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Systems of linear inequalities in practice

Systems of linear inequalities in practice
10.6
Hello, and welcome back to the steps in practice. We’re dealing today with systems of linear inequalities. And in Exercise 1, we’ve got 50 euros and we can buy books whose price is 1.29 euro each or we can buy a DVD at the price of 2.10 each. It’s quite cheap, isn’t it?
46.8
Now if b is the number of books that we buy, and d is the number of DVDs then the cost, our total cost will be b times 1.29 plus 2.10 d. So our constraint is that the cost must be less than or equal than 50. So the constraint is 1.29 b plus 2.1 d less than or equal to 50. This is a linear inequality. Also, the number of books and DVDs must be positive. So at the same time, we must impose b greater than or equal to 0, d greater than or equal to 0. So this is a system of linear inequalities. And we shall denote by S the set of solutions of the system.
114.8
Now, in truth b and d have to be also integers. You cannot buy a square root of 2 number of books, for instance. So in our case also b and d must be natural numbers. Now let us draw the solutions of this system S. Well, when we have equality, we’ve got a line. The line 1.29 b plus 2.10d equal to 50, let us call it L. And we impose that 2.10 d is less than 50 minus 1.29 b. That is d less or equal than 50 over 2.10 minus 1.29 divided by 2.10 b.
183.5
So the points of our interest of the set S are the points that belong to the set of b and d positive, and that are below the line L. So let us draw the line L. We shall take the vertical axis as the axis of the number of DVDs, and the horizontal axis as the number of books b. Now in order to draw the line L, it is enough to find two points of the line. Well, let us find the intersection of the line L with the vertical and horizontal axis. If we want intersection with the vertical axis, we must take b equal to 0.
237.1
Now for b equal to 0 a point b,d belongs to the line whenever 2.10 d is equal to 50. So if d is equal to 50 divided by 2.10. This is a little bit less than 24.
259.3
So we can draw here, if this is 10 and this is 20, 24 will be more or less here. And that line will pass through the point here. At the same time, the intersection of the line L with the horizontal axis is when d is equal to 0. And if we take d equal to 0, we get 1.29 b equal to 50, which gives b equal to 50 divided by 1.29, which is a little bit less than 39.
300.7
So if this is 10, this is 20, this is 30, and this is 40, well, the point of the line L will be more or less here. So the line L is the line that we draw now. This is the line L. And we have to take the points that are below the line L and at the same time, with b greater than 0. So on this side. And d greater than or equal to 0. So above the horizontal line. So this is the set S. And also we have to take into account that b and d are natural numbers. So actually we’re not taking all the points of this set, just the points with integer coordinates.
363.2
So for instance, 10, 10 and every point with d equal to 10 and b being an integer from 0 to 10. And so on. So we’ve got all the points with integer coordinates that belong to the set S are the solutions to our problem.
390.4
So this is the last video of our course. In the next step, you will find some revision exercises as usual, in the form of quizzes. We want to invite you to do them. We are sure that you will learn a lot of things. And we also did– not only how to clean a window like that one, which was a quite difficult task. If you wish, in our next MOOC, we deal with some more advanced topics like exponential functions, and logarithms, and trigonometric functions also.
446.6
In this setting, we consider all the stuff that you saw here– equalities, inequalities, between these functions.
459.2
And then also, some plane and solid geometry.
470.3
So we wish you good luck for your studies. And we really hope to see you soon in the next MOOC. Bye.
The following exercises are solved in this step. We invite you to try to solve them before watching the video.
In any case, you will find below a PDF file with the solutions.

Exercise 1.

Francis receives a gift card for the amount of 50 euros from an online store, that allows him to buy books at \(1.29\) euros each, or DVDs at \(2.10\) euros each. Set up a system of linear inequalities that represents the situation, and identify the range of possible purchases by means of a graph.

Discussion

In the comments section below, share a real-life example that you can model as a system of linear inequalities.
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Precalculus: the Mathematics of Numbers, Functions and Equations

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