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Writing a System of Linear Inequalities

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Today, we're going to talk about writing systems of linear inequalities. Remember, a system of linear inequalities is just two or more inequalities that have the same variables and are considered at the same time. So we'll first do an example of writing a system of linear inequalities from a real world situation. And then, we'll do an example writing a system of inequalities from a graph.

So for my first example, I've got a graph of three linear inequalities. And I'm going to show how you can write a system of inequalities that is represented by this graph. So the first shaded region I'm going to look at is the one in red. I see is cut off by a vertical line through negative 3 on my x-axis.

And so, first, I see that it's a solid line, which means that our inequality symbol is either going to be less than or equal to, or greater than or equal to, because the solid online indicates that we include x values of negative 3 in our solution set. And then I see that, because it's shaded to the right of negative 3, which is values of x that are greater than negative 3, our inequality symbol is going to be greater than or equal to. So the inequality for the shaded region in red is going to be x, is greater than or equal to negative 3.

Let's look at the shaded region that's in green. So this is cut off by a horizontal line through 2 on my y-axis. Again, it's a solid line, so I know my inequality symbol is going to be either less than or equal to, or greater than or equal to. And I see that it is shaded below 2, which means that it's going to include values that are less than 2. So my inequality that represents the region shaded in green is going to be y is less than or equal to 2.

And my final shaded region is the one in blue. This is cut off by a line with a y-intercept of negative 5 and a slope of 3/2. So I first see that my line is, again, solid. So the inequality symbol is either going to be less than or equal to, or greater than or equal to.

And then I see that it's shaded below that line, so the inequality symbol is going to be less than or equal to. So I'm going to start with y is less than or equal to. Again, this inequality could be represented in slope intercept form with a slope of 3/2 and a y-intercept of negative 5.

So let's see how we can take a real world situation and develop a system of linear inequalities that represents that situation. So let's say that I'm planning a party. And I go to the store to buy bags of chips and cases of soda for the party.

I go to the store with $30. And I know, at the store, that bags of chips cost $3 each and cases of soda cost $4 each. I also know that I do not need more than three cases of soda. So let's see how we can write a system of inequalities to represent this situation.

So first, I'm going to define my two variables. I'm going to say that x is the number of bags of chips that I buy. And I'm going to let y be the number of cases of soda that I buy.

All right. So first, let's write an inequality to show that I don't need more than three cases of soda. So if y is the number of cases of soda that I buy, I know that y should not be more than 3. But it could be equal to 3 or less than 3, so that means that my first inequality will be y is less than or equal to 3.

Then let's write an inequality to show how much we can spend between our bags of chips and cases of soda, because we know that we only have $30 that we can spend. And so, if I look at my cost of chips, since they each cost $3, I'm going to multiply 3 by however many bags of chips that I buy. So the beginning of this inequality will start with 3 times x, which will tell me the total amount that I'm spending on my bags of chips

I'm going to add to that the amount of money that I'll be paying for my cases of soda. Since each case of soda costs $4, I'm going to multiply 4 by however many cases of soda I buy. So that's going to give me 4 times y, as y is the number of cases of soda. And again, this will tell me the total amount of money I'm spending on my cases of soda.

So between the amount of money I spend of my bags of chips and the amount of money I spend on my cases of soda, I only have $30 to spend, so these two amounts added together have to be less than or equal to 30. So these two inequalities represent a system of inequalities for my situation. They have the same variables or same variable. And they're going to be considered at the same time, because the variables are defined in the same way.

So go over our key points from today. As usual, make sure you have them in your notes, if you don't already, so you can refer to them later.

You can write a system of linear inequalities from a graph by identifying the equation for its boundary line and determining the inequality symbol needed. The symbols less than and less than or equal to indicate a shaded region below the line. And the symbols greater than and greater than or equal to indicate a shaded region above the line. The symbols greater than or equal to and less than or equal to indicate a solid boundary line. And the symbols greater than and less than indicate a dotted boundary line.

So I hope that these key points and examples helped you understand a little bit more about writing systems of inequalities. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks, for watching.