In Exercises 46–55, graph the solution set of each system of ineq...

Question

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.

This

is a piecewise function. Refer to the textbook.

Accepted Answer

everyone in this problem? We are asked to consider the given system of inequalities and indicate if the inequality has a solution by graphing it. Okay. The inequalities were given here Are X plus Y bigger than or equal to one and x minus Y less than or equal to one. Ok, so let's go down to the bottom to give ourselves some space to work. Alright. We're just gonna rewrite the inequalities we were given. So we were given X plus Y bigger than or equal to one an x minus Y. Less than or equal to one. Okay. So we want to go ahead and graph these, you know the easiest way to graph these is start by graphing the equality and we know how to graph lines when we have Y equals something. Those are easier lines. First a graph or we know how to do that. Okay, so let's just write these in that format so that it's what we're used to. So this one is gonna be why bigger than or equal to 1 -1. Okay, and in this one we're gonna have to move the y to the right hand side. Okay, So we're gonna end up with y. And again bigger than or equal to because y is going to be on the right hand side. So we get that sign and then we get X -1. Alright, so let's start by graphing the lines. We'll draw a graph or a little plot here. We have X. We have Y. Then we'll just 1234. Alright, so the first one? We're going to graph Is this why greater than or equal to one minus X. Ok, let's start by graphing. Why is equal to one minus X. Okay. So if we have Y is equal to one minus X. When why is zero for the X intercept when y is zero? We get one minus X and we get X is equal to one. So we have the 10.10. And then we have the Y intercept when X is zero and we get y is equal to one. Yes, we have the 10.1. So we're gonna have these two points. Okay so zero and 1 and 10. We have a negative slope. And so this and let's draw this. Maybe in blue. The wine is gonna look something like this. Okay, this is a line. Y is equal to 1 -6. And now we need to figure out what side of this line are inequality contains in this case we have why? Greater than or equal to this line. Okay. If we want why bigger than we want to be above the line. Okay, so we're going to consider everything here, above the line. Okay now if you're a little confused there and you just want to double check that you're on the right side of the line. Okay? What you can do is choose a point. Okay so say we chose this 0.0.0. Okay if we choose 00, we're going to have zero bigger than or equal to one. That's false zero is not bigger than or equal to one. Which means that that 10.0 is not included. Okay? So that means that we're on the other side of the line. So if you ever get confused, you can always choose a point on either side. Test it out and that's going to tell you whether or not you're in the right spot. All right, let's move on here to the other inequality Why bigger than or equal to x minus one. Okay. And again, we're going to start we're going to do it over here with the equality Y is equal to x minus one and we get an X intercept when y is zero of x equals one. Again. So we have 2.10. Okay, So this point here is common and then we have a Y intercept when x is zero of y equals negative one. So we have the 10.0 negative one. So drawing this in red with this .10, we have the 0 -1. We have a positive slope. So our line is going to look something like this. Okay, this is the line y equals X -1. And now we need to figure out what side of this line our inequality contains And again we have why bigger than or equal to x minus one. So we want to be above that line. So our inequality is going to contain everything on this half of the plane. Okay, and again, if you want to check a point, you can check 00, We'll get zero bigger than or equal to negative one, which is true. And so we know that the part of the plane that contains 00 is included. Now if we want to figure out if the system has a solution. Okay, well the system is going to have a solution where these two overlap. Okay. Both of these have to be true for the system to have a solution. And we see that in this triangle here we get both red and blue. That means that this is a solution for both inequalities. And so this triangle here that continues up into infinity is going to be the solution to that inequality. Okay, alright, so let's go up and look at our possible answer choices. And what we see here is that this bottom triangle is empty. So if we compare this to the answer choices. Okay, we see that we have D. That bottom triangle is empty and we have the overlap in that upper triangle. That's it for this one. Thanks everyone for watching. See you in the next video

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.
This
is a piecewise function. Refer to the textbook.

Key Concepts

Systems of Inequalities

Graphing Techniques

Piecewise Functions

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