In Exercises 1–26, graph each inequality. (x−2)^2+(y+1)^2

Question

In Exercises 1–26, graph each inequality. (x−2)^2+(y+1)^2<9

Accepted Answer

Hey everyone in this problem, we are asked to graph the following inequality given X plus five squared plus y minus four squared is less than 49. So let's just go to the bottom here and give ourselves some room to work this out. Okay, we're gonna rewrite the inequality. We were given x plus five squared plus y minus four squared is less than 49. Alright, so the first thing you want to do, we have an equation like this and we're asked to graph it or an inequality like this I should say, is to graph the equality. Okay, so we want to change that inequality to an equal sign graph that first and then that's going to divide the plane up into sections. Okay? And then we need to determine which of those sections are included in our inequality, in which are not included in our inequality. Alright, so what we want to do first is graph the equality like I said x plus five squared plus y minus four squared is equal to 49. Now you'll notice that this looks a bit like an equation for a circle. Okay, we expect that this is going to be a circle. So let's just write the general equation for a circle so that we can compare it with the equation. Were given the general equation of a circle, recall is going to be x minus H squared plus why minus k squared is equal to r squared. Okay, Where H k is going to be the center and r is going to be the radius. So we compare this to our equation, we see that we have x minus h. Y minus K. And then we have r squared. So we want to kind of rewrite our equation a little bit. So this first term we're going to rewrite as x minus negative five. Okay so that we have it in the form x minus something X minus negative five is x plus five. So we're not changing the equation And then we have Y -4 squared. And we want to write the right hand side in terms of something squared, we have 49 and we know that 49 is going to be seven squared. So now that we've written in this form we can compare and see that we have an h. Value of negative five we have a k. Value of positive four and we have an R. Value of seven. So this is going to be a circle of radius seven centered At the 0.-5 and four. Alright so let's draw that and go from there. So 12345. So this is gonna be negative five. And we need to draw positive four. So this is gonna be the center of our circle. Now it has a radius of seven. So we know that it's going to go from negative five in the X direction all the way up to positive two. In the X direction. Okay. In the Negative extraction from -5 to -12. And then in the Y direction it's gonna go up to positive 11. I'm gonna say that here. Or graph is going to be a little bit squished because we're out of space and then in the negative Y direction it's going to go to negative three. Ok, now when we look at our inequality we have less than 49. We don't have less than or equal to So we're not actually going to include this boundary. So we're going to draw it as a dotted line. So dotted line, something like this. Mhm. Okay. All right, so this is our circle. Imagine this is a perfect circle. Um not my poor drawing, but this is an idea of what our circle looks like. Okay, now we want the inequality where this circle is less than 49. So what that means is we want to take the interior of the circle. So the interior of the circle is what's going to be included in our inequality. No, if you I couldn't remember that or it just you couldn't figure out if you were supposed to be the inside of the circle or the outside of the circle, Okay, you can always choose a point. So you could choose the point, say negative two, negative two. Okay, if you choose the point negative two and negative two and you plug this into the equation, you're gonna have negative two plus five squared. Let's write this up. Point negative two. Negative two. We know that that's inside of the circle. I'm sorry, negative to positive two, negative to positive two. We know that that's inside our circle. We plug that into our equation. We're going to get negative two plus five squared plus 2 -4 Squared. On the left hand side. This is going to be equal to three squared. So we're gonna get nine -2 squared. So we're going to get four. This is equal to 13 which is less than 49. Okay so that point satisfies our inequality which means at that point is included. Okay, so that means that we include this middle part where that point is located. Alright, so let's go back up to our answer choices. This is what the graph of our inequality should look like. We go back up. Mhm. And we said that the interior of the circle should be included. So we already know we're looking at answer choice either B or D. And we found that the center of our circle should be at negative 54. So that matches with answer choice B. So we have answer B. The graph of our inequality. Thanks everyone for watching. I hope this video helped see you in the next one.