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

Question

Accepted Answer

Hey everyone welcome back in this problem. We are asked to graph the following inequality and were given x squared plus y squared is greater than or equal to nine. Let's move down to the bottom of the page and just give ourselves some more room to work. We're gonna rewrite what we were given X squared plus y squared is greater than or equal to nine. Now, when we're given the inequality like this and we're asked to graph it, the first thing we want to do is we want to consider the equality. Okay, so we want to consider X squared plus y squared equals nine. And this is going to define the boundary of our inequality. It's gonna split the plane up into different pieces and once we have that boundary then we can figure out which part of the plane is going to be included in our inequality, in which one isn't. Alright, so our equality X squared plus y squared is equal to nine. Okay, now recall that the general equation of a circle is x minus h squared plus y minus k squared is equal to r squared. Okay, We have x squared plus y squared equals something. So we know that this is the equation of a circle. Now, H and K are both zero in this problem, which means our circle is going to be centered at the origin. Now let's just break the right hand side so that it's something squared. Okay, so X squared plus y squared is equal to Well we know that nine is going to be equal to three squared. So x squared plus y squared is equal to three squared. This tells us that the radius of our circle is going to be three. Okay, so we have our equals three. So we have a circle of radius three centered at the origin. So let's start by drawing that. So our X axis or y axis and I've just drawn a kind of three in each direction. Now, first things first when we're drawing this boundary, do we include the boundary? In which case we draw it as a solid line or are we not including the boundary? In which case we draw it as a dot or dash line? In this case we have X squared plus Y squared is greater than or equal to nine. Okay, so we have that equals +29. So we include the boundary. So we're going to draw it as a solid line. So we draw this circle. Okay, this is our equality. This is the boundary of our inequality and it has a radius three. So this is 33 negative three and negative three. And it's centered at the origin. Now the question is, what do we include? Do we include inside of the circle or do we include outside of the circle? Now we want x squared plus y squared that circle to be greater than or equal tonight. Okay, bigger than we want the circle to be bigger than. So we want the outside of this circle. Everything around it on the outside. And we want circles that are bigger. Now, if you have having trouble figuring out do we take the outside or the inside? You can always choose a point. Okay. So let's say we chose the 0.0.0 inside of the circle. Okay, if we take the 0.0 and we substitute this into our inequality, we're going to have zero squared plus zero squared, which gives us zero, which is less than nine. Okay, It is not greater than or equal to +90. Greater than or equal to nine. It is not That means that it doesn't satisfy our inequality. So that point isn't included. So we know that the zero isn't included. Which means that inside of the circle isn't included. And you could do the same with the point on the exterior and you would find that it is included. Okay. So you can always test points to figure out whether we're including the outside of the circle or the inside of the circle. Alright, so let's go back up to our answer traces now that we've drawn our inequality. Now, the first thing we found was that we had a circle centered at the origin of radius three. Ok, so we can already eliminate A and B. Because the circle is a radius nine there. So we're looking at options. Crd Now we found that we should be including the exterior of the circle, not the interior of the circle. And so we have option D. For the graph of our inequality. 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≤1

Key Concepts

Inequalities

Graphing Circles

Shading Regions

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