is equal to-- what's
$1.05 divided by 3?

00:10:24.700 --> 00:10:27.980
So 3 goes into $1.05.

00:10:27.980 --> 00:10:30.340
It goes into 1 zero times.

00:10:30.340 --> 00:10:33.290
0 times 3 is 0.

00:10:33.290 --> 00:10:35.190
1 minus 0 is 1.

00:10:35.190 --> 00:10:36.620
Bring down a 0.

00:10:36.620 --> 00:10:38.800
3 goes into 10 three times.

00:10:38.800 --> 00:10:41.220
3 times 3 is 9.

00:10:41.220 --> 00:10:42.380
Subtract.

00:10:42.380 --> 00:10:44.170
10 minus 9 is 1.

00:10:44.170 --> 00:10:46.360
Bring down the 5.

00:10:46.360 --> 00:10:49.170
3 goes into 15 five times.

00:10:49.170 --> 00:10:51.740
5 times 3 is 15.

00:10:51.740 --> 00:10:52.630
Subtract.

00:10:52.630 --> 00:10:54.150
We have no remainder.

00:10:54.150 --> 00:10:57.910
So y is equal to $0.35.

00:10:57.910 --> 00:11:01.020
So the cost of a Fruit
Roll-Up is $0.35.

00:11:01.020 --> 00:11:03.930
Now we can substitute back into
either of these equations

00:11:03.930 --> 00:11:07.720
to figure out the cost
of a candy bar.

00:11:07.720 --> 00:11:12.470
So let's use this bottom
equation right here.

00:11:12.470 --> 00:11:14.680
Which was originally, if you
remember before I multiplied

00:11:14.680 --> 00:11:22.840
it by negative 1, it was 3x
plus y is equal to $1.79.

00:11:22.840 --> 00:11:29.590
So that means that 3x plus the
cost of a Fruit Roll-Up, 0.35

00:11:29.590 --> 00:11:32.090
is equal to $1.79.

00:11:32.090 --> 00:11:39.470
If we subtract 0.35 from both
sides, what do we get?

00:11:39.470 --> 00:11:42.110
The left-hand side-- you're just
left with the 3x; these

00:11:42.110 --> 00:11:45.530
cancel out-- is equal
to-- let's see, this

00:11:45.530 --> 00:11:47.750
is $1.79 minus $0.35.

00:11:47.750 --> 00:11:52.000
That's $1.44.

00:11:52.000 --> 00:11:59.670
And 3 goes into $1.44, I think
it goes-- well, 3 goes into

00:11:59.670 --> 00:12:03.250
$1.44, it goes into
1 zero times.

00:12:03.250 --> 00:12:06.280
1 times 3 is 0.

00:12:06.280 --> 00:12:07.540
Bring down the 1.

00:12:07.540 --> 00:12:08.190
Subtract.

00:12:08.190 --> 00:12:09.380
Bring down the 4.

00:12:09.380 --> 00:12:11.970
3 goes into 14 four times.

00:12:11.970 --> 00:12:14.130
4 times 3 is 12.

00:12:14.130 --> 00:12:17.360
I'm making this messy.

00:12:17.360 --> 00:12:20.260
14 minus 12 is 2.

00:12:20.260 --> 00:12:21.750
Bring down the 4.

00:12:21.750 --> 00:12:23.440
3 goes into 24 eight times.

00:12:23.440 --> 00:12:24.820
8 times 3 is 24.

00:12:24.820 --> 00:12:26.250
No remainder.

00:12:26.250 --> 00:12:30.740
So x is equal to 0.48.

00:12:30.740 --> 00:12:31.660
So there you have it.

00:12:31.660 --> 00:12:34.760
We figured out, using
elimination, that the cost of

00:12:34.760 --> 00:12:39.030
a candy bar is equal to $0.48,
and that the cost of a Fruit

00:12:39.030 --> 00:12:42.800
Roll-Up is equal to $0.35.


Solving linear systems by substitution

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:04.530
In the last video, we saw what
a system of equations is.

00:00:04.530 --> 00:00:06.970
And in this video, I'm going
to show you one algebraic

00:00:06.970 --> 00:00:09.390
technique for solving systems of
equations, where you don't

00:00:09.390 --> 00:00:12.630
have to graph the two lines and
try to figure out exactly

00:00:12.630 --> 00:00:13.560
where they intersect.

00:00:13.560 --> 00:00:15.610
This will give you an exact
algebraic answer.

00:00:15.610 --> 00:00:17.290
And in future videos,
we'll see more

00:00:17.290 --> 00:00:18.290
methods of doing this.

00:00:18.290 --> 00:00:19.760
So let's say you had
two equations.

00:00:19.760 --> 00:00:27.910
One is x plus 2y is equal to 9,
and the other equation is

00:00:27.910 --> 00:00:35.140
3x plus 5y is equal to 20.

00:00:35.140 --> 00:00:37.810
Now, if we did what we did in
the last video, we could graph

00:00:37.810 --> 00:00:38.350
each of these.

00:00:38.350 --> 00:00:39.010
These are lines.

00:00:39.010 --> 00:00:42.630
You could put them in either
slope-intercept form or

00:00:42.630 --> 00:00:43.560
point-slope form.

00:00:43.560 --> 00:00:45.290
They're in standard
form right now.

00:00:45.290 --> 00:00:47.530
And then you could graph each
of these lines, figure out

00:00:47.530 --> 00:00:50.040
where they intersect, and that
would be a solution to that.

00:00:50.040 --> 00:00:52.710
But it's sometimes hard to
find, to just by looking,

00:00:52.710 --> 00:00:54.280
figure out exactly where
they intersect.

00:00:54.280 --> 00:00:56.890
So let's figure out a way to
algebraically do this.

00:00:56.890 --> 00:00:59.150
And what I'm going to do is
the substitution method.

00:00:59.150 --> 00:01:02.400
I'm going to use one of the
equations to solve for one of

00:01:02.400 --> 00:01:05.440
the variables, and then I'm
going to substitute back in

00:01:05.440 --> 00:01:06.880
for that variable over here.

00:01:06.880 --> 00:01:08.570
So let me show you what
I'm talking about.

00:01:08.570 --> 00:01:11.500
So let me solve for x using
this top equation.

00:01:11.500 --> 00:01:16.440
So the top equation says x
plus 2y is equal to 9.

00:01:16.440 --> 00:01:20.410
I want to solve for x, so let's
subtract 2y from both

00:01:20.410 --> 00:01:22.520
sides of this equation.

00:01:22.520 --> 00:01:28.500
So I'm left with x is
equal to 9 minus 2y.

00:01:28.500 --> 00:01:30.780
This is what this first equation
is telling me.

00:01:30.780 --> 00:01:32.960
I just rearranged
it a little bit.

00:01:32.960 --> 00:01:34.710
The first equation
is saying that.

00:01:34.710 --> 00:01:39.350
So in order to satisfy both of
these equations, x has to

00:01:39.350 --> 00:01:41.890
satisfy this constraint
right here.

00:01:41.890 --> 00:01:47.080
So I can substitute this
back in for x.

00:01:47.080 --> 00:01:49.430
We're saying, this top equation
says, x has to be

00:01:49.430 --> 00:01:50.320
equal to this.

00:01:50.320 --> 00:01:52.690
Well, if x has to be equal
to that, let's

00:01:52.690 --> 00:01:54.820
substitute this in for x.

00:01:54.820 --> 00:01:59.990
So this second equation
will become 3 times x.

00:01:59.990 --> 00:02:07.300
And instead of an x, I'll write
this thing, 9 minus 2y.

00:02:07.300 --> 00:02:15.090
3 times 9 minus 2y, plus
5y is equal to 20.

00:02:15.090 --> 00:02:16.540
That's why it's called the
substitution method.

00:02:16.540 --> 00:02:18.250
I just substituted for x.

00:02:18.250 --> 00:02:20.730
And the reason why that's
useful is now I have one

00:02:20.730 --> 00:02:23.830
equation with one unknown,
and I can solve for y.

00:02:23.830 --> 00:02:25.060
So let's do that

00:02:25.060 --> 00:02:27.810
3 times 9 is 27.

00:02:27.810 --> 00:02:35.150
3 times negative 2 is negative
6y, plus 5y is equal to 20.

00:02:35.150 --> 00:02:39.180
Add the negative 6y plus the 5y,
add those two terms. You

00:02:39.180 --> 00:02:45.750
have 27-- let's see, this will
be-- minus y is equal to 20.

00:02:45.750 --> 00:02:48.870
Let's subtract 27
from both sides.

00:02:48.870 --> 00:02:51.260
And you get-- let me
write it out here.

00:02:51.260 --> 00:02:56.110
So let's subtract 27
from both sides.

00:02:56.110 --> 00:02:59.250
The left-hand side, the 27's
cancel each other out.

00:02:59.250 --> 00:03:03.420
And you're left with negative y
is equal to 20 minus 27, is

00:03:03.420 --> 00:03:04.770
negative 7.

00:03:04.770 --> 00:03:07.170
And then we can multiply both
sides of this equation by

00:03:07.170 --> 00:03:12.450
negative 1, and we get
y is equal to 7.

00:03:12.450 --> 00:03:20.320
So we found the y value of the
point of intersection of these

00:03:20.320 --> 00:03:22.320
two lines. y is equal to 7.

00:03:22.320 --> 00:03:26.550
Let me write over here, so I
don't have to keep scrolling

00:03:26.550 --> 00:03:29.660
down and back up.
y is equal to 7.

00:03:29.660 --> 00:03:32.290
Well, if we know y, we can
now solve for x. x is

00:03:32.290 --> 00:03:34.780
equal to 9 minus 2y.

00:03:34.780 --> 00:03:36.120
So let's do that.

00:03:36.120 --> 00:03:42.010
x is equal to 9 minus 2
times y, 2 times 7.

00:03:42.010 --> 00:03:49.210
Or x is equal to 9 minus 14, or
x is equal to negative 5.

00:03:49.210 --> 00:03:51.640
So we've just, using
substitution, we've been able

00:03:51.640 --> 00:03:55.600
to find a pair of x
and y points that

00:03:55.600 --> 00:03:57.020
satisfy these equations.

00:03:57.020 --> 00:04:00.990
The point x is equal to negative
5, y is equal to 7,

00:04:00.990 --> 00:04:02.520
satisfy both of these.

00:04:02.520 --> 00:04:04.830
And you can try it out.

00:04:04.830 --> 00:04:10.270
Negative 5 plus 2 times 7,
that's negative 5 plus 14,

00:04:10.270 --> 00:04:11.990
that is indeed 9.

00:04:11.990 --> 00:04:12.910
You do this equation.

00:04:12.910 --> 00:04:18.170
3 times negative 5 is negative
15, plus 5 times y,

00:04:18.170 --> 00:04:19.670
plus 5 times 7.

00:04:19.670 --> 00:04:24.310
So negative 15 plus
35 is indeed 20.

00:04:24.310 --> 00:04:26.350
So this satisfies
both equations.

00:04:26.350 --> 00:04:28.640
If you were to graph both of
these equations, they would

00:04:28.640 --> 00:04:33.090
intersect at the point
negative 5 comma 7.

00:04:33.090 --> 00:04:38.000
Now let's use our newly
found skill to do

00:04:38.000 --> 00:04:40.830
an actual word problem.

00:04:40.830 --> 00:04:46.530
Let's say that they tell
us that the sum of

00:04:46.530 --> 00:04:55.130
two numbers is 70.

00:04:55.130 --> 00:05:00.710
And they differ-- or maybe we
could say their difference--

00:05:00.710 --> 00:05:06.820
they differ by 11.

00:05:06.820 --> 00:05:08.070
What are the numbers?

00:05:14.900 --> 00:05:16.980
So let's do this word problem.

00:05:16.980 --> 00:05:18.460
So let's define some
variables.

00:05:18.460 --> 00:05:29.210
Let's let x be the larger
number, and let y be the

00:05:29.210 --> 00:05:30.420
smaller number.

00:05:30.420 --> 00:05:33.180
I'm just arbitrarily creating
these variables.

00:05:33.180 --> 00:05:34.640
One of them is larger
than the other.

00:05:34.640 --> 00:05:39.940
They differ by 11.

00:05:39.940 --> 00:05:44.020
Now, this first statement, the
sum of the two numbers is 70.

00:05:44.020 --> 00:05:51.340
That tells us that x plus
y must be equal to 70.

00:05:51.340 --> 00:05:56.350
That second statement, that
they differ by 11.

00:05:56.350 --> 00:05:58.450
That means the larger number
minus the smaller

00:05:58.450 --> 00:06:00.290
number must be 11.

00:06:00.290 --> 00:06:06.320
That tells us that x minus
y must be equal to 11.

00:06:06.320 --> 00:06:06.960
So there we have it.

00:06:06.960 --> 00:06:08.510
We have two equations
and two unknowns.

00:06:08.510 --> 00:06:10.530
We have a system of
two equations.

00:06:10.530 --> 00:06:15.040
We can now solve it using
the substitution method.

00:06:15.040 --> 00:06:18.190
So let's solve for x on this
equation right here.

00:06:18.190 --> 00:06:21.160
So if you add y to both
sides of this

00:06:21.160 --> 00:06:26.610
equation, what do you get?

00:06:26.610 --> 00:06:28.700
On the left-hand side, you just
get an x, because these

00:06:28.700 --> 00:06:29.890
cancel out.

00:06:29.890 --> 00:06:33.000
And then on the right-hand side,
you get x is equal to 11

00:06:33.000 --> 00:06:36.630
plus y, or y plus 11.

00:06:36.630 --> 00:06:39.240
So we get x is equal
to 11 plus y

00:06:39.240 --> 00:06:40.830
using the second equation.

00:06:40.830 --> 00:06:44.070
And then we can substitute it
back into this top equation.

00:06:44.070 --> 00:06:46.770
So instead of writing x plus
y is equal to 70, we can

00:06:46.770 --> 00:06:49.850
substitute this in for x.

00:06:49.850 --> 00:06:53.210
We've already used the second
equation, the magenta one, now

00:06:53.210 --> 00:06:55.360
we have to use the
top constraint.

00:06:55.360 --> 00:07:00.600
So if we substitute this in, we
get y plus 11-- remember,

00:07:00.600 --> 00:07:04.990
this is what x was, we're
substituting that in for x--

00:07:04.990 --> 00:07:08.700
plus y is equal to 70.

00:07:08.700 --> 00:07:11.710
This is x.

00:07:11.710 --> 00:07:14.870
And that constraint was given to
us by this second equation,

00:07:14.870 --> 00:07:16.650
or by this second statement.

00:07:16.650 --> 00:07:20.020
I just substituted this x with
y plus 11, and I was able to

00:07:20.020 --> 00:07:22.480
do that because that's the
constraint the second

00:07:22.480 --> 00:07:24.420
equation gave us.

00:07:24.420 --> 00:07:26.280
So now let's just solve for y.

00:07:26.280 --> 00:07:32.230
We get y plus 11, plus
y is equal to 70.

00:07:32.230 --> 00:07:36.470
That's 2y plus 11
is equal to 70.

00:07:36.470 --> 00:07:45.430
And then if we subtract 11 from
both sides, we get 2y is

00:07:45.430 --> 00:07:46.570
equal to-- what is that?

00:07:46.570 --> 00:07:48.220
59?

00:07:48.220 --> 00:07:50.160
You subtract 10 from
70, you get 60, so

00:07:50.160 --> 00:07:51.530
it's going to be 59.

00:07:51.530 --> 00:07:56.120
So y is equal to 59 over 2.

00:07:56.120 --> 00:08:00.240
Or another way to write it,
you could write that as 59

00:08:00.240 --> 00:08:10.110
over 2 is the same thing as--
let's see-- 25-- 29.5.

00:08:10.110 --> 00:08:15.380
y is equal to 29.5.

00:08:15.380 --> 00:08:17.650
Now, what is x going
to be equal to?

00:08:17.650 --> 00:08:21.090
Well, we already figured out
x is equal to y plus 11.

00:08:21.090 --> 00:08:26.120
So x is going to be equal to
29.5-- that's what y is, we

00:08:26.120 --> 00:08:30.835
just figured that out-- plus 11,
which is equal to-- so you

00:08:30.835 --> 00:08:32.549
add 10, you get 39.5.

00:08:32.549 --> 00:08:36.809
You add another 1,
you get 40.5.

00:08:36.809 --> 00:08:37.990
And we're done.

00:08:37.990 --> 00:08:40.690
If you wanted to find the
intersection of these two

00:08:40.690 --> 00:08:48.140
lines, it would intersect at
the point 40.5 comma 29.5.

00:08:48.140 --> 00:08:51.435
And you could have used this
equation to solve for x and

00:08:51.435 --> 00:08:52.200
then substituted in this one.

00:08:52.200 --> 00:08:53.756
You could have used this
equation to solve for y and

00:08:53.756 --> 00:08:55.250
then substituted in this one.

00:08:55.250 --> 00:08:57.245
You could use this equation
to solve for y and then

00:08:57.245 --> 00:08:58.760
substitute into that equation.

00:08:58.760 --> 00:09:02.770
The important thing is, is
you use both constraints.

00:09:02.770 --> 00:09:05.850
Now let's just verify that
this actually works out.

00:09:05.850 --> 00:09:07.330
What's the sum of these
two numbers?

00:09:07.330 --> 00:09:11.600
40.5 plus 29.5, that
indeed is 70.

00:09:11.600 --> 00:09:15.250
And the difference between
the two is indeed 11.

00:09:15.250 --> 00:09:17.550
They're exactly 11 apart.

00:09:17.550 --> 00:09:20.320
Anyway, hopefully you
found that useful.


Solving linear systems by graphing

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:06.530
Let's say I have the equation
y is equal to x plus 3.

00:00:06.530 --> 00:00:10.570
And I want to graph all of the
sets, all of the coordinates x

00:00:10.570 --> 00:00:13.750
comma y that satisfy this
equation right there.

00:00:13.750 --> 00:00:16.000
And we've done this
many times before.

00:00:16.000 --> 00:00:19.000
So we draw our axis, our axes.

00:00:19.000 --> 00:00:20.250
That's my y-axis.

00:00:22.630 --> 00:00:24.620
This is my x-axis.

00:00:27.470 --> 00:00:31.070
And this is already in
mx plus b form, or

00:00:31.070 --> 00:00:32.540
slope-intercept form.

00:00:32.540 --> 00:00:35.840
The y-intercept here is y
is equal to 3, and the

00:00:35.840 --> 00:00:37.690
slope here is 1.

00:00:37.690 --> 00:00:39.840
So this line is going
to look like this.

00:00:39.840 --> 00:00:43.910
We intersect at 0 comma
3-- 1, 2, 3.

00:00:43.910 --> 00:00:45.190
At 0 comma 3.

00:00:45.190 --> 00:00:47.890
And we have a slope of 1,
so every 1 we go to the

00:00:47.890 --> 00:00:49.070
right, we go up 1.

00:00:49.070 --> 00:00:51.870
So the line will look
something like that.

00:00:55.610 --> 00:00:57.760
It's a good enough
approximation.

00:00:57.760 --> 00:00:59.770
So the line will
look like this.

00:00:59.770 --> 00:01:03.380
And remember, when I'm drawing
a line, every point on this

00:01:03.380 --> 00:01:07.280
line is a solution
to this equation.

00:01:07.280 --> 00:01:11.210
Or it represents a pair
of x and y that

00:01:11.210 --> 00:01:13.100
satisfy this equation.

00:01:13.100 --> 00:01:22.300
So maybe when you take x is
equal to 5, you go to the

00:01:22.300 --> 00:01:25.010
line, and you're going to see,
gee, when x is equal to 5 on

00:01:25.010 --> 00:01:28.160
that line, y is equal
to 8 is a solution.

00:01:28.160 --> 00:01:29.370
And it's going to
sit on the line.

00:01:29.370 --> 00:01:32.550
So this represents the solution
set to this equation,

00:01:32.550 --> 00:01:34.820
all of the coordinates
that satisfy y is

00:01:34.820 --> 00:01:36.660
equal to x plus 3.

00:01:36.660 --> 00:01:39.000
Now let's say we have
another equation.

00:01:39.000 --> 00:01:43.040
Let's say we have an equation
y is equal to

00:01:43.040 --> 00:01:45.210
negative x plus 3.

00:01:45.210 --> 00:01:49.540
And we want to graph all of the
x and y pairs that satisfy

00:01:49.540 --> 00:01:50.130
this equation.

00:01:50.130 --> 00:01:51.630
Well, we can do the
same thing.

00:01:51.630 --> 00:01:55.340
This has a y-intercept also
at 3, right there.

00:01:55.340 --> 00:01:57.250
But its slope is negative 1.

00:01:57.250 --> 00:01:58.810
So it's going to look
something like this.

00:02:02.210 --> 00:02:04.505
Every time you move to the
right 1, you're going

00:02:04.505 --> 00:02:05.870
to move down 1.

00:02:05.870 --> 00:02:07.980
Or if you move to the right a
bunch, you're going to move

00:02:07.980 --> 00:02:10.020
down that same bunch.

00:02:10.020 --> 00:02:14.720
So that's what this equation
will look like.

00:02:14.720 --> 00:02:18.860
Every point on this line
represents a x and y pair that

00:02:18.860 --> 00:02:21.070
will satisfy this equation.

00:02:21.070 --> 00:02:24.770
Now, what if I were to ask you,
is there an x and y pair

00:02:24.770 --> 00:02:28.740
that satisfies both of
these equations?

00:02:28.740 --> 00:02:33.280
Is there a point or coordinate
that satisfies both equations?

00:02:33.280 --> 00:02:34.420
Well, think about it.

00:02:34.420 --> 00:02:36.850
Everything that satisfies this
first equation is on this

00:02:36.850 --> 00:02:40.500
green line right here, and
everything that satisfies this

00:02:40.500 --> 00:02:43.460
purple equation is on the
purple line right there.

00:02:43.460 --> 00:02:45.520
So what satisfies both?

00:02:45.520 --> 00:02:48.210
Well, if there's a point that's
on both lines, or

00:02:48.210 --> 00:02:51.190
essentially, a point of
intersection of the lines.

00:02:51.190 --> 00:02:55.150
So in this situation, this
point is on both lines.

00:02:55.150 --> 00:02:56.570
And that's actually
the y-intercept.

00:02:56.570 --> 00:03:00.660
So the point 0, 3 is on
both of these lines.

00:03:00.660 --> 00:03:05.090
So that coordinate pair, or that
x, y pair, must satisfy

00:03:05.090 --> 00:03:06.020
both equations.

00:03:06.020 --> 00:03:06.880
And you can try it out.

00:03:06.880 --> 00:03:11.890
When x is 0 here, 0 plus
3 is equal to 3.

00:03:11.890 --> 00:03:14.610
When x is 0 here, 0 plus
3 is equal to 3.

00:03:14.610 --> 00:03:17.880
It satisfies both of
these equations.

00:03:17.880 --> 00:03:21.500
So what we just did, in a
graphical way, is solve a

00:03:21.500 --> 00:03:22.786
system of equations.

00:03:26.080 --> 00:03:27.330
Let me write that down.

00:03:31.330 --> 00:03:34.550
And all that means is we
have several equations.

00:03:34.550 --> 00:03:37.650
Each of them constrain
our x's and y's.

00:03:37.650 --> 00:03:41.790
So in this case, the first one
is y is equal to x plus 3, and

00:03:41.790 --> 00:03:48.140
then the second one is y is
equal to negative x plus 3.

00:03:48.140 --> 00:03:50.950
This constrained it to a line
in the xy plane, this

00:03:50.950 --> 00:03:52.810
constrained our solution
set to another

00:03:52.810 --> 00:03:54.510
line in the xy plane.

00:03:54.510 --> 00:03:57.600
And if we want to know the x's
and y's that satisfy both of

00:03:57.600 --> 00:04:01.620
these, it's going to be the
intersection of those lines.

00:04:01.620 --> 00:04:04.070
So one way to solve these
systems of equations is to

00:04:04.070 --> 00:04:08.270
graph both lines, both
equations, and then look at

00:04:08.270 --> 00:04:09.420
their intersection.

00:04:09.420 --> 00:04:12.690
And that will be the solution
to both of these equations.

00:04:12.690 --> 00:04:14.460
In the next few videos, we're
going to see other ways to

00:04:14.460 --> 00:04:16.850
solve it, that are maybe more

00:04:16.850 --> 00:04:18.430
mathematical and less graphical.

00:04:18.430 --> 00:04:20.730
But I really want you to
understand the graphical

00:04:20.730 --> 00:04:23.310
nature of solving systems
of equations.

00:04:23.310 --> 00:04:24.560
Let's do another one.

00:04:27.980 --> 00:04:36.270
Let's say we have y is
equal to 3x minus 6.

00:04:36.270 --> 00:04:37.620
That's one of our equations.

00:04:37.620 --> 00:04:42.410
And let's say the other equation
is y is equal to

00:04:42.410 --> 00:04:45.860
negative x plus 6.

00:04:45.860 --> 00:04:48.220
And just like the last video,
let's graph both of these.

00:04:51.360 --> 00:04:54.740
I'll try to do it as
precisely as I can.

00:04:59.950 --> 00:05:01.530
There you go.

00:05:01.530 --> 00:05:03.260
Let me draw some.

00:05:03.260 --> 00:05:08.680
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:05:08.680 --> 00:05:12.710
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:05:12.710 --> 00:05:16.410
And then 1, 2, 3, 4,
5, 6, 7, 8, 9, 10.

00:05:16.410 --> 00:05:19.790
I should have just copied and
pasted some graph paper here,

00:05:19.790 --> 00:05:21.470
but I think this'll
do the job.

00:05:21.470 --> 00:05:23.580
So let's graph this purple
equation here.

00:05:23.580 --> 00:05:27.380
Y-intercept is negative 6, so
we have-- let me do another

00:05:27.380 --> 00:05:27.660
[? slash-- ?]

00:05:27.660 --> 00:05:30.380
1, 2, 3, 4, 5, 6.

00:05:30.380 --> 00:05:32.670
So that's y is equal
to negative 6.

00:05:32.670 --> 00:05:35.150
And then the slope is 3.

00:05:35.150 --> 00:05:38.510
So every time you move
1, you go up 3.

00:05:38.510 --> 00:05:41.370
You moved to the right 1,
your run is 1, your

00:05:41.370 --> 00:05:46.830
rise is 1, 2, 3.

00:05:46.830 --> 00:05:47.580
That's 3, right?

00:05:47.580 --> 00:05:49.120
1, 2, 3.

00:05:49.120 --> 00:05:51.640
So the equation, the line
will look like this.

00:05:51.640 --> 00:05:53.960
And it looks like I intersect
at the point 2

00:05:53.960 --> 00:05:55.080
comma 0, which is right.

00:05:55.080 --> 00:05:58.270
3 times 2 is 6, minus 6 is 0.

00:05:58.270 --> 00:06:00.830
So our line will look something

00:06:00.830 --> 00:06:03.700
like that right there.

00:06:03.700 --> 00:06:04.560
That's that line there.

00:06:04.560 --> 00:06:05.770
What about this line?

00:06:05.770 --> 00:06:08.700
Our y-intercept is plus 6.

00:06:08.700 --> 00:06:11.850
1, 2, 3, 4, 5, 6.

00:06:11.850 --> 00:06:14.620
And our slope is negative 1.

00:06:14.620 --> 00:06:18.630
So every time we go 1 to the
right, we go down 1.

00:06:26.500 --> 00:06:33.240
And so this will intersect at--
well, when y is equal to

00:06:33.240 --> 00:06:34.100
0, x is equal to 6.

00:06:34.100 --> 00:06:36.280
1, 2, 3, 4, 5, 6.

00:06:36.280 --> 00:06:37.880
So right over there.

00:06:37.880 --> 00:06:39.620
So this line will
look like that.

00:06:39.620 --> 00:06:44.300
The graph, I want to get it
as exact as possible.

00:06:44.300 --> 00:06:46.730
And so we're going to ask
ourselves the same question.

00:06:46.730 --> 00:06:49.440
What is an x, y pair
that satisfies

00:06:49.440 --> 00:06:51.500
both of these equations?

00:06:51.500 --> 00:06:54.470
Well, you look at it here, it's
going to be this point.

00:06:54.470 --> 00:07:00.785
This point lies on both lines.

00:07:00.785 --> 00:07:04.560
And let's see if we can figure
out what that point is.

00:07:04.560 --> 00:07:07.060
Just eyeballing the graph here,
it looks like we're at

00:07:07.060 --> 00:07:10.370
1, 2, 3 comma 1, 2, 3.

00:07:10.370 --> 00:07:12.810
It looks like this is the same
point right there, that this

00:07:12.810 --> 00:07:14.820
is the point 3 comma 3.

00:07:14.820 --> 00:07:17.370
I'm doing it just on inspecting
my hand-drawn

00:07:17.370 --> 00:07:19.080
graphs, so maybe it's
not the exact--

00:07:19.080 --> 00:07:20.250
let's check this answer.

00:07:20.250 --> 00:07:24.260
Let's see if x is equal to
3, y equals 3 definitely

00:07:24.260 --> 00:07:26.630
satisfies both these
equations.

00:07:26.630 --> 00:07:31.790
So if we check it into the first
equation, you get 3 is

00:07:31.790 --> 00:07:35.740
equal to 3 times 3, minus 6.

00:07:35.740 --> 00:07:39.030
This is 9 minus 6, which
is indeed 3.

00:07:39.030 --> 00:07:41.350
So 3 comma 3 satisfies
the top equation.

00:07:41.350 --> 00:07:44.120
And let's see if it satisfies
the bottom equation.

00:07:44.120 --> 00:07:49.800
You get 3 is equal to negative
3 plus 6, and negative 3 plus

00:07:49.800 --> 00:07:51.740
6 is indeed 3.

00:07:51.740 --> 00:07:54.487
So even with our hand-drawn
graph, we were able to inspect

00:07:54.487 --> 00:07:57.420
it and see that, yes, we were
able to come up with the point

00:07:57.420 --> 00:08:00.370
3 comma 3, and that
does satisfy

00:08:00.370 --> 00:08:01.450
both of these equations.

00:08:01.450 --> 00:08:04.810
So we were able to solve this
system of equations.

00:08:04.810 --> 00:08:07.160
When we say system of equations,
we just mean many

00:08:07.160 --> 00:08:09.510
equations that have
many unknowns.

00:08:09.510 --> 00:08:12.060
They don't have to be, but they
tend to have more than

00:08:12.060 --> 00:08:12.610
one unknown.

00:08:12.610 --> 00:08:16.310
And you use each equation as a
constraint on your variables,

00:08:16.310 --> 00:08:18.860
and you try to find the
intersection of the equations

00:08:18.860 --> 00:08:21.300
to find a solution
to all of them.

00:08:21.300 --> 00:08:24.420
In the next few videos, we'll
see more algebraic ways of

00:08:24.420 --> 00:08:27.300
solving these than drawing their
two graphs and trying to

00:08:27.300 --> 00:08:29.910
find their intersection
points.


Introduction to graphing inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.160
Let's graph ourselves
some inequalities.

00:00:03.160 --> 00:00:06.050
So let's say I had the
inequality y is less than or

00:00:06.050 --> 00:00:11.520
equal to 4x plus 3.

00:00:11.520 --> 00:00:15.500
On our xy coordinate plane, we
want to show all the x and y

00:00:15.500 --> 00:00:19.650
points that satisfy this
condition right here.

00:00:19.650 --> 00:00:22.210
So a good starting point might
be to break up this less than

00:00:22.210 --> 00:00:24.570
or equal to, because we
know how to graph y is

00:00:24.570 --> 00:00:26.140
equal to 4x plus 3.

00:00:26.140 --> 00:00:31.510
So this thing is the same thing
as y could be less than

00:00:31.510 --> 00:00:38.870
4x plus 3, or y could be
equal to 4x plus 3.

00:00:38.870 --> 00:00:41.170
That's what less than
or equal means.

00:00:41.170 --> 00:00:43.200
It could be less
than or equal.

00:00:43.200 --> 00:00:46.015
And the reason why I did that on
this first example problem

00:00:46.015 --> 00:00:47.730
is because we know how
to graph that.

00:00:47.730 --> 00:00:48.980
So let's graph that.

00:00:52.060 --> 00:00:55.170
Try to draw a little bit
neater than that.

00:00:55.170 --> 00:00:57.800
So that is-- no, that's
not good.

00:00:57.800 --> 00:01:02.460
So that is my vertical
axis, my y-axis.

00:01:02.460 --> 00:01:07.920
This is my x-axis,
right there.

00:01:07.920 --> 00:01:09.920
And then we know the
y-intercept, the

00:01:09.920 --> 00:01:11.580
y-intercept is 3.

00:01:11.580 --> 00:01:16.510
So the point 0, 3-- 1, 2,
3-- is on the line.

00:01:16.510 --> 00:01:18.405
And we know we have
a slope of 4.

00:01:21.330 --> 00:01:25.430
Which means if we go 1 in the
x-direction, we're going to go

00:01:25.430 --> 00:01:26.980
up 4 in the y.

00:01:26.980 --> 00:01:29.280
So 1, 2, 3, 4.

00:01:29.280 --> 00:01:31.920
So it's going to
be right here.

00:01:31.920 --> 00:01:33.236
And that's enough
to draw a line.

00:01:33.236 --> 00:01:35.210
We could even go back
in the x-direction.

00:01:35.210 --> 00:01:37.290
If we go 1 back in the
x-direction, we're

00:01:37.290 --> 00:01:38.500
going to go down 4.

00:01:38.500 --> 00:01:40.930
1, 2, 3, 4.

00:01:40.930 --> 00:01:43.480
So that's also going to be
a point on the line.

00:01:43.480 --> 00:01:47.670
So my best attempt at drawing
this line is going to look

00:01:47.670 --> 00:01:52.590
something like-- this
is the hardest part.

00:01:52.590 --> 00:01:55.610
It's going to look something
like that.

00:01:55.610 --> 00:01:57.270
That is a line.

00:01:57.270 --> 00:01:58.470
It should be straight.

00:01:58.470 --> 00:02:00.240
I think you get the idea.

00:02:00.240 --> 00:02:05.540
That right there is the graph
of y is equal to 4x plus 3.

00:02:05.540 --> 00:02:07.440
So let's think about what it
means to be less than.

00:02:07.440 --> 00:02:09.789
So all of these points
satisfy this

00:02:09.789 --> 00:02:11.750
inequality, but we have more.

00:02:11.750 --> 00:02:13.410
This is just these
points over here.

00:02:13.410 --> 00:02:17.830
What about all these where
y ix less than 4x plus 3?

00:02:17.830 --> 00:02:20.790
So let's think about
what this means.

00:02:20.790 --> 00:02:22.410
Let's pick up some
values for x.

00:02:22.410 --> 00:02:26.140
When x is equal to 0,
what does this say?

00:02:26.140 --> 00:02:29.390
When x is equal to 0, then that
means y is going to be

00:02:29.390 --> 00:02:33.150
less than 0 plus 3.
y is less than 3.

00:02:33.150 --> 00:02:40.620
When x is equal to negative 1,
what is this telling us?

00:02:40.620 --> 00:02:44.590
4 times negative 1 is negative
4, plus 3 is negative 1. y

00:02:44.590 --> 00:02:46.840
would be less than negative 1.

00:02:46.840 --> 00:02:50.950
When x is equal to 1, what
is this telling us?

00:02:50.950 --> 00:02:54.110
4 times 1 is 4, plus 3 is 7.

00:02:54.110 --> 00:02:57.340
So y is going to
be less than 7.

00:02:57.340 --> 00:02:59.130
So let's at least try
to plot these.

00:02:59.130 --> 00:03:04.670
So when x is equal to-- let's
plot this one first. When x is

00:03:04.670 --> 00:03:07.805
equal to 0, y is less than 3.

00:03:11.350 --> 00:03:15.230
So it's all of these points
here-- that I'm shading in in

00:03:15.230 --> 00:03:19.310
green-- satisfy that
right there.

00:03:19.310 --> 00:03:22.200
If I were to look at this one
over here, when x is negative

00:03:22.200 --> 00:03:26.140
1, y is less than negative 1.

00:03:26.140 --> 00:03:30.850
So y has to be all of these
points down here.

00:03:30.850 --> 00:03:36.470
When x is equal to 1,
y is less than 7.

00:03:36.470 --> 00:03:39.500
So it's all of these
points down here.

00:03:39.500 --> 00:03:44.960
And in general, you take any
point x-- let's say you take

00:03:44.960 --> 00:03:47.020
this point x right there.

00:03:47.020 --> 00:03:50.980
If you evaluate 4x plus 3,
you're going to get the point

00:03:50.980 --> 00:03:52.120
on the line.

00:03:52.120 --> 00:03:55.950
That is that x times 4 plus 3.

00:03:55.950 --> 00:03:58.860
Now the y's that satisfy it,
it could be equal to that

00:03:58.860 --> 00:04:01.100
point on the line, or it
could be less than.

00:04:01.100 --> 00:04:03.840
So it's going to go
below the line.

00:04:03.840 --> 00:04:06.870
So if you were to do this for
all the possible x's, you

00:04:06.870 --> 00:04:11.130
would not only get all the
points on this line which

00:04:11.130 --> 00:04:15.990
we've drawn, you would get all
the points below the line.

00:04:15.990 --> 00:04:19.279
So now we have graphed
this inequality.

00:04:19.279 --> 00:04:23.480
It's essentially this line, 4x
plus 3, with all of the area

00:04:23.480 --> 00:04:25.490
below it shaded.

00:04:25.490 --> 00:04:29.530
Now, if this was just a less
than, not less than or equal

00:04:29.530 --> 00:04:32.670
sign, we would not include
the actual line.

00:04:32.670 --> 00:04:36.090
And the convention to do that is
to actually make the line a

00:04:36.090 --> 00:04:37.860
dashed line.

00:04:37.860 --> 00:04:44.880
This is the situation if we were
dealing with just less

00:04:44.880 --> 00:04:46.690
than 4x plus 3.

00:04:46.690 --> 00:04:49.410
Because in that situation, this
wouldn't apply, and we

00:04:49.410 --> 00:04:50.620
would just have that.

00:04:50.620 --> 00:04:53.550
So the line itself wouldn't have
satisfied it, just the

00:04:53.550 --> 00:04:55.010
area below it.

00:04:55.010 --> 00:04:57.190
Let's do one like that.

00:04:57.190 --> 00:05:06.230
So let's say we have y is
greater than negative x

00:05:06.230 --> 00:05:09.530
over 2 minus 6.

00:05:09.530 --> 00:05:11.940
So a good way to start-- the
way I like to start these

00:05:11.940 --> 00:05:14.830
problems-- is to just graph
this equation right here.

00:05:14.830 --> 00:05:18.780
So let me just graph-- just for
fun-- let me graph y is

00:05:18.780 --> 00:05:24.200
equal to-- this is the same
thing as negative 1/2 minus 6.

00:05:24.200 --> 00:05:30.890
So if we were to graph it, that
is my vertical axis, that

00:05:30.890 --> 00:05:34.130
is my horizontal axis.

00:05:34.130 --> 00:05:36.920
And our y-intercept
is negative 6.

00:05:36.920 --> 00:05:40.170
So 1, 2, 3, 4, 5, 6.

00:05:40.170 --> 00:05:42.020
So that's my y-intercept.

00:05:42.020 --> 00:05:43.960
And my slope is negative 1/2.

00:05:43.960 --> 00:05:49.390
Oh, that should be an x there,
negative 1/2 x minus 6.

00:05:49.390 --> 00:05:53.950
So my slope is negative 1/2,
which means when I go 2 to the

00:05:53.950 --> 00:05:55.395
right, I go down 1.

00:05:55.395 --> 00:05:58.775
So if I go 2 to the right,
I'm going to go down 1.

00:06:02.590 --> 00:06:05.870
If I go 2 to the left, if
I go negative 2, I'm

00:06:05.870 --> 00:06:06.880
going to go up 1.

00:06:06.880 --> 00:06:09.040
So negative 2, up 1.

00:06:09.040 --> 00:06:11.530
So my line is going
to look like this.

00:06:11.530 --> 00:06:15.200
My line is going to
look like that.

00:06:15.200 --> 00:06:17.400
That's my best attempt
at drawing the line.

00:06:17.400 --> 00:06:19.400
So that's the line
of y is equal to

00:06:19.400 --> 00:06:22.080
negative 1/2 x minus 6.

00:06:22.080 --> 00:06:24.730
Now, our inequality is not
greater than or equal, it's

00:06:24.730 --> 00:06:28.320
just greater than negative x
over 2 minus 6, or greater

00:06:28.320 --> 00:06:30.850
than negative 1/2 x minus 6.

00:06:30.850 --> 00:06:35.230
So using the same logic as
before, for any x-- so if you

00:06:35.230 --> 00:06:39.070
take any x, let's say that's
our particular x we want to

00:06:39.070 --> 00:06:43.260
pick-- if you evaluate negative
x over 2 minus 6,

00:06:43.260 --> 00:06:44.970
you're going to get that
point right there.

00:06:44.970 --> 00:06:47.720
You're going to get the
point on the line.

00:06:47.720 --> 00:06:50.620
But the y's that satisfy this
inequality are the y's

00:06:50.620 --> 00:06:51.650
greater than that.

00:06:51.650 --> 00:06:54.920
So it's going to be not that
point-- in fact, you draw an

00:06:54.920 --> 00:06:58.930
open circle there-- because you
can't include the point of

00:06:58.930 --> 00:07:00.750
negative 1/2 x minus 6.

00:07:00.750 --> 00:07:02.750
But it's going to be all the
y's greater than that.

00:07:05.300 --> 00:07:06.670
That'd be true for any x.

00:07:06.670 --> 00:07:07.900
You take this x.

00:07:07.900 --> 00:07:11.260
You evaluate negative 1/2 or
negative x over 2 minus 6,

00:07:11.260 --> 00:07:13.660
you're going to get this
point over here.

00:07:13.660 --> 00:07:19.100
The y's that satisfy it are
all the y's above that.

00:07:19.100 --> 00:07:22.660
So all of the y's that satisfy
this equation, or all of the

00:07:22.660 --> 00:07:25.990
coordinates that satisfy this
equation, is this entire area

00:07:25.990 --> 00:07:28.310
above the line.

00:07:28.310 --> 00:07:30.070
And we're not going to
include the line.

00:07:30.070 --> 00:07:33.300
So the convention is to make
this line into a dashed line.

00:07:33.300 --> 00:07:35.850
And let me draw-- I'm trying
my best to turn it into a

00:07:35.850 --> 00:07:36.430
dashed line.

00:07:36.430 --> 00:07:41.070
I'll just erase sections of
the line, and hopefully it

00:07:41.070 --> 00:07:43.160
will look dashed to you.

00:07:43.160 --> 00:07:45.590
So I'm turning that solid line
into a dashed line to show

00:07:45.590 --> 00:07:48.500
that it's just a boundary, but
it's not included in the

00:07:48.500 --> 00:07:51.210
coordinates that satisfy
our inequality.

00:07:51.210 --> 00:07:53.760
The coordinates that satisfy our
equality are all of this

00:07:53.760 --> 00:08:00.430
yellow stuff that I'm shading
above the line.

00:08:00.430 --> 00:08:03.260
Anyway, hopefully you
found that helpful.


Absolute value inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.730
I now want to solve some
inequalities that also have

00:00:03.730 --> 00:00:05.550
absolute values in them.

00:00:05.550 --> 00:00:08.900
And if there's any topic in
algebra that probably confuses

00:00:08.900 --> 00:00:10.260
people the most, it's this.

00:00:10.260 --> 00:00:12.890
But if we kind of keep our head
on straight about what

00:00:12.890 --> 00:00:15.620
absolute value really means, I
think you will find that it's

00:00:15.620 --> 00:00:16.650
not that bad.

00:00:16.650 --> 00:00:20.810
So let's start with a nice,
fairly simple warm-up problem.

00:00:20.810 --> 00:00:27.370
Let's start with the absolute
value of x is less than 12.

00:00:27.370 --> 00:00:28.740
So remember what I told
you about the

00:00:28.740 --> 00:00:30.420
meaning of absolute value.

00:00:30.420 --> 00:00:33.890
It means how far away
you are from 0.

00:00:33.890 --> 00:00:37.390
So one way to say this is, what
are all of the x's that

00:00:37.390 --> 00:00:41.120
are less than 12 away from 0?

00:00:41.120 --> 00:00:42.370
Let's draw a number line.

00:00:45.490 --> 00:00:49.120
So if we have 0 here, and we
want all the numbers that are

00:00:49.120 --> 00:00:52.100
less than 12 away from 0, well,
you could go all the way

00:00:52.100 --> 00:00:56.570
to positive 12, and you could go
all the way to negative 12.

00:00:56.570 --> 00:01:02.820
Anything that's in between these
two numbers is going to

00:01:02.820 --> 00:01:06.360
have an absolute value
of less than 12.

00:01:06.360 --> 00:01:10.550
It's going to be less
than 12 away from 0.

00:01:10.550 --> 00:01:15.020
So this, you could say, this
could be all of the numbers

00:01:15.020 --> 00:01:19.310
where x is greater
than negative 12.

00:01:19.310 --> 00:01:26.220
Those are definitely going to
have an absolute value less

00:01:26.220 --> 00:01:31.050
than 12, as long as they're
also-- and, x has to

00:01:31.050 --> 00:01:33.520
be less than 12.

00:01:33.520 --> 00:01:37.170
So if an x meets both of these
constraints, its absolute

00:01:37.170 --> 00:01:39.890
value is definitely going
to be less than 12.

00:01:39.890 --> 00:01:42.050
You know, you take the absolute
value of negative 6,

00:01:42.050 --> 00:01:43.860
that's only 6 away from 0.

00:01:43.860 --> 00:01:46.670
The absolute value of negative
11, only 11 away from 0.

00:01:46.670 --> 00:01:49.520
So something that meets both
of these constraints will

00:01:49.520 --> 00:01:50.630
satisfy the equation.

00:01:50.630 --> 00:01:52.440
And actually, we've solved it,
because this is only a

00:01:52.440 --> 00:01:53.810
one-step equation there.

00:01:53.810 --> 00:01:56.670
But I think it lays a good
foundation for the next few

00:01:56.670 --> 00:01:59.030
problems. And I could actually
write it like this.

00:01:59.030 --> 00:02:02.280
In interval notation, it would
be everything between negative

00:02:02.280 --> 00:02:06.880
12 and positive 12, and not
including those numbers.

00:02:06.880 --> 00:02:10.690
Or we could write it like this,
x is less than 12, and

00:02:10.690 --> 00:02:13.100
is greater than negative 12.

00:02:13.100 --> 00:02:15.180
That's the solution
set right there.

00:02:15.180 --> 00:02:18.420
Now let's do one that's a little
bit more complicated,

00:02:18.420 --> 00:02:21.340
that allows us to think
a little bit harder.

00:02:21.340 --> 00:02:28.480
So let's say we have the
absolute value of 7x is

00:02:28.480 --> 00:02:31.260
greater than or equal to 21.

00:02:31.260 --> 00:02:33.710
So let's not even think about
what's inside of the absolute

00:02:33.710 --> 00:02:36.020
value sign right now.

00:02:36.020 --> 00:02:38.180
In order for the absolute
value of anything to be

00:02:38.180 --> 00:02:41.260
greater than or equal to
21, what does it mean?

00:02:41.260 --> 00:02:44.040
It means that whatever's inside
of this absolute value

00:02:44.040 --> 00:02:47.640
sign, whatever that is inside of
our absolute value sign, it

00:02:47.640 --> 00:02:52.840
must be 21 or more
away from 0.

00:02:52.840 --> 00:02:54.910
Let's draw our number line.

00:02:54.910 --> 00:02:57.010
And you really should visualize
a number line when

00:02:57.010 --> 00:02:58.970
you do this, and you'll never
get confused then.

00:02:58.970 --> 00:03:01.660
You shouldn't be memorizing
any rules.

00:03:01.660 --> 00:03:03.860
So let's draw 0 here.

00:03:03.860 --> 00:03:08.980
Let's do positive 21, and let's
do a negative 21 here.

00:03:08.980 --> 00:03:12.470
So we want all of the numbers,
so whatever this thing is,

00:03:12.470 --> 00:03:17.400
that are greater than
or equal to 21.

00:03:17.400 --> 00:03:20.860
They're more than
21 away from 0.

00:03:20.860 --> 00:03:23.920
Their absolute value
is more than 21.

00:03:23.920 --> 00:03:27.000
Well, all of these negative
numbers that are less than

00:03:27.000 --> 00:03:30.090
negative 21, when you take their
absolute value, when you

00:03:30.090 --> 00:03:33.750
get rid of the negative sign,
or when you find their

00:03:33.750 --> 00:03:37.160
distance from 0, they're all
going to be greater than 21.

00:03:37.160 --> 00:03:39.110
If you take the absolute value
of negative 30, it's going to

00:03:39.110 --> 00:03:40.900
be greater than 21.

00:03:40.900 --> 00:03:47.070
Likewise, up here, anything
greater than positive 21 will

00:03:47.070 --> 00:03:50.440
also have an absolute value
greater than 21.

00:03:50.440 --> 00:03:54.160
So what we could say is 7x needs
to be equal to one of

00:03:54.160 --> 00:03:58.530
these numbers, or 7x needs to
be equal to one of these

00:03:58.530 --> 00:03:59.930
numbers out here.

00:03:59.930 --> 00:04:06.230
So we could write 7x needs to
be one of these numbers.

00:04:06.230 --> 00:04:07.580
Well, what are these numbers?

00:04:07.580 --> 00:04:10.810
These are all of the numbers
that are less than or equal to

00:04:10.810 --> 00:04:20.070
negative 21, or 7x-- let me do a
different color here-- or 7x

00:04:20.070 --> 00:04:22.160
has to be one of
these numbers.

00:04:22.160 --> 00:04:26.290
And that means that 7x has to
be greater than or equal to

00:04:26.290 --> 00:04:28.070
positive 21.

00:04:28.070 --> 00:04:29.870
I really want you to
kind of internalize

00:04:29.870 --> 00:04:31.460
what's going on here.

00:04:31.460 --> 00:04:34.590
If our absolute value is greater
than or equal to 21,

00:04:34.590 --> 00:04:38.480
that means that what's inside
the absolute value has to be

00:04:38.480 --> 00:04:42.370
either just straight up greater
than the positive 21,

00:04:42.370 --> 00:04:45.070
or less than negative 21.

00:04:45.070 --> 00:04:47.170
Because if it's less than
negative 21, when you take its

00:04:47.170 --> 00:04:51.110
absolute value, it's going to
be more than 21 away from 0.

00:04:51.110 --> 00:04:52.090
Hopefully that make sense.

00:04:52.090 --> 00:04:54.120
We'll do several of these
practice problems, so it

00:04:54.120 --> 00:04:56.190
really gets ingrained
in your brain.

00:04:56.190 --> 00:04:58.650
But once you have this set up,
and this just becomes a

00:04:58.650 --> 00:05:01.960
compound inequality, divide both
sides of this equation by

00:05:01.960 --> 00:05:08.130
7, you get x is less than
or equal to negative 3.

00:05:08.130 --> 00:05:12.590
Or you divide both sides of
this by 7, you get x is

00:05:12.590 --> 00:05:14.710
greater than or equal to 3.

00:05:14.710 --> 00:05:15.520
So I want to be very clear.

00:05:15.520 --> 00:05:17.920
This, what I drew here, was
not the solution set.

00:05:17.920 --> 00:05:21.070
This is what 7x had
to be equal to.

00:05:21.070 --> 00:05:23.470
I just wanted you to visualize
what it means to have the

00:05:23.470 --> 00:05:27.460
absolute value be greater
than 21, to be more than

00:05:27.460 --> 00:05:30.470
21 away from 0.

00:05:30.470 --> 00:05:33.950
This is the solution set. x
has to be greater than or

00:05:33.950 --> 00:05:37.570
equal to 3, or less than
or equal to negative 3.

00:05:37.570 --> 00:05:43.370
So the actual solution set to
this equation-- let me draw a

00:05:43.370 --> 00:05:49.870
number line-- let's say that's
0, that's 3, that is negative

00:05:49.870 --> 00:05:54.365
3. x has to be either greater
than or equal to 3.

00:05:58.410 --> 00:05:59.620
That's the equal sign.

00:05:59.620 --> 00:06:02.000
Or less than or equal
to negative 3.

00:06:08.210 --> 00:06:08.960
And we're done.

00:06:08.960 --> 00:06:10.500
Let's do a couple
more of these.

00:06:10.500 --> 00:06:13.520
Because they are, I think,
confusing, but if you really

00:06:13.520 --> 00:06:17.490
start to get the gist of what
absolute value is saying, they

00:06:17.490 --> 00:06:19.930
become, I think, intuitive.

00:06:19.930 --> 00:06:23.840
So let's say that we have
the absolute value-- let

00:06:23.840 --> 00:06:24.570
me get a good one.

00:06:24.570 --> 00:06:32.310
Let's say the absolute value of
5x plus 3 is less than 7.

00:06:32.310 --> 00:06:34.780
So that's telling us that
whatever's inside of our

00:06:34.780 --> 00:06:40.030
absolute value sign has to be
less than 7 away from 0.

00:06:40.030 --> 00:06:43.360
So the ways that we can be less
than 7 away from 0-- let

00:06:43.360 --> 00:06:47.770
me draw a number line-- so the
ways that you can be less than

00:06:47.770 --> 00:06:52.870
7 away from 0, you could be less
than 7, and greater than

00:06:52.870 --> 00:06:54.740
negative 7.

00:06:54.740 --> 00:06:54.910
Right?

00:06:54.910 --> 00:06:56.970
You have to be in this range.

00:06:56.970 --> 00:07:00.240
So in order to satisfy this
thing in this absolute value

00:07:00.240 --> 00:07:03.590
sign, it has to be-- so the
thing in the absolute value

00:07:03.590 --> 00:07:07.170
sign, which is 5x plus 3--
it has to be greater than

00:07:07.170 --> 00:07:16.310
negative 7 and it has to be less
than 7, in order for its

00:07:16.310 --> 00:07:19.060
absolute value to
be less than 7.

00:07:19.060 --> 00:07:23.680
If this thing, this 5x plus 3,
evaluates anywhere over here,

00:07:23.680 --> 00:07:26.810
its absolute value, its
distance from 0, will

00:07:26.810 --> 00:07:28.420
be less than 7.

00:07:28.420 --> 00:07:30.270
And then we can just
solve these.

00:07:30.270 --> 00:07:32.810
You subtract 3 from
both sides.

00:07:32.810 --> 00:07:36.160
5x is greater than
negative 10.

00:07:36.160 --> 00:07:40.010
Divide both sides by 5. x is
greater than negative 2.

00:07:40.010 --> 00:07:42.200
Now over here, subtract
3 from both sides.

00:07:42.200 --> 00:07:45.960
5x is less than 4.

00:07:45.960 --> 00:07:51.300
Divide both sides by 5, you
get x is less than 4/5.

00:07:51.300 --> 00:07:53.560
And then we can draw
the solution set.

00:07:53.560 --> 00:07:58.270
We have to be greater than
negative 2, not greater than

00:07:58.270 --> 00:08:01.120
or equal to, and
less than 4/5.

00:08:01.120 --> 00:08:03.230
So this might look like a
coordinate, but this is also

00:08:03.230 --> 00:08:05.590
interval notation, if we're
saying all of the x's between

00:08:05.590 --> 00:08:07.770
negative 2 and 4/5.

00:08:07.770 --> 00:08:10.780
Or you could write it all of the
x's that are greater than

00:08:10.780 --> 00:08:15.140
negative 2 and less than 4/5.

00:08:15.140 --> 00:08:19.160
These are the x's that satisfy
this equation.

00:08:19.160 --> 00:08:20.810
And I really want you
to internalize this

00:08:20.810 --> 00:08:22.110
visualization here.

00:08:22.110 --> 00:08:28.830
Now, you might already be seeing
a bit of a rule here.

00:08:28.830 --> 00:08:31.110
And I don't want you to just
memorize it, but I'll give it

00:08:31.110 --> 00:08:32.590
to you just in case
you want it.

00:08:32.590 --> 00:08:36.809
If you have something like f of
x, the absolute value of f

00:08:36.809 --> 00:08:41.409
of x is less than, let's
say, some number a.

00:08:41.409 --> 00:08:41.730
Right?

00:08:41.730 --> 00:08:42.600
So this was the situation.

00:08:42.600 --> 00:08:44.810
We have some f of
x less than a.

00:08:44.810 --> 00:08:48.080
That means that the absolute
value of f of x, or f of x has

00:08:48.080 --> 00:08:50.600
to be less than a away from 0.

00:08:50.600 --> 00:08:56.960
So that means that f of x has to
be less than positive a or

00:08:56.960 --> 00:08:59.730
greater than negative a.

00:08:59.730 --> 00:09:05.090
That translates to that, which
translates to f of x greater

00:09:05.090 --> 00:09:10.460
than negative a and f
of x less than a.

00:09:10.460 --> 00:09:12.280
But it comes from
the same logic.

00:09:12.280 --> 00:09:15.540
This has to evaluate to
something that is less than a

00:09:15.540 --> 00:09:17.380
away from 0.

00:09:17.380 --> 00:09:20.280
Now, if we go to the other side,
if you have something of

00:09:20.280 --> 00:09:24.810
the form f of x is
greater than a.

00:09:24.810 --> 00:09:28.660
That means that this thing has
to evaluate to something that

00:09:28.660 --> 00:09:30.730
is further than a away from 0.

00:09:30.730 --> 00:09:34.330
So that means that f of x is
either just straight up

00:09:34.330 --> 00:09:40.720
greater than positive a, or f of
x is less than negative a.

00:09:40.720 --> 00:09:40.880
Right?

00:09:40.880 --> 00:09:44.380
If it's less than negative a,
maybe it's negative a minus

00:09:44.380 --> 00:09:50.980
another 1, or negative
5 plus negative a.

00:09:50.980 --> 00:09:52.525
Then, when you take its
absolute value, it'll

00:09:52.525 --> 00:09:53.730
become a plus 5.

00:09:53.730 --> 00:09:56.490
So its absolute value is going
to be greater than a.

00:09:56.490 --> 00:09:59.300
So I just want to-- you could
memorize this if you want, but

00:09:59.300 --> 00:10:01.760
I really want you to think about
this is just saying, OK,

00:10:01.760 --> 00:10:05.100
this has to evaluate, be less
than a away from 0, this has

00:10:05.100 --> 00:10:07.440
to be more than a away from 0.

00:10:07.440 --> 00:10:10.310
Let's do one more, because
I know this can be

00:10:10.310 --> 00:10:11.780
a little bit confusing.

00:10:11.780 --> 00:10:14.040
And I encourage you to watch
this video over and over and

00:10:14.040 --> 00:10:16.490
over again, if it helps.

00:10:16.490 --> 00:10:23.020
Let's say we have the absolute
value of 2x-- let me do

00:10:23.020 --> 00:10:24.040
another one over here.

00:10:24.040 --> 00:10:25.810
Let's do a harder one.

00:10:25.810 --> 00:10:31.460
Let's say the absolute value
of 2x over 7 plus 9 is

00:10:31.460 --> 00:10:35.460
greater than 5/7.

00:10:35.460 --> 00:10:40.370
So this thing has to evaluate to
something that's more than

00:10:40.370 --> 00:10:43.090
5/7 away from 0.

00:10:43.090 --> 00:10:47.610
So this thing, 2x over 7 plus
9, it could just be straight

00:10:47.610 --> 00:10:49.740
up greater than 5/7.

00:10:49.740 --> 00:10:53.220
Or it could be less than
negative 5/7, because if it's

00:10:53.220 --> 00:10:56.180
less than negative 5/7, its
absolute value is going to be

00:10:56.180 --> 00:10:57.840
greater than 5/7.

00:10:57.840 --> 00:11:04.970
Or 2x over 7 plus 9 will be
less than negative 5/7.

00:11:04.970 --> 00:11:07.540
We're doing this case
right here.

00:11:07.540 --> 00:11:10.290
And then we just solve both
of these equations.

00:11:10.290 --> 00:11:13.750
See if we subtract-- let's just
multiply everything by 7,

00:11:13.750 --> 00:11:16.290
just to get these denominators
out of the way.

00:11:16.290 --> 00:11:21.730
So if you multiply both sides by
7, you get 2x plus 9 times

00:11:21.730 --> 00:11:24.800
7 is 63, is greater than 5.

00:11:24.800 --> 00:11:25.750
Let's do it over here, too.

00:11:25.750 --> 00:11:32.630
You'll get 2x plus 63 is
less than negative 5.

00:11:32.630 --> 00:11:37.010
Let's subtract 63 from both
sides of this equation, and

00:11:37.010 --> 00:11:39.200
you get 2x-- let's see.

00:11:39.200 --> 00:11:44.470
5 minus 63 is 58, 2x
is greater than 58.

00:11:44.470 --> 00:11:47.230
If you subtract 63 from both
sides of this equation, you

00:11:47.230 --> 00:11:59.040
get 2x is less than
negative 68.

00:11:59.040 --> 00:12:00.345
Oh, I just realized I
made a mistake here.

00:12:00.345 --> 00:12:04.090
You subtract 63 from both sides
of this, 5 minus 63 is

00:12:04.090 --> 00:12:05.350
negative 58.

00:12:05.350 --> 00:12:07.970
I don't want to make a careless
mistake there.

00:12:07.970 --> 00:12:09.420
And then divide both
sides by 2.

00:12:09.420 --> 00:12:12.180
You get, in this case, x is
greater than-- you don't have

00:12:12.180 --> 00:12:14.910
to swap the inequality, because
we're dividing by a

00:12:14.910 --> 00:12:22.440
positive number-- negative 58
over 2 is negative 29, or,

00:12:22.440 --> 00:12:29.110
here, if you divide both sides
by 2, or, x is less than

00:12:29.110 --> 00:12:33.520
negative 34.

00:12:33.520 --> 00:12:36.180
68 divided by 2 is 34.

00:12:36.180 --> 00:12:39.160
And so, on the number line,
the solution set to that

00:12:39.160 --> 00:12:41.880
equation will look like this.

00:12:41.880 --> 00:12:43.470
That's my number line.

00:12:43.470 --> 00:12:46.460
I have negative 29.

00:12:46.460 --> 00:12:49.890
I have negative 34.

00:12:49.890 --> 00:12:53.380
So the solution is, I can either
be greater than 29, not

00:12:53.380 --> 00:12:58.550
greater than or equal to, so
greater than 29, that is that

00:12:58.550 --> 00:13:06.470
right there, or I could be
less than negative 34.

00:13:06.470 --> 00:13:10.410
So any of those are going to
satisfy this absolute value

00:13:10.410 --> 00:13:11.660
inequality.


Compound inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:03.470
Let's do some compound
inequality problems, and these

00:00:03.470 --> 00:00:07.450
are just inequality problems
that have more than one set of

00:00:07.450 --> 00:00:08.060
constraints.

00:00:08.060 --> 00:00:10.020
You're going to see what I'm
talking about in a second.

00:00:10.020 --> 00:00:15.110
So the first problem I have is
negative 5 is less than or

00:00:15.110 --> 00:00:22.300
equal to x minus 4, which is
also less than or equal to 13.

00:00:22.300 --> 00:00:25.640
So we have two sets of
constraints on the set of x's

00:00:25.640 --> 00:00:27.426
that satisfy these equations.

00:00:27.426 --> 00:00:31.270
x minus 4 has to be greater than
or equal to negative 5

00:00:31.270 --> 00:00:36.240
and x minus 4 has to be less
than or equal to 13.

00:00:36.240 --> 00:00:40.180
So we could rewrite this
compound inequality as

00:00:40.180 --> 00:00:49.040
negative 5 has to be less than
or equal to x minus 4, and x

00:00:49.040 --> 00:00:57.800
minus 4 needs to be less
than or equal to 13.

00:00:57.800 --> 00:00:59.990
And then we could solve each of
these separately, and then

00:00:59.990 --> 00:01:02.220
we have to remember this "and"
there to think about the

00:01:02.220 --> 00:01:05.069
solution set because it has to
be things that satisfy this

00:01:05.069 --> 00:01:07.200
equation and this equation.

00:01:07.200 --> 00:01:09.660
So let's solve each of
them individually.

00:01:09.660 --> 00:01:12.430
So this one over here,
we can add 4 to both

00:01:12.430 --> 00:01:13.680
sides of the equation.

00:01:17.090 --> 00:01:21.840
The left-hand side, negative
5 plus 4, is negative 1.

00:01:21.840 --> 00:01:26.120
Negative 1 is less than
or equal to x, right?

00:01:26.120 --> 00:01:28.850
These 4's just cancel out here
and you're just left with an x

00:01:28.850 --> 00:01:30.620
on this right-hand side.

00:01:30.620 --> 00:01:37.120
So the left, this part right
here, simplifies to x needs to

00:01:37.120 --> 00:01:40.780
be greater than or equal to
negative 1 or negative 1 is

00:01:40.780 --> 00:01:42.280
less than or equal to x.

00:01:42.280 --> 00:01:43.580
So we can also write
it like this.

00:01:43.580 --> 00:01:46.110
X needs to be greater than
or equal to negative 1.

00:01:46.110 --> 00:01:46.950
These are equivalent.

00:01:46.950 --> 00:01:48.800
I just swapped the sides.

00:01:48.800 --> 00:01:51.506
Now let's do this other
condition here in green.

00:01:55.960 --> 00:01:57.910
Let's add 4 to both sides
of this equation.

00:02:01.760 --> 00:02:04.330
The left-hand side,
we just get an x.

00:02:04.330 --> 00:02:07.380
And then the right-hand
side, we get 13 plus

00:02:07.380 --> 00:02:09.840
14, which is 17.

00:02:09.840 --> 00:02:13.800
So we get x is less than
or equal to 17.

00:02:13.800 --> 00:02:17.970
So our two conditions, x has to
be greater than or equal to

00:02:17.970 --> 00:02:22.310
negative 1 and less than
or equal to 17.

00:02:22.310 --> 00:02:24.460
So we could write this
again as a compound

00:02:24.460 --> 00:02:25.700
inequality if we want.

00:02:25.700 --> 00:02:29.470
We can say that the solution
set, that x has to be less

00:02:29.470 --> 00:02:34.950
than or equal to 17 and greater
than or equal to

00:02:34.950 --> 00:02:35.550
negative 1.

00:02:35.550 --> 00:02:38.750
It has to satisfy both
of these conditions.

00:02:38.750 --> 00:02:43.676
So what would that look
like on a number line?

00:02:43.676 --> 00:02:46.250
So let's put our number
line right there.

00:02:46.250 --> 00:02:48.590
Let's say that this is 17.

00:02:48.590 --> 00:02:50.090
Maybe that's 18.

00:02:50.090 --> 00:02:51.040
You keep going down.

00:02:51.040 --> 00:02:52.180
Maybe this is 0.

00:02:52.180 --> 00:02:55.620
I'm obviously skipping a bunch
of stuff in between.

00:02:55.620 --> 00:02:58.650
Then we would have a negative
1 right there, maybe a

00:02:58.650 --> 00:02:59.920
negative 2.

00:02:59.920 --> 00:03:03.630
So x is greater than or equal
to negative 1, so we would

00:03:03.630 --> 00:03:04.610
start at negative 1.

00:03:04.610 --> 00:03:07.000
We're going to circle it in
because we have a greater than

00:03:07.000 --> 00:03:08.680
or equal to.

00:03:08.680 --> 00:03:13.440
And then x is greater than that,
but it has to be less

00:03:13.440 --> 00:03:17.580
than or equal to 17.

00:03:17.580 --> 00:03:21.170
So it could be equal to
17 or less than 17.

00:03:21.170 --> 00:03:23.650
So this right here is a solution
set, everything that

00:03:23.650 --> 00:03:25.710
I've shaded in orange.

00:03:25.710 --> 00:03:28.850
And if we wanted to write it in
interval notation, it would

00:03:28.850 --> 00:03:35.030
be x is between negative 1 and
17, and it can also equal

00:03:35.030 --> 00:03:37.120
negative 1, so we put
a bracket, and it

00:03:37.120 --> 00:03:39.510
can also equal 17.

00:03:39.510 --> 00:03:43.350
So this is the interval notation
for this compound

00:03:43.350 --> 00:03:45.330
inequality right there.

00:03:45.330 --> 00:03:46.580
Let's do another one.

00:03:49.920 --> 00:03:51.980
Let me get a good
problem here.

00:03:51.980 --> 00:03:56.620
Let's say that we have
negative 12.

00:03:56.620 --> 00:03:58.640
I'm going to change the problem
a little bit from the

00:03:58.640 --> 00:04:00.270
one that I've found here.

00:04:00.270 --> 00:04:08.230
Negative 12 is less than 2 minus
5x, which is less than

00:04:08.230 --> 00:04:10.230
or equal to 7.

00:04:10.230 --> 00:04:12.950
I want to do a problem that has
just the less than and a

00:04:12.950 --> 00:04:14.630
less than or equal to.

00:04:14.630 --> 00:04:16.709
The problem in the book that
I'm looking at has an equal

00:04:16.709 --> 00:04:18.600
sign here, but I want to remove
that intentionally

00:04:18.600 --> 00:04:20.500
because I want to show you
when you have a hybrid

00:04:20.500 --> 00:04:22.390
situation, when you have
a little bit of both.

00:04:22.390 --> 00:04:28.310
So first we can separate this
into two normal inequalities.

00:04:28.310 --> 00:04:31.940
You have this inequality
right there.

00:04:31.940 --> 00:04:37.530
We know that negative 12 needs
to be less than 2 minus 5x.

00:04:37.530 --> 00:04:43.230
That has to be satisfied, and--
let me do it in another

00:04:43.230 --> 00:04:46.810
color-- this inequality also
needs to be satisfied.

00:04:46.810 --> 00:04:50.740
2 minus 5x has to be less than
7 and greater than 12, less

00:04:50.740 --> 00:04:56.530
than or equal to 7 and greater
than negative 12, so and 2

00:04:56.530 --> 00:05:02.140
minus 5x has to be less
than or equal to 7.

00:05:02.140 --> 00:05:05.290
So let's just solve this the
way we solve everything.

00:05:05.290 --> 00:05:08.050
Let's get this 2 onto the
left-hand side here.

00:05:08.050 --> 00:05:11.730
So let's subtract 2 from both
sides of this equation.

00:05:11.730 --> 00:05:15.500
So if you subtract 2 from both
sides of this equation, the

00:05:15.500 --> 00:05:19.560
left-hand side becomes negative
14, is less than--

00:05:19.560 --> 00:05:23.830
these cancel out-- less
than negative 5x.

00:05:23.830 --> 00:05:27.140
Now let's divide both
sides by negative 5.

00:05:27.140 --> 00:05:29.360
And remember, when you multiply
or divide by a

00:05:29.360 --> 00:05:32.140
negative number, the inequality
swaps around.

00:05:32.140 --> 00:05:35.880
So if you divide both sides by
negative 5, you get a negative

00:05:35.880 --> 00:05:39.980
14 over negative 5, and you have
an x on the right-hand

00:05:39.980 --> 00:05:43.140
side, if you divide that by
negative 5, and this swaps

00:05:43.140 --> 00:05:47.920
from a less than sign to
a greater than sign.

00:05:47.920 --> 00:05:53.560
The negatives cancel out, so you
get 14/5 is greater than

00:05:53.560 --> 00:05:58.580
x, or x is less than 14/5,
which is-- what is this?

00:05:58.580 --> 00:06:01.380
This is 2 and 4/5.

00:06:01.380 --> 00:06:04.320
x is less than 2 and 4/5.

00:06:04.320 --> 00:06:08.090
I just wrote this improper
fraction as a mixed number.

00:06:08.090 --> 00:06:10.620
Now let's do the other
constraint

00:06:10.620 --> 00:06:12.630
over here in magenta.

00:06:12.630 --> 00:06:15.210
So let's subtract 2 from both
sides of this equation, just

00:06:15.210 --> 00:06:16.800
like we did before.

00:06:16.800 --> 00:06:19.910
And actually, you can do these
simultaneously, but it becomes

00:06:19.910 --> 00:06:20.770
kind of confusing.

00:06:20.770 --> 00:06:23.360
So to avoid careless mistakes, I
encourage you to separate it

00:06:23.360 --> 00:06:24.650
out like this.

00:06:24.650 --> 00:06:27.300
So if you subtract 2 from both
sides of the equation, the

00:06:27.300 --> 00:06:30.680
left-hand side becomes
negative 5x.

00:06:30.680 --> 00:06:33.100
The right-hand side, you have
less than or equal to.

00:06:33.100 --> 00:06:37.620
The right-hand side becomes
7 minus 2, becomes 5.

00:06:37.620 --> 00:06:40.780
Now, you divide both sides
by negative 5.

00:06:40.780 --> 00:06:42.370
On the left-hand side,
you get an x.

00:06:42.370 --> 00:06:46.450
On the right-hand side, 5
divided by negative 5 is

00:06:46.450 --> 00:06:47.600
negative 1.

00:06:47.600 --> 00:06:50.440
And since we divided by a
negative number, we swap the

00:06:50.440 --> 00:06:51.380
inequality.

00:06:51.380 --> 00:06:53.310
It goes from less than or
equal to, to greater

00:06:53.310 --> 00:06:54.610
than or equal to.

00:06:54.610 --> 00:06:56.820
So we have our two
constraints.

00:06:56.820 --> 00:07:01.510
x has to be less than 2 and 4/5,
and it has to be greater

00:07:01.510 --> 00:07:03.720
than or equal to negative 1.

00:07:03.720 --> 00:07:05.600
So we could write
it like this.

00:07:05.600 --> 00:07:10.300
x has to be greater than or
equal to negative 1, so that

00:07:10.300 --> 00:07:13.390
would be the lower bound on our
interval, and it has to be

00:07:13.390 --> 00:07:14.940
less than 2 and 4/5.

00:07:20.720 --> 00:07:22.590
And notice, not less
than or equal to.

00:07:22.590 --> 00:07:24.510
That's why I wanted to show you,
you have the parentheses

00:07:24.510 --> 00:07:26.810
there because it can't be
equal to 2 and 4/5.

00:07:26.810 --> 00:07:29.580
x has to be less
than 2 and 4/5.

00:07:29.580 --> 00:07:31.502
Or we could write this way.

00:07:31.502 --> 00:07:37.010
x has to be less than 2 and
4/5, that's just this

00:07:37.010 --> 00:07:40.540
inequality, swapping the
sides, and it has to be

00:07:40.540 --> 00:07:44.670
greater than or equal
to negative 1.

00:07:44.670 --> 00:07:47.210
So these two statements
are equivalent.

00:07:47.210 --> 00:07:52.040
And if I were to draw it
on a number line, it

00:07:52.040 --> 00:07:53.490
would look like this.

00:07:53.490 --> 00:08:00.410
So you have a negative 1, you
have 2 and 4/5 over here.

00:08:00.410 --> 00:08:01.850
Obviously, you'll have
stuff in between.

00:08:01.850 --> 00:08:03.580
Maybe, you know, 0
sitting there.

00:08:03.580 --> 00:08:06.640
We have to be greater than or
equal to negative 1, so we can

00:08:06.640 --> 00:08:08.100
be equal to negative 1.

00:08:08.100 --> 00:08:10.220
And we're going to be greater
than negative 1, but we also

00:08:10.220 --> 00:08:12.700
have to be less than
2 and 4/5.

00:08:12.700 --> 00:08:14.780
So we can't include
2 and 4/5 there.

00:08:14.780 --> 00:08:18.100
We can't be equal to 2 and 4/5,
so we can only be less

00:08:18.100 --> 00:08:22.590
than, so we put a empty circle
around 2 and 4/5 and then we

00:08:22.590 --> 00:08:24.960
fill in everything below that,
all the way down to negative

00:08:24.960 --> 00:08:27.580
1, and we include negative 1
because we have this less than

00:08:27.580 --> 00:08:29.120
or equal sign.

00:08:29.120 --> 00:08:31.820
So the last two problems I did
are kind of "and" problems.

00:08:31.820 --> 00:08:34.419
You have to meet both of
these constraints.

00:08:34.419 --> 00:08:36.025
Now, let's do an "or" problem.

00:08:38.799 --> 00:08:42.620
So let's say I have these
inequalities.

00:08:42.620 --> 00:08:49.910
Let's say I'm given-- let's say
that 4x minus 1 needs to

00:08:49.910 --> 00:08:58.770
be greater than or equal to
7, or 9x over 2 needs to

00:08:58.770 --> 00:09:00.290
be less than 3.

00:09:00.290 --> 00:09:03.460
So now when we're saying "or,"
an x that would satisfy these

00:09:03.460 --> 00:09:06.400
are x's that satisfy either
of these equations.

00:09:06.400 --> 00:09:09.250
In the last few videos or in the
last few problems, we had

00:09:09.250 --> 00:09:11.770
to find x's that satisfied
both of these equations.

00:09:11.770 --> 00:09:14.260
Here, this is much
more lenient.

00:09:14.260 --> 00:09:16.780
We just have to satisfy
one of these two.

00:09:16.780 --> 00:09:19.140
So let's figure out the solution
sets for both of

00:09:19.140 --> 00:09:21.680
these and then we figure out
essentially their union, their

00:09:21.680 --> 00:09:22.980
combination, all of
the things that'll

00:09:22.980 --> 00:09:25.100
satisfy either of these.

00:09:25.100 --> 00:09:27.010
So on this one, on the one
on the left, we can

00:09:27.010 --> 00:09:29.490
add 1 to both sides.

00:09:29.490 --> 00:09:31.440
You add 1 to both sides.

00:09:31.440 --> 00:09:35.480
The left-hand side just becomes
4x is greater than or

00:09:35.480 --> 00:09:39.840
equal to 7 plus 1 is 8.

00:09:39.840 --> 00:09:42.120
Divide both sides by 4.

00:09:42.120 --> 00:09:46.120
You get x is greater
than or equal to 2.

00:09:46.120 --> 00:09:48.790
Or let's do this one.

00:09:48.790 --> 00:09:51.770
Let's see, if we multiply both
sides of this equation by 2/9,

00:09:51.770 --> 00:09:53.070
what do we get?

00:09:53.070 --> 00:09:56.070
If you multiply both sides by
2/9, it's a positive number,

00:09:56.070 --> 00:09:58.570
so we don't have to do anything
to the inequality.

00:09:58.570 --> 00:10:06.760
These cancel out, and you get
x is less than 3 times 2/9.

00:10:06.760 --> 00:10:10.610
3/9 is the same thing as
1/3, so x needs to

00:10:10.610 --> 00:10:12.460
be less than 2/3.

00:10:12.460 --> 00:10:17.280
So or x is less than 2/3.

00:10:17.280 --> 00:10:18.840
So that's our solution set.

00:10:18.840 --> 00:10:23.240
x needs to be greater than or
equal to 2, or less than 2/3.

00:10:23.240 --> 00:10:24.340
So this is interesting.

00:10:24.340 --> 00:10:27.920
Let me plot the solution
set on the number line.

00:10:31.240 --> 00:10:33.380
So that is our number line.

00:10:33.380 --> 00:10:41.630
Maybe this is 0, this is 1, this
is 2, 3, maybe that is

00:10:41.630 --> 00:10:42.940
negative 1.

00:10:42.940 --> 00:10:46.910
So x can be greater than
or equal to 2.

00:10:46.910 --> 00:10:49.960
So we could start-- let me
do it in another color.

00:10:49.960 --> 00:10:53.320
We can start at 2 here and it
would be greater than or equal

00:10:53.320 --> 00:10:58.690
to 2, so include everything
greater than or equal to 2.

00:10:58.690 --> 00:11:01.530
That's that condition
right there.

00:11:01.530 --> 00:11:03.440
Or x could be less than 2/3.

00:11:06.550 --> 00:11:11.120
So 2/3 is going to be right
around here, right?

00:11:11.120 --> 00:11:13.680
That is 2/3.

00:11:13.680 --> 00:11:16.850
x could be less than 2/3.

00:11:16.850 --> 00:11:19.150
And this is interesting.

00:11:19.150 --> 00:11:21.490
Because if we pick one of these
numbers, it's going to

00:11:21.490 --> 00:11:23.020
satisfy this inequality.

00:11:23.020 --> 00:11:24.840
If we pick one of these numbers,
it's going to satisfy

00:11:24.840 --> 00:11:25.850
that inequality.

00:11:25.850 --> 00:11:28.500
If we had an "and" here, there
would have been no numbers

00:11:28.500 --> 00:11:32.420
that satisfy it because you
can't be both greater than 2

00:11:32.420 --> 00:11:34.630
and less than 2/3.

00:11:34.630 --> 00:11:37.020
So the only way that there's
any solution set here is

00:11:37.020 --> 00:11:40.740
because it's "or." You can
satisfy one of the two

00:11:40.740 --> 00:11:41.920
inequalities.

00:11:41.920 --> 00:11:44.480
Anyway, hopefully you,
found that fun.