Welcome back, everyone. So just in case you were struggling with this problem, I'm here to help. So we're going to graph this inequality, x2+y-12≤9. So let's get started here with our steps. First, we're going to have to figure out whether you're dealing with a solid or a dashed line, and we do this by looking at the equation or the inequality symbol. So remember, if there's a bar underneath the symbol, that means it's going to be a solid line. That's exactly what we have here. So we're going to have a solid line, not dashed, and we're going to have to graph this by switching the inequality symbol with an equal sign. So we're basically just going to have to graph the inequality x2+y-12=9. All right?
So what does this look like? You may have been trying to rearrange this and trying to solve for a while, and you'll get some weird square roots. But, actually, if you remember here that we've dealt with these kinds of formulas before, this basically is kind of like the equation x-h2+y-k2= r2. If you remember this, this is the equation for circles. So circles over here have, like, an x2 and y2 equal to something on the right side, and these numbers h and k just represent the vertex. So what this looks like, this equation here, is it's a circle, and the center is going to be at the coordinates that are inside of the parentheses. So here, if we have no numbers inside of the parentheses, that means it's 0. So center is at the coordinate 0 comma, and then this number over here gives us our k value. And remember, when we see a minus sign, that means the k is actually positive. So this is a circle with a center at 0 comma 1. All right? So that's what this formula is—the circle is going to look like. So the circle center is going to be here at 0 comma 1. But remember, we actually need the radius of the circle, and that's given to us by this information over here, the number that's on the right side. So clearly, we can see that this r2 is equal to 9. And so the radius r is equal to the square root of 9, which is just equal to 3. So remember, what we're going to have to do is we're going to have to go up, down, and to the right and left by 3 units. We're going to have to connect all of those points over here. Alright? So this is going to be, 1, 2, 3, and we're going to have to graph a circle that connects all of these points together. All right? And once I do this, it's going to look something like this. All right. So that's sort of like my crude circle there. So remember, we're going to use a solid line for this, but this is basically what our graph is going to look like.
Now let's take a look at the second step over here, which is we're going to have to test a point. Because even if you got this far, you may not have noticed where to shade this graph. Do we always go below the circle? Do we go above it? What's going on here? So let's go ahead and test the points, that's on the x or y axis. All right? So what I'm going to do is I'm going to test the points. I'm just going to test this point over here, which is the point 2, 0. All right? So let's test this out. So what about 2, 0? So what this is saying here is that you're going to have to do 2 squared plus, and then you're going to have to do 0 minus 1 squared is less than or equal to the number 9. All right. So let's actually just move this over here and test this out. So this is going to be 4 plus then we've got 0 minus 1, which is negative 1, squared, which is positive 1. 4 plus 1 is less than or equal to 9. So in other words, is 5 less than or equal to 9? That is actually a true statement. So 5 is less than or equal to 9. So that means we're going to have to shade the point that includes that point, within the graph. So but does that mean that we actually have to graph everything that's below the circle? Well, actually, no. And the way you can sort of test this out is you can sort of test out now this point over here that is still below the circle but outside of it. And what you actually see here is that if you test this point over here, you'll see that the inequality actually fails. And so what happens here is we actually have to shade the area that's actually within the circle. So these types of inequalities are a little bit trickier because you can't just shade everything that's below the circle or the center of the circle. You actually have to test all these points, which you'll see is that only the points within the circle will actually make the inequality true. All right? So that's how to graph these types of inequalities with equations of circles. Let me know if you have any questions, and thanks for watching.