Graph each system of inequalities.$y>x^{2}-4$$y<-x^{2}+3$...

Solve each system of inequalities by graphing.
$$
\begin{array}{l}{x \geq 2} \\ {y > 3}\end{array}
$$

This problem wants to solve this system of inequalities by graphing. So first we can adjust the inequalities a bit, so they're easier to graph. So for the blue one, if you take squared of both sides, then you get why minus three is greater than or equal to squared of X plus two. Or why minus three is less than or equal to negative square root of experts to because of pleasure mine. It's when we take the square root. Then for each, you can add three to both sides. She then gives us that why is greater than 23 plus root X plus two? Or why is less than equal to three minus screwed of experts to now for the green equation, you can just subtract for my stick. We can subtract four on both sides. That gives us X squared. Minus four is less than or equal to why and flipping this round. We get that why is greater than two X squared? Minus four now, before we graph, it's good to find the in terms most important intercept. So for the green equation, this is going to be the X intercepts. So when y is equal to zero. We're gonna just makes inequality, I think easier X squared minus four out of four on both sides. And we find that X squared is he to four. Take the square root and we get that X is equal to plus or minus two. Those air the intrinsic acts intersects for the blue equation. Since it it is a radical, we want to find the Why intercepts instead. In order to do this, it's easier to just use the original equation and plug in Y equals zero. We're sorry to plug in X equals zero and then we do this by why minus three squared is equal to zero plus two. Take the square root on both sides. Why minus three is equal to root. Two plus Remind us, of course, and then add three. And the why intercepts are three plus or minus squared. So now we are ready to graphs Want to draw my graft out here? We know that we're gonna start with the green equation because that's a little bit easier. We know that a quadratic generally looks like so, but this one has transformation, so gets moved down by four, gets transformed downward by four. So instead, it is going to look like this with the intercepts at two. And negative too. And I want to take away this original graph. Then for the blue equation. We know that why equal to broad ical X squared of X plus or minus looks something like this. And this one also the transformation, as you can see, because we have the plus two inside the radical and plus three on the outside. That means that we're gonna move this up by three and to the left by two. And the new graph is gonna look like this race This one with the intercept at three plus square to to and three minds to swear to Jim, this is not exactly drawn to schedule. But so then we want to see where shade So first for the green one. If we take a test, your point at the origin, which is always an easy one to dio. And we plug in these values into this equation then So it's clicking 00 to the 1st 1 Then zero square. It's less than or equal to zero plus four zero is less than equal to four that is correct. Which means that this here towards the origin is Theo area for the green of the inequality that the inequality equals. Show no for the blue. We're going to do the same thing we do. Tester point at the origin. If you plug this in zero minus three squared is greater than equal to zero plus two. It's becomes a negative. Three squared is greater than or equal to two. The native three sort of the 19 nineties Great of you to to that is also correct. So to shape and for the blue equation, it's gonna be on the outside here, and that is going to be our solution ist overlapping part. I'm gonna just out one and read this whole area and then extending upwards is the solution to the inequality.

Already 37. It looks like we have crapola a little and air inequality so that just do. But Rambla, which is why is greater than or equal to X squared minus four, then our linear one iss. Why, it's less than or equal. Teoh ex wants to institute linear first. So this is in standard form. That makes it a bit easier. I have a Y intercept of negative too in the slope of one. So zero negative too. Look up. One and over one. There you go. Easier to get slept that way one. So I'm going to draw my line that goes through is a solid line Because of the organic or two I'm going to check with 00 so zero less than equal to see where I'm honest too. Y zero sum Ciro Miss less than Nicole tonight to that is not true. So I do not want to shade 00 out. I want to shade the other side and 00 So shade that in just gonna be all of this area in here. Gonna be this half plane, all right? And now is did with our little rat. Look. See what he has to say for himself. Now the problem is gonna have a Vertex and zero negative four. And I'm gonna find the intercepts of it by doing that To do that under the move by for over and I have expired and then I'm gonna square root. Besides, so I have my intercepts. Or 20 in negative to zero, which is here and here. And my vertex 01 Teoh Negative four just down here. And think you might be able to draw a gravel that will fit into that using that little tool Start with new removing you make you a little bit wider. Yeah, Quit it a little bit wider steel area. And now we want to see Is that going to be shaded in or out? So I'm gonna check 00 again. So zero greater than equal to zero squared minus four zero is greater than equal to negative force. I'm going to shade inside of the parappa, which means it'll get my right thing here area I'm gonna shade inside Parappa It would go up and up forever. And so I'm looking for the intersection, which is gonna be this part right here in the middle? No, but you can see that yellow begin to blue. You see, He What color can you see? What about the orange? All those colors together make Katelyn ground. And so on your stations Were they intersect? There you

Already 38. Well, that has a family shape. Then it she was one of those yet get creative. We have y squared Last thing are equal to texts, and we have Why? Greater than or equal to s. Okay. It was like we're gonna do some tables and value your if I put one in for X, I get one in for Why? Yeah, I thought that it to indirect. So I get to infer why. So it's like we just have a line that's going up that way. Don't take them on their right line, Gloria. So we're gonna go across there. I'm gonna make it look green because I like things to match color wise. Did Nereo and we still wanna check who can't check with 00 So I'm gonna check with negative 11 So it wise one x is negative one that makes the true statement. So I'm going to shade that half playing rain for that linear inequality. And they were to go back to the 2nd 1 and let's see if I put it be easier Capital One. And for why I get a one in for X. It happened it to infer why I get a four for X if I put a negative tune in for X. What? Why I get a four for X. I've got 11 for 2123412 Negative for two clips. 1234 Sorry, not for negative. It's for negative to breath in the wrong one under for negative 212 11 I've been to in for why they get negative to fourth and I get negative. Something's hinky. Should be cement. Er, what am I doing? One square, it's one two squared is four. No is for for two. Negative, too. Let's do another quite and see what's going on. Oh, that's what it is consistent with critics up 00 in. Yeah, So if I put negative one in for why they get it one forex. Okay, so I had my Vertex in their own place, and that's just on a good thing. When you put your Vertex in her own place, this messes everything up, just telling you you want to vertex in the right place. So I'm gonna put this. We'll travel a Remember, this product was a little bit weird because it is going the other direction because it's the wind that squared. I don't know if I can shrink it are. Yeah, just a little bit more. Every guest was gonna go out that way and then let's check your shading and see and I can't check with 00 goes, it goes through 00 Can't check with 11 cause guster 110 no. I'm gonna check with this one. I'm gonna check with negative one one. So one square is less than or equal to negative one until you know. So I'm not gonna shake. There have been a shade over here, so I'm going to shade this way and it's going to go out that way. So, you know, it looks to lie. Oh, my See, that's double shaded. It said a department. See that any baby foot take that stick. Only that shaded. I think it would be easier since it's such a a a TV part to do this because there's a little bit of across the air. So is that really, really big? Right? There is the only participation that

Already. 39. Another circle. Okay, so we have X square. Plus y squared is less than or equal. The 16 is our circle. And then we have why righter than or equal to X squared minus four. Is our problem started? Problems aren't pink. Okay, so the vertex of my probably zero negative four. If I move the four over, I have four equals. X squared square written sides. You get X. It's plus or minus two somewhat. Intercepts are 20 and negative. 2001234 and 20 and negative to zero. Let's see if we can. It's weird. Let's see if we can draw our rocket attack. She Kanda. See, there we go. So Zach grapple that it's pretty close. So let's go ahead and, uh, make it pink. So it met is its points because, like things to match, There we go. Now let's decide the shading zero. Is it greater than or equal to zero squared months for and zero iss greater than negative. Four plots were shading inside the problem. Look, eighth graders like to call it a parabola. That was funny. The 1st 5000 times I heard it. And after a while, it's just annoying. You really run? When your math teacher called a parabola doctor told you to do that, Okay. And then I lost my green Only a second. Now this circle has a center of 00 at a radius of four men. But we get that by doing this square root this constant. So I've got 00 1234 1231234 1234 123 four In the safe metal circles working for me Yeah, did that with your Pennsylvania And then must decide our shading on this little puppy Nick Green and then our shading cf 00 count. Since you're all square CRR square, is it less than or equal to 16? Until you? Yes. So we're shade in there because 00 made a truce. Technip. So I'm gonna shade inside. I guess I could have just told it to feel like, but then I couldn't see that when it feels that it really feels and you can't see anything. So it looks to me like our, uh, solution is gonna deep from here down here and up to there. It's gonna be that piece of pie right there. It's Gunflint