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Using Linear Inequalities in Real World Scenarios

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Today, we're going to talk about using linear inequalities in real-world scenarios. So we'll do some examples looking at how you can represent a real-world scenario both with an inequality, and then we'll look at how you could graph that inequality on a coordinate grid.

So let's see how we can take a real-world situation and represent it as an inequality, both algebraically and as a graph.

So let's suppose you want to rent a car. The cost to rent the car starts at $50 and then it costs an additional $10 for every day that you rent the car. So we're going to start by defining two variables.

I'm going to let y be equal to the cost, or budget, of renting the car. And I'm going to let x be equal to the number of days that I rent the car. So I'm going to start by coming up with an expression for the cost to rent the car.

So I know that the cost at starts at $50, and that it costs an additional $10 for every day that I rent the car. So I'm going to multiply 10 by the number of days that I rent the car, which is my variable x. So in addition to the $50, I'm going to have $10 times however many days that I rent the car.

And now I want the amount of money that I spend on the car to be less than the overall cost, or my budget, that I have to spend on the car. So I want the overall amount to spend on the car to be smaller than the cost or the budget for renting the car. So this situation can be represented by the inequality y is greater than or equal to 10x plus 50. So now let's see how to graph this inequality.

On my graph, I have days on my x, or horizontal, axis and I have cost in dollars on my y, or vertical, axis. I know that my inequality-- or actually, the equation y equals 10x plus 50 would be a line that's in slope-intercept form, so I can begin to graph this inequality by looking at the y-intercept and the slope. So I see that the y-intercept is 50 and the slope is 10, or 10/1.

On my graph, I would place a point at 50, which makes sense because at day 0, or when I first rent the car, I've only spend $50. Just the initial fee for renting the car.

Then I know the slope is 10, or 10/1. So from 50, I'm going to go up to 10 and over 1. This point also makes sense because after one day, I should have spent $60. The $50 initially plus an additional $10 for the one day that I rented the car. So now I can connect these two points to create a line.

And thinking about my inequality, I know that the inequality symbol greater than or equal to-- or less than or equal to is going to be a solid line. So I'm using a solid line on my graph. And I know that the inequality symbol greater than or equal to, or it could also be strictly greater than, is going to have a graph that is shaded above the line. So I'm going to shade above my line.

So the shaded region on my graph represents the points, the x and y-values that are going to satisfy my inequality. And for example, these points are the number of days that I can rent a car while staying within a specific budget, or underneath a specific cost.

So for example, let's say that I want to know if I can rent my car for 7 days if my budget is $100. I can use my graph to figure that out.

So if I go to 7 on my x-axis and up to 100 on my y-axis, I can see that this point is not within my shaded region. And because it's not within this shaded region, these x and y-values will not satisfy the inequality, which would mean that no, under this scenario I cannot rent my car for 7 days if my budget is only $100.

What if I'm wondering if I can rent the car for 2 days if my budget is $75? So I can use my graph to answer that question.

I'll find 2 on my x-axis and 75 on my y-axis and place a point. And I can see that this point is in my shaded region, which means that the values of x and y do satisfy the inequality. And that yes, I can rent the car for 2 days if my budget is $75.

So let's go over our key points from today. As usual, make sure you get them in your notes if you don't already so you can refer to them later. A linear inequality can be used for scenarios limited by certain constraints, such as a budget. Now, linear inequality can be graphed to model the scenario and observe the solution region.

So I hope that these key points and examples helped you understand a little bit more about using linear inequalities with real-world scenarios. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.