 This video will talk about nonlinear inequalities and their inequalities, so the solution to a nonlinear inequality is a shaded region. We're going to graph these to find out what happens. Just like we've done with other inequalities that weren't linear, we're going to have boundaries and test points again. So when we're graphing these, what we're trying to do is determine the boundaries, and that'll usually be the graph of the function that we have. And then we have to test points, and those test points will either be like if it's a circle inside or outside the circle, or if it's a parabola inside the parabola or outside the parabola. So again, it says the functions will become the boundaries, and we need to test points on either side. Alright, we have the system of inequalities here, and we want to do them by hand and graphically, because sometimes a calculator dies, so we want to make sure we can do it both ways. So I'm going to solve this one for y, and I'm just going to take the four to the other side. It doesn't change my inequality because I didn't divide. So y is greater than x squared minus four. Well, I know that to be a quadratic. Parent family is the parabola, but this says minus four, which means I'm going to go down four on that graph, and I will have my vertex here, and my graph will look something like this. I also know that the zeros of this thing are my x-intercepts, so mine doesn't look very symmetrical, but this point here is two, and this point here is negative two. So I want to test this point right here. Zero, zero is inside. So I'm going to test zero, zero, and that tells me that zero is it greater than zero squared minus four. And it would tell me that zero is greater than negative four, and zero is greater than negative four. There's one thing I had to do before I can shade, but I do know this is true. So this one works, but you'll notice my inequality is less than. It's not less than or equal to, so I need to come in here and get my little eraser, and I need to make this a dotted graph. The boundary, it is the parabola, but I can't include the parabola. So I'm going to graph and shade everything that's in between inside that parabola, but just not including the parabola. So I'm done with that first inequality, and now I have to go to the second one. And this one I can see is a circle. For now I don't have to solve it for why. If I want to check it graphically in my calculator I will, but for now I don't. I just know that it has a center that is at zero zero, and it has a radius of, remember it's the square root of this. So it's the square root of 34. The square root of 25 is 5, and the square root of 36 is 6. So we know it's approximately 5.7. It actually ends up being 5.8, so almost 6. So if I'm going to put my circle in here, there's my center, zero zero. So I'm going to go out 5.8 over here, and that'll be that part of my circle. And then I'm going to go over 5.8 over here. That's a negative 5.8 over here, and a positive 5.8 over here, and I'm going to go up. Looks like maybe 5.8 would be right here. It's just a sketch, but this time I do have a solid circle. It might always look like eggs. And I didn't put my negative part down here. So I'm going to check again, zero zero. I'm going to have zero squared plus zero squared is that less than or equal to 34. And it is. So that means I want to include that zero zero. So I want to graph everything inside this circle. So my final solution is everywhere that I have blue and purple, I will be able to have a final graph of that all colored in black. And that's everything inside the circle and inside my parabola, but not including my parabola, although I can include the circle. So I can include that circle arc and everything in between, but just not quite to my parabola. It's the dotted line, so I'm going to show you just shy inside it so you realize that it's not including that parabola. And if I do it graphically, there's a couple of things you need to know. What I'm really going to put in here is y is equal to x squared minus 4. And then over here, but then if you look to the left of my y equal, I can arrow over there. And I see the dots right there that are blinking. I want to make them look something like this. When I have greater than, I want a triangle that looks like it's in the top right hand corner. So I just press enter until I see that triangle that looks like it's in the top right hand corner. And I just passed it because my calculator slows. So let me try again. They're where normal. So one, two, and it looks like I'm there. Now I'm ready to go down and look at my second equation. And this one is going to have to be a y equal equation, but it's going to be less than or equal to. So I want to shade in the bottom. I'm just going to come down. And so I'm still on those dotted line. And I'm going to press enter one, two, and then three gets me to the bottom triangle. And now we need to go and solve this equation, which is kind of interesting. I know it's going to be y equal and we'll figure out what it's equal to. So we want this to be x squared plus y squared is less than or equal to 34. So let's say that that's y squared less than or equal to 34 minus x squared. Well, it doesn't say y squared. I can't make it y squared. That's a sub two, not a squared. So I have to take the square root of both sides. So the square root of y squared and the square root of this side. And this gives me y less than or equal to plus or minus the square root of 34 minus x squared. Well, I really don't have to put both of those in my calculator. And the second one gets kind of tricky with inequalities. I really just need to do one part of my circle. And then I can see whether it's going to sketch inside my circle or outside my circle. Because those are the only two options. So I'm going to come in here and say that my y is just the square root of 34 minus x squared. And come over here to my calculator and say second x squared and 34 minus x squared. Close my parentheses. So it should work in a standard window. So I just hit graph. And here's my graph. I've already had it graphed. But it's shaded inside my parabola, just like we thought. And it's shaded inside my circle. If I could, I would draw the rest of my circle down here. And I can see it does go below, but that's not considering the bottom of my circle. So it's only inside my circle and my parabola and double hashing here where it looks like a checkerboard is just my inside circle. It's inside my parabola, also inside the circle.