In Exercises 1–26, graph each inequality. x^2+y^225

Question

In Exercises 1–26, graph each inequality. x^2+y^2>25

Accepted Answer

Hey everyone in this problem. We are asked to graph the following inequality and were given x squared plus y squared is less than 81. Let's move down to the bottom and give ourselves some room to work this out. We're just gonna rewrite what we were given, X squared plus y squared is less than 81. Now we're given an inequality like this and ask to graph it. The first thing we want to graph is the equality that's going to set some sort of boundary that's going to divide the plane up. And once we have that boundary we can decide, decide which part of that plane is going to be included in our inequality. So that said we wanna graph x squared plus y squared equals 81 1st. Now this looks like a standard circle. The equation and recall that the circle general equation for a circle can be given by x squared minus H squared plus y minus k squared is equal to r squared. Where h and k are going to be the center of our circle. The X and y coordinate of our center of a circle and r is the radius. So in this case we have H and k equal to zero. Okay, So we have H equals zero and K equals zero because we just have x squared and y squared. Now the right hand side we can write as nine squared OK. Nine squared gives us 81. So we can write this as nine squared, which tells us that the radius R of our circle is going to be nine. So we know that the boundary is going to be a circle centered at the origin with radius nine. So let's start by drawing that. Okay, so we're just gonna draw a tick that's going to represent nine all the way around 9. 9, -9 and -9 are x and Y axes. And then our circle is gonna be centered at zero. Now the question is do we include the boundary with a solid line or do we not include the boundary? In which case we draw it as a dotted line? If we look at our inequality we have X squared plus Y squared is less than 81. Okay, strictly less than 81. Not less than or equal to 81. So we don't want to include the boundary, which means we're going to draw it as a dotted line. So we draw this as a dotted line like so and this is going to be the boundary. This is our circle looks a little bit like an oval, but imagine this is a nice perfect circle. All right. Now, the question, do we include what's outside of this boundary or do we include what's inside of this boundary? Well, the problem is giving us x squared plus y squared is less than 81. Okay, so we want the circles that are less than this one. So we want everything inside. Mhm. All right. Now, if you couldn't remember or you're having a hard time figuring out which part to include. What you can do is you can also test a particular point. Okay? So you can test the point say 00. If we take the 00.0 then X squared plus Y squared is equal to zero which is less than 81. So that point satisfies the inequality K zero is less than 81. Which means that that points included. So we include 00. Which means that we're including inside of that boundary of that circle that we drew. Alright, so this is what our inequality the graph of our inequality looks like. Let's go back up to the top and choose one of these answers. Now all of these have circles centered at the origin of Radius nine. What we found is that our circle should have a dotted or dashed boundary because we're not including the boundary case. We're looking at option A or B. And we found that we want to include what's inside of the circle. So we're looking at option A. So the answer here is going to be a Where we have that circle centered at the origin of Radius nine. We don't include the boundary but we include everything else inside of that circle. Thanks everyone for watching. I hope this video helped see you in the next one

In Exercises 1–26, graph each inequality. x^2+y^2>25

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