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Lecture Comments (4)

3 answers

Last reply by: Professor Selhorst-Jones
Sun Jul 24, 2016 12:59 PM

Post by Joshua Jacob on July 23, 2014

Professor Vincent,
Could you explain the second equation in blue for example 2?
I graphed it out and even graphed it in my calculator and the shaded area is under the line, not above it.
Thanks for the amazing lectures!

Systems of Linear Inequalities

  • Remember, we solve an inequality in much the same way we solve an equation. However, instead of giving just a single answer, an inequality has an infinite variety of solutions. Also, remember, when doing algebra with an inequality, multiplying or dividing both sides by a negative number will flip the direction of the inequality sign.
  • When we graph the answers to a linear inequality, we shade in the entire region that satisfies the inequality. Furthermore, the boundary line changes depending on if the inequality is strict ( < or > ) or not strict ( ≤ or ≥ ). If it is strict, the line is dashed, while if it is not strict, it is a solid line.
  • To help figure out where to shade, begin by just graphing the boundary of the inequality (as if it were an equation). Then, once you've drawn the boundary (with the appropriate solid or dashed line), test some point on one side of the boundary. If the point satisfies the inequality, shade that side. If not, shade the other side.
  • When finding the solutions to a system of linear inequalities, it is best to graph each inequality in the system. When we worked with linear equalities (=), we could use substitution or elimination, but that was because we were solving for a single answer. With a system of inequalities, though, we aren't going to get just one answer-we're going to have infinitely many (or none, if the shadings don't overlap). Thus, graph each inequality, shade each appropriately as above, and the solutions to the system are where all the shadings overlap.
  • An excellent application of linear inequalities is linear programming. It helps us optimize systems and make the best choice given certain requirements. We start with a linear objective function that we are trying to maximize or minimize (such as profit or cost). The objective function is based off some number of variables. The variables in the objective function also have various constraints (given as a system of linear inequalities). This gives a region of feasible solutions that are allowed by the constraints. Our objective is to find the maximum (or minimum) of our objective function within those feasible solutions.
  • The theory of linear programming says that if there is a max/min for the objective function within the constraints, it occurs at a vertex ("corner") of the feasible solutions. Thus, to find the max/min of an objective function, we begin by finding the locations of the vertices by solving for intersection points. Once we know all the vertices, we can try each of them in our objective function, and whichever is highest/lowest is our max/min.

Systems of Linear Inequalities

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:04
  • Inequality Refresher-Solutions 0:46
    • Equation Solutions vs. Inequality Solutions
    • Essentially a Wide Variety of Answers
  • Refresher--Negative Multiplication Flips 1:43
  • Refresher--Negative Flips: Why? 3:19
    • Multiplication by a Negative
    • The Relationship Flips
  • Refresher--Stick to Basic Operations 4:34
  • Linear Equations in Two Variables 6:50
  • Graphing Linear Inequalities 8:28
    • Why It Includes a Whole Section
    • How to Show The Difference Between Strict and Not Strict Inequalities
    • Dashed Line--Not Solutions
    • Solid Line--Are Solutions
  • Test Points for Shading 11:42
    • Example of Using a Point
    • Drawing Shading from the Point
  • Graphing a System 14:53
    • Set of Solutions is the Overlap
    • Example
  • Solutions are Best Found Through Graphing 18:05
  • Linear Programming-Idea 19:52
    • Use a Linear Objective Function
    • Variables in Objective Function have Constraints
  • Linear Programming-Method 22:09
    • Rearrange Equations
    • Graph
    • Critical Solution is at the Vertex of the Overlap
    • Try Each Vertice
  • Example 1 24:58
  • Example 2 28:57
  • Example 3 33:48
  • Example 4 43:10

Transcription: Systems of Linear Inequalities

Hi--welcome back to

Today, we are going to talk about systems of linear inequalities.0002

In the previous lesson, we worked with systems of linear equations.0005

In this lesson, we will work with systems of linear inequalities.0009

Remember: equations are based on equals signs--left side = right side.0012

But inequalities are relationships based on less than (I should perhaps use something different),0016

less than or equal, greater than or equal, or greater than.0023

Since we haven't really worked with inequalities before in this course, we will first have a brief refresher.0028

After that, we will consider systems of linear inequalities and how their solutions are based on all the inequalities at the same time.0033

Finally, we will see how systems of linear inequalities can be applied through linear programming, which will allow us to solve optimization problems.0039

Let's go: when solving a linear equation in one variable, we always find exactly one solution.0046

For example, if we have x + 1 = 0, we subtract by 1 on both sides, and we get x = -1.0051

So, on the number line, we would see that the answer to this is when x is -1 on the number line.0056

On the other hand, when we solve a linear inequality in one variable, we find an infinite variety of solutions.0063

So, if we had x + 1 ≥ 0, we subtract 1 on both sides, and we get x ≥ -1, which means we have x equal to -1 as one of the solutions;0070

but we also are allowed to go anywhere above that.0080

+3 would be an answer to this; +52 would be an answer to this; -1/2 would be an answer to this.0083

Everything from -1 on up is an answer to this--everything there will satisfy our inequality.0089

This idea of a wide variety of answers will be important to us as we work with systems of linear inequalities, so we want to keep this in mind.0096

Mostly, when you are doing algebra with an inequality, it is just the same as doing it with an equation.0104

We can do any arithmetic operation--addition, subtraction, multiplication, and division--on both sides, just like normal algebra.0109

However, there is one special thing to be aware of: if you use algebra to multiply or divide by a negative number, the relationship flips.0117

For example, if we have -x + 1, we can subtract 1 on both sides, and we get -x < -1.0126

At this point, we want to make our x positive, so we multiply both sides by -1.0131

That causes our inequality symbol to flip to the other way; so we go from less than to greater than now.0136

So, we now have x > 1; this is really important to remember.0146

You don't want to forget about this when you are working with inequalities.0151

Any time you multiply or divide by a negative number, it will cause0153

your less than/greater than/less than or equal to/greater than or equal to/whatever you are dealing with...0157

it will flip; your inequality symbol will flip to the other direction.0163

It is really easy to miss this flip; it is a really easy mistake to make, so just be careful with this.0167

Every time you are doing either multiplication or division, and it is a negative number that you are dealing with--0172

negative anything that you are multiplying on both sides--you want to make sure you remember about this flipping thing.0178

Notice that we can also get the same thing as we have here if we just added x to both sides.0184

If we had had +x and +x, we would have had 1 < x, which is equivalent to saying x > 1.0188

They are just the same statement, but two different ways of saying it.0197

What causes this to happen--why do we see this negative flipping?0200

You probably learned about it in algebra for a very long time, but you might not have a great understanding of it.0204

So, let's actually get this: consider if we had some a and b, where a is less than b.0208

On our number line, a would come before b; a would be closer to the 0 than b would be.0213

So, what would it look like if we multiplied a and b by -1?0219

Well, that would cause this mirroring around the 0; they would pop to the side opposite, depending on the distance that they originally were from the 0.0223

So, -b and -a appear here; but notice that -b is now farther to the left than -a.0232

Just as b was farther to the right, it is now going to be farther to the left.0238

So, there is still a relationship, but now the relationship has flipped.0242

Because b was more positive than a--b was originally more right than a--we know that -b has to be more negative,0245

has to be more left, than -a; thus, we have this -a > -b,0252

because originally a is less than b, but now -b is less than -a, so that causes our sign to effectively flip,0258

because -b < -a is the same thing as saying -a > -b.0266

Great; that is what is going on.0274

You want to stick to basic operations; when you are working with inequalities, you want to try to keep using addition, subtraction, multiplication, and division.0276

You can use more complicated algebra, like raising both sides to a power or taking a square root or taking a logarithm or taking some trigonometric function.0284

But they can cause little problems to come up.0293

Consider if you have x2 > 0; now, that does not imply that x is greater than 0.0296

We might think, "x2 > 0, so I will take the square root of both sides."0302

√x2 > ±0...well, that is just 0, so we have x > 0, right?0306

No, that is not true at all: over here, x = -3 is true--it checks out: (-3)2 is greater than 0, because 9 is greater than 0.0312

So, that is great; but -3 > 0--that is completely false; that is not a true statement.0322

So, we do not have this ability to just toss out other, more complicated algebra things0330

without really thinking about all of the implications that are going to happen here.0335

So, this idea here doesn't work, because we are dealing with an inequality.0339

We can't just take square roots, like we can when we are dealing with equations, which is how we built up all that previous theory.0344

So, when you have an inequality, and you want to do something more than just a basic operation on both sides,0350

you really have to be careful and think about what you are doing.0355

That means that it is best to stick to basic arithmetic;0359

so we really just want to stick with addition, subtraction, multiplication, and division,0363

not because it is impossible to do this other stuff--not because you wouldn't get it--0367

but just because it is easy to make a mistake doing it.0370

You can end up making mistakes, because you haven't thought about absolutely all of the ramifications.0373

So, you want to stick to the basic operations, because it is easy to see what happens.0378

And remember: even with the basic operations, when we have multiplication and division,0381

there is still a little bit of danger there, because when you do it with a negative number, it causes flipping of your inequality.0385

So, with all of this in mind, it is really just best to stick to basic arithmetic.0391

Happily, since we are only going to be working with linear inequalities, that means we can get through everything0395

just using these basic operations--addition, subtraction, multiplication, and division.0400

They are going to be enough to manipulate our linear inequalities into forms that make it easier for us to understand.0404

All right, that finishes our refresher; now we are ready to actually talk about linear inequalities in two variables.0409

A linear inequality in two (or more) variables tells us about a relationship between those variables.0415

Consider -x + y ≥ 1: we can make it easier to interpret by just having y on one side and other stuff involving x on the other.0420

So, we add x to both sides: + x, + x; and we get y ≥ x + 1.0429

This gives us a relationship between x and y--how they are connected.0435

If x is equal to 1, we know that y has to be greater than or equal to 2.0439

If x is 1, then y has to be greater than or equal to 1 + 1, so y must be greater than or equal to 2.0444

Similarly, if x were equal to -9, then that would mean that y is greater than or equal to -8.0451

So, depending on what the x is, we will change the requirements on our y.0456

It is a relationship: it is not this definite single static defined thing.0459

It is this relationship of "as x shifts, y has to shift in certain ways to accommodate the shifting x."0464

Now, notice: unlike a linear equality, where a single x would give us just a single y, a single x in an inequality gives infinitely many possible y's.0470

x = 1 means y is greater than or equal to 2; so x = 1 means that y = 2 is good; y = 5 is good; y = 47 is good; y = 1 billion is good.0482

We have this stack as possibilities, as long as they follow that inequality relationship.0493

If x equals 1, y still can't be equal to -3; so it is not that anything goes--it just has to follow this inequality relationship.0497

But if it does, there is an infinite variety of things that it could end up being.0505

The best way to understand inequalities is by graphing them.0509

This gives us a visual reference to see all of the possible solutions.0512

So, for example, if we had y ≥ x + 1, we could graph it like this.0515

Because it is a greater than or equal to, everything on our actual line is allowed as a solution,0521

because y = x + 1 is going to be true by "greater than or equal."0530

That "or equal" means that we can have equality; the line itself will be true.0534

But then, in addition to that, everything above the line is also going to be true.0538

Everything in here is going to end up being true; and you should notice that it doesn't just stop at the edge there.0544

Once again, it is like how we don't draw arrows to show that it keeps going.0550

We have gotten used to assuming that the graph keeps going beyond the edges of our graphing window.0553

We are going to also be assuming that our shading goes past the edges; it doesn't stop suddenly--it goes forever, infinitely, up.0557

So, everything above the line is a solution, because if y is greater than or equal to x + 1, then consider at x = 0:0566

then 1 is true for y, but so would 2 be true for y...0574

Let's use a different color, just so we can see this a little bit easier.0577

2 would be true for y; 3 would be true for y; 4 would be true for y; 5 would be true for y; 6 would be true for y; and anything above that.0580

So, as long as we are above this line with our y-value for a given x-value, we will end up being true; the y > x + 1 is true.0587

So, a graph is a really great way to get across all of these possibilities, because it is not just this single line.0597

It is certainly not just a single point; it is this huge variety--it is this large area of possibilities.0602

This brings up a question: what does the line become if the inequality is strict--less than or greater than--0609

as opposed to not strict, which is less than or equal to, or greater than or equal to?0615

So, if we have a strict inequality, less than or greater than (it is purely less than, or it is purely greater than),0619

we show this by changing how we draw the line on the graph.0628

So, if it is a strict graph, like y > x + 1, we show that it is strict by using a dashed line.0631

It is a little bit hard to see here; but notice that the line is actually dashed, like this.0639

So, that says that we are saying that we are not including the line; we are only including the stuff above the line; the line itself is not allowed.0644

On the flip side, if it is a not-strict inequality, like greater than or equal to, we show it by using a solid line,0653

saying that the line itself is actually going to be an answer, as well.0661

If you are on the line, that is an allowed answer, in addition to all of the stuff that is above it.0665

So, we show that the points on the line are not solutions to our inequality with a dashed line, when it is a strict inequality.0671

We show that they are solutions with a solid line when it is a not-strict inequality.0684

And then, we just shade the appropriate side.0692

In both of these cases, we have greater than and greater than or equal to; so they both had the part that was above it.0694

Now, we don't necessarily have a very great way to see how to shade which way.0702

You will get a sense of if it is up, or if it is down, as you work with these more.0707

You can pay attention to what the y is, and if it is less than or greater than.0710

But a really great way to be sure of which side to shade is by using a test point.0713

You can shade by seeing how this test point gets affected.0719

Would this test point satisfy our inequality, or would it fail to satisfy it?0723

If the point satisfies the inequality (you just choose some point--it is completely your choice--0729

you choose some point, and if it satisfies that inequality), then the whole side does,0735

because we have to shade in an entire side.0739

So, if one point on one side satisfies, that entire side has to satisfy.0741

On the flip, if the point does not satisfy the inequality, then the other side will satisfy the inequality.0745

So, if your point fails to satisfy, the opposite side will satisfy it.0755

This makes it easy for us to shade: for example, if we have y < -5/2x + 3, we could try the test point (0,0).0759

(0,0) is almost always a great test point, because you can plug in 0's really, really easily.0771

Let's see: would that end up working out, if we have y < -5/2x + 3?0777

We plug in 0 for y; we plug in 0 for our x; plus we 0 less than 3?0783

Sure enough, that checks out; so our test point is good.0792

If our test point is good, that means that the entire side must be good; so we just shade this side to say anything below that line is going to be true.0795

And notice: how did we get this line in the first place?0808

We just take our inequality; we turn it into a line, as if it had been an equation, y = -5/2x + 3;0810

and then, we graph that, because we know that that is going to be our line of demarcation.0817

It is going to be the place where the shading changes over.0821

And then, it gets dashed, because it is a strict inequality; so that is why we have that dashed line.0824

And then, we shade off of that line that we drew.0830

Alternatively, we could have tested a different point, like, let's say, (4,4).0834

Maybe we had wanted to test (4,4); well, if we had tested that, it would be 4 for our y, less than -5/2 times 4, plus 3.0838

So, 4 < -5/2(4) that gets us -10...+ 3; so is 4 less than -7?0848

No, that fails to be true; so this side cannot be the side that get shaded in; that side is not going to end up being true.0865

But really, I recommend using the point (0,0).0873

You could use any point; you could use (1,0), (-5,3)...whatever point is useful for your specific case.0875

But (0,0) is almost always going to be want to choose something that is definitely on one side or the other.0881

You don't want to choose something on the line itself.0886

But (0,0), as long as it is not on one of your lines, is going to be a really good test point for using.0888

All right, if you are graphing a system of inequalities, you do it in the same way.0893

You graph each of the lines as if they were equalities; and you use either a dashed line0898

(if it is a strict inequality, less than or greater than); and you use a solid line if it is a not-strict inequality0904

(less than or equal, or greater than or equal); and then you shade in each inequality appropriately.0912

And wherever the shadings overlap is the set of solutions to the system.0917

So, let's see how that would work out here.0921

For this one, y < 2/3x - 2, that is our red dashed line--dashed because it is a less than.0922

So, let's try the test point (0,0) for that: (0,0)...we plug that in; we have 0 < 2/3(0) - 2.0930

So, 0 < -2 is not true; 0 is greater than -2; so this fails.0945

So, that means that the side opposite our test point is going to have to be what we shade in for the red one.0955

Our test point was in the very center, in the origin; so we are going to be opposite that side.0961

So, this is going to be what we shade in for the red line: y < 2/3x - 2 is true for everything in the shading, but not on the dashed line itself.0966

Then, if we have y ≥ -2x + 1, let's use that same test point; it is not on that line, either.0979

So, (0,0): 0 ≥ -2(0) + 1; so 0 ≥ 1; is that true?0985

No, 0 is not greater than or equal to 1; so that fails here, as well; 0 is not greater than 1.0998

That fails; so by the same logic, we are going to have to be opposite that side, greater than or equal;1006

so we are going to be above it; so we shade in the side that is opposite.1013

Also, you can think of this as being that y is greater, so it must be above the line;1019

and the red one was "y is less than," so it must be below it.1023

But the test points are a nice way to be absolutely sure of it.1026

And notice how we have overlapping; we have red and blue overlapping in this bottom-right portion.1029

So, that means the part that really, truly makes up the answers is everything in here.1034

And that is going to keep going out forever, as long as it is in there.1043

Everything on the blue line actually would be an answer, but things on the red dashed line...1047

The blue solid line here is going to be answers; but the red dashed line here, since it is dashed, fails to be answers,1052

because it doesn't satisfy our inequality y < 2/3x - 2, because that is strict.1064

So, if it is on the blue line in that shaded portion, it is an answer; if it is on the dashed red line, it is not an answer.1070

And there we are--we can see how the system comes together; we can see all of the solutions to it at once.1077

All right, now: solutions are best found through graphing.1086

Graphing is usually the best way to understand the solutions to a system of linear inequalities, as long as it is in two dimensions.1089

For dealing with really high numbers of dimensions, it gets kind of hard to graph.1095

But if we are in two dimensions, it is a good way to understand what is going on.1098

It gives us a way to visually comprehend what we are seeing.1101

It lets us see all of the requirements that the system places on the variables.1105

So, this system of inequalities makes certain requirements of our variables.1110

And by graphing it, we are able to see all of the requirements at the same time.1116

Now, with a linear equality--if we had a linear equality (an equals sign between the left and right side),1119

we could use substitution or elimination, because we were solving for a single answer.1126

A linear equality is solving for one answer.1131

But with a system of inequalities, we aren't going to get just one answer; we are going to have infinitely many answers, because we have shadings.1135

As long as we have overlapping shadings for our system, that entire area of overlap,1145

where all of our inequalities overlap together--that is going to be all of our answers.1150

And because it is an area, there are infinitely many things inside of that area.1155

Now, it is possible for us to have no answers whatsoever, if the shadings don't overlap--1158

if one side shades up, and the other side shades down, and they never touch each other.1163

That is never going to have any answers, because the two can never agree; they are inconsistent.1168

But as long as they end up agreeing in some portion--some amount of area, inside of that area we will have infinitely many answers.1172

So, when working with a system of inequalities, you will almost always want to graph,1178

because you can graph it and then shade in the solutions, and you are able to see what is going on,1183

and get an intuitive, visual sense of how this thing is coming together.1188

All right, now we are able to talk about linear programming: an excellent application of linear inequalities is linear programming.1193

It is not quite like computer programming; but it helps us optimize...1201

It is actually quite different than computer programming; but in any case,1205

it helps us optimize systems and make the best choice, given certain requirements.1208

What does it mean to optimize? Well, we start with a linear objective function.1214

We have something that is our objective, something that is sort of looking at the variables that we are dealing with.1218

And we are trying to either maximize the objective function, or we are trying to minimize the objective function.1224

So, we might want to try to maximize something like profit; or we might want to try to minimize something like cost.1229

These are useful things; business is something where linear programming will definitely pop up.1237

For example, we could have an objective function like z = 2x + 7y.1241

So, if we have z = 2x + 7y, it uses x; it uses y; it is an objective function.1247

We get some number out of 2x + 7y; we want to try to maximize or minimize this quantity, z.1252

Now, notice: if there are no restrictions on x and y, z will be at its biggest when x is flying out to infinity and y is flying out to infinity.1263

It will be minimized when x is flying to negative infinity and y is flying to negative infinity.1273

So, we need to have some limitations on what x and y are going to be for us to actually be able to find any maximum or minimum, for that to be meaningful.1277

And indeed, when you are working with linear programming, the variables in the objective function,1285

these variables x and y, are going to have various constraints on them--things that keep them from being able to go anywhere.1289

And those constraints will be given as a system of linear inequalities.1295

So, we might have constraints like x + 3y must be less than or equal to 8,1300

2x + y must be less than or equal to -4, and -x + 2y must be greater than or equal to -3.1304

So, our objective will be to find the maximum or minimum of our objective function1309

(remember: our objective function, in this case, was our z = 2x + 7y) while we are obeying these constraints.1315

We can't break these constraints; so we have to stick within these constraints,1321

but we are trying to get z to be the biggest or the smallest thing we possibly can get out of it.1325

So, once we know what our objective function is (z in the previous slide), the first step is to graph the system of linear inequalities.1331

To do that, we probably want to put them in a form that we can easily graph.1337

So, x + 3y ≤ 8--we can convert that into y ≤ -1/3x + 8/3: we subtract x and divide by 3.1340

We can convert 2x + y ≤ -4 into -2x - 4 by subtracting 2x.1351

And -x + 2y ≥ -3--we add x to both sides and divide by 2; so we have things that are easy for us to graph.1358

We have slope; we have the y-intercept at this point; so we can graph these.1366

We graph these, and we get this; and we also shade in, and according to our shading,1370

we find out that what is inside of there is going to be what is allowed by these constraints.1375

So, the shaded area is the location of all feasible solutions--and by feasible, we mean those that are allowed by the constraints,1381

those that are possible things that we can even begin to consider using.1388

So, the location of our optimum value, whether it is a maximum or a minimum, must be inside of this portion.1392

It has to be inside of this shaded thing or on the lines, because these were all less than or equal or greater than or equal.1399

They are all not strict, so we can be on the lines, as well.1405

So, since we can be on the lines, it has to be somewhere either on the lines or inside of that shaded portion.1408

We know that for sure; otherwise, it won't have followed the constraint, so we can't even look at it for trying to use our objective function.1413

Here is the critical idea: the theory of linear programming tells us that, if the solution exists--1421

if there is some optimum maximum or minimum--if we can maximize or minimize our objective function--1427

if that exists, then it occurs at a vertex of the set of feasible solutions.1434

Now, a vertex of the set is one of the corners of the set, where two or more of them intersect.1441

That is going to be one of the vertices; a vertex occurs at a corner.1451

Now, we can find the locations of the vertices by solving for intersection points.1455

We can see where the red line intersects the blue line, where the green line intersects the blue line, or where the green line intersects the red line.1459

We are just using what we learned from systems of linear equations to be able to figure out where two lines intersect each other.1466

We can figure out where these vertices are located.1472

Once we know all of the vertices, we just try each of them in our objective function.1475

We can figure out each of these vertices from what we learned in the previous lesson.1480

We can understand what is going on from what we have seen in this lesson.1483

And then, from there, we just plug them into our objective function;1486

and whichever ends up coming out to be highest or lowest ends up being our maximum or our minimum.1489

We will see this process done in Example 3 and in Example 4.1494

All right, we are ready for some examples.1498

The first one: Give the system of linear inequalities that produces the below graph.1501

So, the first thing: let's figure out what are the lines that we are seeing drawn here.1505

Before we even worry about it greater than? Is it less than? Is it less than or equal to? Is it greater than or equal to?...1509

before we even worry about that, let's just figure out what equation could draw each of these lines.1516

For our red one right here, we see that it is a horizontal line; it is at y height of -2.1521

So, that would be made up by y = -2; that equation would draw that line.1527

Of course, it is not going to be a less than or equal or a greater than or equal, because it was dashed.1532

But we know that that line would be drawn by y = -2; we will come back to figuring out what it is later.1538

Our green one is going to be set at a horizontal location of 4; it has no slope whatsoever (it is vertical).1543

So, that is going to be x = 4; we have set our horizontal location at 4, and our y is allowed to freely roam up and down.1551

Finally, the only one that might even be slightly difficult is this blue dashed line, which goes from (0,4) to (2,0).1559

If that is the case, we can figure out what the slope of our blue one is.1568

It manages to go down four; it has a vertical change of -4 over a horizontal change of +2.1572

It goes from a height of 4 to a height of 0 by going 2 steps to the right.1580

That means it has a slope of -2; we also see that its y-intercept is right here at 4.1583

So, its y-intercept is 4, so we have y = -2x + 4.1589

At this point, we have equations that would produce each of these lines.1597

So now, it is a question of what symbol goes in there; we can't use equals, because that would be an actual line.1601

We need inequalities; so now it is a question of less than, greater than, less than or equal to, or greater than or equal to, or each of these.1607

All right, we are going to swap out that equals sign for something that can actually be used as an inequality.1616

So, the first step: the red has to shade above, because we have stuff above it,1622

because if we go below it, that fails to be true; so it must be above it.1628

That means that y has to be strictly greater than (because it is a dashed line) -2.1633

If we want to be absolutely sure about this, we can try the test point (0,0).1645

If we plug in 0 > -2, that works out; so we know that that side is correct; that is the side that we want.1648

For the green one, we see that it is one the left side of that.1655

So, if it is on the left side, we know that the horizontal values that we are allowed to use must be less than 4.1659

Is it less than, or less than or equal to? It is less than or equal to, because it is a solid line: less than or equal to 4.1665

We can also check out a test point: if we plug in (0,0), then we have 0 ≤ 4; yes, that is true.1673

So, we know that that is correct for that shading.1679

And then finally, our blue one: for this one, we know that it is shading up.1681

So, if it is shading up, then it must be that our y-values are greater than1688

(and it is going to be strictly greater, because it was a dashed blue line) -2x + 4.1693

If we want to check that with a test point, we could try a test point, just like we did with the others.1702

This one is a little bit harder to do in our head, since it involves more things.1707

0 is greater than -2(0) + 4; 0 > 4; that test point fails, so indeed, it has to be shaded on this side--we chose the correct one.1711

I'm sorry; I didn't mean to make that look like a greater than or equal to.1723

We chose the correct symbol: y > -2x + 4, so in total, these are the inequalities that make that system be graphed like that.1725

Great; the second example: Graph the solutions to the system below.1737

The first thing we want to do--we probably want to convert this into a form where we can easily graph these.1741

Divide by -3; since we are dividing by a negative number, the sign flips.1747

We have y ≤ -2; next, 3x + y < 3, so y <...subtracting 3x...-3x...+3...we don't have to worry about flipping, because it is addition or subtraction.1751

And then finally, 6x - 4y...-4y < -6x - 4; divide by -4: flips because we divided by a negative number, so -6/-4 becomes +3/2x; -4/-4 becomes positive 1.1766

At this point, we have enough for us to graph this thing; so let's draw some axes in.1784

We will make a tick-mark spacing; it is just a length of 1, so 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5.1795

Great; at this point, we can draw each one of these in.1818

We will draw them in as lines first; and then, from there, we will figure out how to shade.1821

y ≤ -2; we use a solid line, because it is less than or equal, so it is not strict.1826

At height -2, we are going to have a line, because we said y ≤ 2; so we graph in the line as if it was y = -2.1832

We have just set a vertical height of -2, and our x is allowed to freely roam.1842

Next, the red one: y < -3x + 3, so our first point is at the y-intercept of 3.1847

If we go over one step, we will have our slope of -3 kick in; we will go down 3, so we can draw this one in.1853

All right; oops, I should have had that be dashed, because it was a less than, so it is strict; I will dash that with the eraser.1865

And then finally, our green one: I won't forget to dash this one.1879

3/2x + 1; so we have a y-intercept at 1; if we go over 2, we go up 3, because our slope is 3/2: over 2, up 3; 1, 2, 3.1882

We draw this one in; all right, and now let's figure out our shading.1894

y ≤ -2; well, if it is less than or equal to -2, we have to have shading below that; so we shade below that here,1909

because we can use a test point, like (0,0), but we also see that y has to be below -2; otherwise that is going to not be true.1925

So, we shade in below that.1933

For our red one, y has to be less than -3x + 3; once again, the y's will have to be below this value.1934

We can use a test point, like (0,0); if we plug that in, we will have 0 < -3(0) + 3, which means 0 < +3, which is, indeed, true.1940

So, we shade towards that test point of (0,0).1950

And then finally, we have a green one: y > 3/2 + x; 3/2 + x is going to get us this one here.1956

So, that is going to cause us to shade up; if we plug in a test point of (0,0), we have 0 > 3/2 times 0, which means 0 > 1.1975

Oops, if 0 is greater than 1...that would fail to be true, so we are shading the opposite way; we are shading away from our original test point.1987

Our original test point was (0,0), so we are shading in the opposite direction of that.1997

Let's extend our red so we can see a little bit better where we are going.2001

At this point, we see that the only place that ends up having green, red, and blue together is this section over here.2005

Here (where I am shading in with a zigzag purple)--this section over here, and continuing out in that arc, will end up being the answers to our solution.2013

We see the solution graphed in that way.2025

All right, the third example: Find the maximum and minimum values of the objective function z = 2x + 7, given the below constraints.2028

The first thing we do is change them into a form so we can easily graph them and have a better understanding of what is going on.2037

Just like we had before, this is the same example that we were working with2045

when we were talking about the idea of linear programming, maximizing and minimizing objective functions.2048

So, this is the same one that we had before: y ≤ -1/3x + 8/3; 2x + y becomes y ≤ -2x - 4;2054

and then, we have y ≥ -1/2x...sorry, not -1/2; it gets canceled out as it adds over, so +1/2x - 3/2; great.2067

So, let's just give a rough sketch, so we can see what is going on.2082

We don't have to worry about having this perfectly precise.2085

The idea is just to be able to see this, because unlike graphing the solutions to an inequality,2087

where we never really figure out the thing, so we want to have a solid graph; in this case,2093

the graph is just a reference, so we can have a better understanding of how the solving is working.2096

So, -1/3x + 8/3; I will start at around some height of about 3.2101

And so, let's actually put in marks, so we have some rough sense of where we are going here.2109

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5; great.2116

OK, the red one: positive 8/3, so we are almost at the 3; and then 1/ will slowly slope down to...2126

by the time it gets here, it will be about here; you could drop down...and it is going to be a solid line,2134

because it was less than or equal to; so it is like that.2140

And then, y ≤ -2x - 4; at -4...and then, for every one it goes to the left, it will go up 2, 2, 2...2143

Also, it is solid once again, because it was a less than or equal.2158

And then, the final one is also going to be solid; it is less than or equal to 1/2x - 3/2, so halfway between the -1 and the -2.2162

At 1/ there...OK; it is not a perfect graph, but it gives us a rough idea of what we are looking for.2170

We are looking for something that is going to be in this part, including the lines.2181

Now, the theory of linear programming--what we learned about that--the critical idea--was that we know2187

that the answer is going to be (if there is an answer at all) on one of these vertices.2192

For the maximum and the minimum, each will show up as a vertex on this triangle (or whatever kind of picture we have).2198

It is going to show up as a vertex on this image--one of the corners to our set of allowable points/feasible solutions.2205

So, let's figure out what it is: let's do the red and the blue together first.2213

If that is the case, we can treat this as y = -1/3x + 8/3, and y = -2x - 4,2217

because for this, we are trying to figure out where the lines intersect, because we are looking for vertices.2224

-1/3x + 8/3 = -2x - 4; where do our red line and our blue line intersect?2228

Multiply everything by 3; at this point, I am just going to use blue.2240

Multiply everything by 3 for ease: -x + 8 cancels out those fractions; -6x - 12...2242

We add 12 to both sides; we get 20 over here; add x to both sides; we get -5x; divide by -5 on both sides; we get -4 = x.2250

We can plug this into one of the two functions; let's go with the -2x - 4, because that will be easier.2262

Once again, we can use it as an equality, because we are looking for a line intersection.2267

We aren't worried about the inequality effect; we are worried about "as if it were a line," because we only care about the vertex right now.2271

So, we can plug that in, and we will have that y = -2...plug in our x, -4; minus 4; y =...-2(-4) gives us + 8, minus 4; so y = 4.2277

Great; so one of our vertices is (-4,4).2290

Next, what if we had our red intersect our green?2295

-1/3x + 8/3--if that was the line--would be equal to (because we are looking for the intersection) 1/2x - 3/2.2299

So, where do those intersect? Let's multiply everything by 6, just so we can get rid of these fractions; I think that makes it easier to look at.2309

So, multiply everything by 6; I will just choose red for the color I will use here.2316

That gets us -2x plus...8/3 cancels out the 3 portion of the 6, which means it is going to be 2(8), 16.2322

Now, we are multiplying by 6 still, so it is going to give us 3x, multiplying by 6, minus 9, because 3/2 times 6 becomes 9.2330

At this point, we can solve this out: add 2x to both sides; add 9 to both sides.2338

Adding 9 to both sides, we get 25 =...add 2x to both sides...5x/5; 5 = x.2342

We can now plug this into either one of them; to me, the y = 1/2x - 3/2 looks ever so slightly friendlier, so I will plug into that one.2348

So, y = 1/2x, this one right here, so y = 1/2(5) - 3/2; so 5/2 - 3/2 = 2/2, which equals 1.2358

At this point, we have another vertex; 5 = x and y = 1; so that is going to occur where the red and green intersect over here, (5,1).2375

And then finally, let's look at where the blue and the green would intersect, the one that we haven't looked at yet.2390

-2x - 4 = 1/2x - 3/2; multiply everything...2396

I'm sorry; let's change that into colors, just so we can keep our color scheme going.2404

1/2x - 3/2...we will multiply everything by 2 to get rid of that fraction; so -4x - 8 = x - 3.2409

Put our x's together; put our constants together; add 3 to both sides; we get -5.2420

Add positive 4x to both sides; we get 5x; divide by 5 on both sides; we get -1 = x.2424

We can now plug that into either one; this one, to me, looks friendlier, so y = -2(-1) - 4; y = +2 (because the negatives cancel out) - 4; y = -2.2431

At this point, we have y = -2 and -1 = x; so a point here will be (-1,-2).2447

We have found the three vertices to this; now, we need to go use that information.2458

They are at (-4,4), (5,1), (-1,-2); so now, we want to evaluate them in our objective function, z = 2x + 7y,2464

and figure out which one makes z biggest, which one makes z smallest, and which one is just somewhere in the middle.2473

All right, so where are our maximum and minimum?2479

Our vertices are (-4,4), (5,1), (-1,2); we have (-4,4) here, (5,1), and (-1,-2) here.2481

All right, so let's start with (-4,4); we plug that in for z = 2x + 7y.2494

So, what will our z end up coming out to be?2503

2 times our x component is -4; plus 7 times our y component is positive 4, so that gets us -8, plus 7 times 4 (28), which equals 20.2506

So, we get 20 out of (-4,4).2517

Let's try our next one, (5,1): plug that in--we get z = 2 times our x component of 5,2519

plus 7 times our y component of 1 which equals 10 + 7, so we get 17.2527

And then, finally, we plug in our (-1,-2), and that gets us z = 2 times x component, -1, plus y component, 7(-2).2535

So, we get -2 - 14, which is equal to -16.2552

All right; at this point, we can see who is our winner for maximum; that is going to be z = 20, which occurred at (-4,4).2558

So, (-4,4) is the maximum for our objective function.2566

And (-1,-2) put out -16, so that was the minimum value that we ended up seeing.2571

And (5,1) gave us +17, but that is neither a maximum nor a minimum, so we just forget about it.2581

All right, the final example: A car lot needs to choose what cars to order for selling on their lot.2589

They have two options: the Nice, which they purchase at 20000 and sell at 30000, and the Superfine, which they purchase at 50000 and sell at 65000.2596

If they have a total budget of 2.9 million dollars for purchasing cars, and 100 spots on the lot for new cars,2606

how many of each car should they buy to maximize profit?2613

Assume that they manage to sell whatever they purchase.2616

The idea here is that they purchase a car, and then they mark it up and sell it.2619

And so, the difference between those two is how much money they make.2623

But they have limits on how much money they start with for purchasing cars, and how many spots they have to actually put the cars on the lot.2626

So, they have to figure out what is the best combination of cars to purchase, to be able to maximize how much money they make out of it.2633

So, with all of this in mind, let's start setting some things up.2640

First, we are going to have to figure out how many numbers of Nice they buy.2643

Let's say N is going to be the symbol that we use for saying the number of Nice cars that they buy.2647

The Nice cars that they buy will be N, however many that is.2655

And then, the Superfines that they buy--S will symbolize the number of Superfines that they buy.2659

OK, now another thing that we have here is a lot of really big numbers: 30000, 20000, 50000, 65000, 2.9 million dollars.2666

Now, we could work with the numbers as they actually are, and everything would end up working out.2678

But we would have this extra factor of 100 showing up; all of these things are divisible by 1000.2681

So, I am going to say, "Why don't we make things a little bit easier on ourselves?"2686

And since we are dealing with lots of money--we are dealing with all of these big things,2690

measured in the thousands and tens of thousands, let's convert everything.2694

For ease, let's work in terms of thousands of dollars; let's talk about everything in terms of how many K it is--2699

how many thousand (K, kilo, thousand)--how many thousand dollars everything is.2708

OK, that will make things a little bit easier, just in how many 0's we have to write down.2713

So, next, let's figure out how much profit they make off of a single Nice,2718

because what we want to do is maximize the profit; so we need to create some function p2724

that is going to be p = the profit, so we need to come up with some equation that describes p2729

in terms of how many Nice and how many Superfines they buy--how much N and S we have to go around--2735

because we know for sure that they will sell all of them.2740

So, how much profit do they make if they buy such a number of Nice and such a number of Superfines?2742

First, we have to figure out how much profit they make off of selling a Nice and how much profit they make off of selling a Superfine.2747

A Nice: if they buy a Nice, it costs them 20K, 20000; it costs them 20000, which is 20K in our new form.2753

They can sell that for 30K, which would net them a profit of the amount that they sold it for, minus the amount that they had to buy it for.2766

That would net them a profit of 10K.2778

For the Superfines, they cost them 50K to buy; they sell them at 65K; so each one they sell is going to net them a profit of 15K.2782

OK, with this in mind, let's write p in green for American money, since we are dealing with U.S. dollars here.2800

p = 10 times N, the number of Nices that we sell, plus 15 times S, the number of Superfines that we sell,2810

because remember: for each Nice we sell, we manage to make 10K; for each Superfine we sell, we manage to make 15K.2822

So, the total profit is going to be 10N + 15S...K, but we don't have to worry about the K's now.2829

We won't worry about it when we are actually dealing with the numbers--the fact that they have the unit of thousands of dollars.2836

All right; but there are some restrictions here.2842

We were told that there is a restriction of 2.9 million dollars for purchasing cars.2845

They don't just have unlimited amounts of money to buy cars.2852

If they had unlimited amounts of money, they would want to stock their entire car lot with nothing but Superfines.2855

They would just want to fill the thing all up with Superfines; but they have a cost to these things.2860

So, we have to deal with that; so if it is 2.9 million for purchasing cars, then...2865

well, each Nice that we buy is going to be 20N; and each Superfine that we buy is going to cause us to spend 50S.2873

So, 20N + 50S is how much we have spent on cars.2884

Now, they can spend up to 2.9 million; so 20N + 50S has to be less than or equal to their total budget.2887

And their total budget is 2.9 million, which we can write out as 2900, because 2.9 million is the same thing as 2900 thousand.2895

So, since we divided everything by 1000, we are now dealing with 2900.2904

All right, another piece of information that we had was that there are only 100 spots on the lot.2908

There is a maximum number of spots that the can fit cars into.2914

So, if that is the case--if they only have 100 spots to fit cars into--well then, the number of Nices that we are buying,2918

plus the number of Superfines that we are buying, has to be less than or equal to 100,2924

because they can only fit up to 100 of these cars on the lot.2929

So, that is two requirements so far.2932

Now, there are two hidden requirements that we might not see, but they will make it easier to actually graph the thing.2934

Think about this: is it possible to have a negative number of Nices--can they buy negative cars?2942

No, the lowest number of Nices that they can buy, or the lowest number of any car that they can buy, is 0.2947

So, we know that however many Nices that they buy, it has to be greater than or equal to 0.2952

And similarly, S must be greater than or equal to 0.2958

We have four requirements on this: the amount of money that have to spend (20N + 50S has to be less than or equal to 2900);2961

N + S has to be less than or equal to 100 (the number of spots on the lot); and the fact that they can't buy negative cars--2968

N has to be greater than or equal to 0, and S has to be greater than or equal to 0.2974

And our objective function that we are trying to maximize, because we are trying to maximize our profit, is this p = 10N + 15S.2977

We want to maximize p; at this point, we bring to bear the power of linear programming.2987

First, let's convert some of this stuff into things that we can easily graph.2993

So, if we are going to graph this, so we can see what is going on, we have to choose something to be horizontal and something to be vertical.2998

So, just arbitrarily, let's make N the horizontal and S the vertical; N gets to be horizontal; S is vertical.3009

With this idea in mind, we can now talk about ordered pairs that come in the form (N,S), just as (x,y) is (horizontal,vertical).3022

Since we made N horizontal and we made S vertical, it is going to come in (N,S).3031

Now, there is no reason it couldn't be flipped; but we just chose one, and we stick to it.3036

And as long as we stick to it, it will work out.3039

OK, at this point, if that is the case, then since S is the vertical, we normally solve for y (the vertical) in terms of other stuff;3041

so we want to solve for S (the vertical) in terms of other stuff, because that is what we are used to.3047

N + S ≤ 100 means that S has to be less than or equal to -N + 100.3051

20N + 50S ≤ 2900; 50S ≤ -20N + 2900.3058

We divide both sides by 50; we have S ≤ -2/5N + 2900/50 (becomes...give me just a second...58; I had to use a calculator for that one).3068

So, 2900/50 becomes 58; we have S ≤ -2/5N + 58.3084

So, with this in mind, we have enough to be able to see what we are allowed to go for here.3091

If N has to be greater than or equal to 0, then we have this graph...3099

If N has to be greater than or equal to 0, this is actually going to be the green one; so for a second, forget about that mistake.3104

N has to be greater than or equal to 0, so we set N equal to 0 and draw a line.3114

S is allowed to go wherever it wants; everything on this side is fine by our N > 0.3119

S has to be greater than or equal to 0: we set a line S = 0; everything greater than that is fine by S ≥ 0.3126

Now, the interesting parts are actually going to end up being the S ≤ -N + 100...3136

If that is the case, we can get a good sense of this; let's say 100 is up here; at -1, it marches down like this at a 45-degree angle.3141

And we know that S has to be less than or equal to -N + 100, so it will be below this.3152

And S ≤ -2/5N + 58; it is going to start at a y-intercept height of 58, but it goes much less steeply down at -2/5.3159

So, it goes like that; and it also will allow for the part below it.3172

So, the total set of things that is allowed is this highlighted part in yellow.3177

However, the only things that are really going to be interesting are going to be our vertices--the vertex here, here, here, and here.3184

We will circle those in yellow, just to make them a little bit more obvious.3197

But it is probably becoming pretty hard to see through the loudness of those colors.3200

We have an idea of those corners.3203

Now, notice: (0,0) is one of the points--we have an origin corner.3205

Is it possible to make any money if we plug that into p =...0 for N and 0 for S, if we buy no cars to sell?3209

No, we can't make any money; so we only have three things to care about:3217

where red intersects green, where red intersects blue, and where blue intersects purple.3220

So, if that is the case, let's look at where red intersects green first.3227

That is -2/5N + 58 gives us N ≥ 0.3231

If that is the case, if it is N ≥ 0, that is a little bit difficult, because we are set up for solving for S.3241

So, where is that going to intersect? Well, at N ≥ 0, we want to figure out what that comes out to be.3248

If S is equal to -2/5 times 0 plus 58, if we know that this is going to be equal to S, and N is greater than or equal to 0,3259

we can treat it as N = 0, because we are only dealing with lines here.3269

So, we plug that in here; we have -2/5 times 0, plus 58, equals S; so 58 is S, and we have an N of 0.3272

So, the point that we get out of this is going to be N = 0, 58 = S, so (0,58) is one of our points so far.3285

Next, let's look at where the blue intersects the purple.3294

-N + 100 =...the purple had a height of S = 0, no height at all, so that is just going to be 0, so 100 = N.3299

That means we have (100,0), because if we were to plug that into N + S = 100,3311

since we are dealing with the lines now--if we plugged in that, we would have 100 + S = 100; therefore, S = 0.3321

So, we have (100,0); notice that these are the two extremes possible for the car lot to sell.3328

It could sell all Superfines--it could sell nothing but Superfines--and it has a maximum amount of money that it can spend, 2.9 million.3334

So, it can't buy its lot entirely full of Superfines; it could only have 58 Superfines on the lot before it would run out of money to purchase cars with.3342

On the other hand, it could buy 100 of the Nices; and it would still have money left over, but it has run out of spots on the lot.3351

So, it could buy 100 Nices and no Superfines, and have extra money left over;3358

or the lot could buy all Superfines and run out of money before it runs out of spots.3363

It would have 58 Superfines, but still, that gives us another 42 spots left for Nices.3370

So, those are the two extremes possible, so far.3375

Finally, we could also have the possibility of if we have the blue and the red intersect.3378

If we had -N + 100 equal to -2/5N + 58, we can multiply everything by 5, just to make it a little bit easier on us.3385

-5N + 500 = -2N + 58...we turn to a calculator to figure out what 58 times 5 is.3402

We get 290 out of that; we add 5n to both sides; we subtract 290 on both sides.3411

500 - 290 gets us 210; we add 5n to both sides; that gets us 3n.3423

That gets us...dividing by 3...I'm sorry; not 80, but 70, equals N.3429

We use that to figure out how many Superfines that would bring us.3436

By our lot equation of S ≤ -N + 100, we want to be on the line (equals), so S = -N + 100; so S = -70 + 100, or S = 30.3440

The three possibilities are (0,58), (100,0), and (70,30).3457

At this point, we just want to actually plug each one of them in and figure out which of these works best.3466

Let's start with the point that we have arbitrarily made the red point.3473

If we sell nothing but Superfines--we spend all of our money on Superfines--then that is 10 times 0, plus 15 times 58, just so we keep up the same pattern.3477

15 times 58 becomes 870; so we would make a total of 870 thousand dollars at the lot if we bought nothing but Superfines.3490

Next, we will do the one that we arbitrarily made the blue point: the profit out of that one would be 10(100) + 15(0), the number of Superfines.3501

So, the profit that we would make off the lot would be 1000 + 0.3512

So, if we sold nothing but Nices, we would be able to get 1000 thousand, or 1 million, off of our sales.3516

But what if we sold a combination of Nice and Superfine?3526

If we sold a combination, the one we arbitrarily made the green point, then our profit would be 10(70) + 15(30).3529

10 is the profit from the Nices; 15 is the profit from the Superfines; we are selling 70 Nices and 30 Superfines.3541

We know that it is possible, because it is on our vertex.3547

That gets us 700 + 15(30) is 300 + 150; 450; so this is a total profit of 1150 thousand from 70 and 30.3550

So, the best, the optimum solution, for maximum profit, is to sell 70 Nices and 30 Superfines.3567

And if you are curious, that came out to be a profit of 1150 thousand, so the total profit,3582

the maximum possible profit for the lot, will end up being 1150 million dollars.3588

All right, cool; that finishes for linear inequalities.3596

There is this idea of linear programming, and there is also just being able to figure out what the things that our linear inequality allows for are.3600

Linear programming is a great thing that we can use all of our information about linear inequalities for.3605

But you can also just work with linear inequalities.3609

All right, we will see you at later--goodbye!3611