Hi, and welcome. This is Anthony Varela. And in this tutorial, we are going to write a system of linear inequalities. We're going to do this in two ways. First, we're going to look at a graph. So we're going to write the lines that we see as equations. And then we'll figure out which inequality symbol to use based on the solution region that we see. In our second example, we're going to be given a scenario. So we need to define variables and then construct a series of inequality statements.
So let's get to our graph example first. So here, I see a couple of different lines that make up the system of linear inequalities. And the first thing that I want to do is sort of ignore the fact that there are inequalities right now and just write these lines as equations. So let's focus on this blue line first.
To write the equation of this line, well, I notice that the y-intercept is at the origin. And to go from one point to the next, it looks like I have a rise of 2 and a run of 1, rise of 2 run of 1. So my slope is a positive 2. So I can write this line then as y equals 2x, slope of 2, and a y-intercept at y equals 0.
Now let's focus on this red line. Well, I notice that I see a negative slope. I have a y-intercept occurring at y equals 2. And it looks like my slope has a rise of negative 1 and a run of 1, a rise of negative 2 and a run of 1, rise of negative 2, run of 1.
So the slope is negative 2. And I have a y-intercept of positive 2. So this is y equals negative 2x plus 2.
My other line here is a vertical line. And this is at x equals 2. And that's the equation for vertical lines, x equals some value. So this is x equals 2. So now that I have these written as equations, let's take a look at how we can turn them into inequalities.
And the first thing I'm going to look at is my dashed lines and my solid lines. Now if I have dashed lines, this represents my strict inequalities, less than or greater than. We're not including equal to.
My solid lines are what we call the non-strict inequality symbols, less than or equal to or greater than or equal to. We're including equal to. So taking a look at the blue line, so we know that this is y equals 2x as an inequality.
Take a look at the solution region. Well, it highlights everything that's underneath this line. So this is going to be y is less than or equal to 2x. Doing less than or equal to because the solution region shows everything underneath the line. And this is a solid line, so we're going to be including equal to.
For the red line, we see that this is a dashed line. So we're going to be using either less than or greater than. And taking a look at the solution region, this is above the line. So we're going to change our equal sign into a greater than sign.
And for our last line here, x equals 2, we notice that it is a dashed line. So we're going to be using a strict inequality symbol. And taking a look at the solution region, it highlights x values that are greater than 2. So we're going to write this then as x is greater than 2. So that is our system of inequalities that represents this graph here.
So now let's move on to a scenario. And our scenario involves printing posters to advertise an event. So an organization puts together a black and white poster and a color poster. And to print these off, it is $1.25 for every black and white poster. And it's $2.75 for the color poster.
And they have a couple of budget restraints. So they can only spend $250 on both black and white and color posters. And they want to have at least 40 black and white posters around. And they want to have, at most, 50 because they don't want to spend too much of their budget on the color copies.
So we're going to write a system of inequalities to represent this budget and quantity restraint. So the first thing that we want to do is define our variables. So I'm going to define our variables as x and y. And x is going to be the number of black and white posters that we're going to purchase. And y is going to represent the number of color posters that we're going to purchase.
So now we can use the variables x and y to create an expression for the cost. So the expression for the cost would be or 1.25 times x-- that represents the total cost of our black and white posters-- plus 2.75 y. This would be the total cost of all of the color posters.
Now we need to turn this into an inequality keeping our budget in mind. So a budget means that you can spend up to that amount, but you can't go over at least you don't want to go over. So this then means that we're going to be using the less than or equal to inequality symbol.
So this tells us that our cost can certainly be less than $250 It can be exactly $250 but we don't want it to be over $250 So that's why we're using less than or equal to. So that is our inequality that represents this budget constraint.
We also have two other constraints that deals with the quantities. We want at least 40 in black and white posters. And we want, at most, 50 color posters. So thinking about at least and at most, at least means we're using the greater than or equal to inequality symbol because this will then put it at a minimum of value.
It could certainly be over. It can be equal to, but it shouldn't be under. And at most, is kind of like our budget, right? It's allowed to be underneath. It's allowed to be exactly equal to, but it cannot be over. So it's at most a certain value.
So we're going to be using these two inequality symbols for our quantity restraints. So to show that we need at least 40 in black and white posters, x is the number of black and white posters. So we're going to say that x is greater than or equal to 40. And we'll say then that y is less than or equal to 50.
So here is our budget constraints that the total price of our black and white and color posters has to be less than or equal to $250. The number of black and white copies should be at least 40. And the number of color posters that we print should be, at most, 50.
So let's review writing a system of linear inequalities. On a graph, dashed lines represent those strict inequality symbols less than or greater than. Solid lines represent those non-m strict inequality symbols that include equal to. So here then would be how to shade a half plane on a graph. This would be y is greater than or equal to your linear inequality there. And then this would be below the line, so less than.
When looking at vertical lines, shading to the left would be less than, and shading to the right would be greater than. And when you're working from a situation, you want to define your variables. And when you see things like at least, that corresponds to greater than or equal to. And at most corresponds to less than or equal to. So thanks for watching this tutorial on writing a system of linear inequalities. Hope to see you next time.