Inequalities common core algebra 1 homework answer key

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Hello and welcome to another common core algebra one lesson. My name is Kirk weiler, and today we're going to be doing unit four lesson number 11, graphs of linear inequalities. As a reminder, you can find the worksheet and a homework set that go along with this lesson by clicking on the video's description. Or by visiting our website at WWW dot E math instruction dot com. As well, don't forget about the QR codes at the top of each page. Use your cell phone or a smart tablet to scan that code and bring you right to this video. All right, let's begin. So we've spent quite a bit of time talking about how to plot equations. Today, we're going to be talking about graphing or plotting inequalities. And yet, the truth about graphing is the truth about graphing. To graph an inequality simply means to plot or shade all XY pairs that make the inequality true. All right? And not to plot any that make it false. Okay? So let's jump right into an exercise that will introduce us to how to graph inequalities. Let's consider the inequality Y is greater than X plus three. Letter asks us to determine whether each of the following points lies in the solution set and thus the graph of the given inequality. Let's do the first one together. All right, it's pretty simple, right? We've talked about this before. I'm going to put the 7 in for Y I'm going to put the two in for X, right? I'm going to get 7 is greater than 5. And that's true. And because that's a true statement, this does end up being in the graph. Okay? What I'd like you to do is determine whether the other two points lie in the graph of this inequality. All right, let's go through the solutions. For the .01, I put one in for Y, I put zero in for X and I get one is greater than three. That is clearly false. And so the answer is no. Put it up here, where I have a little more space. No, that point doesn't lie in it. Finally, let's do the .1 comma four. I'll put four in here. Greater than one plus three. And that will give me four is greater than four. And that is also no. All right. Letter B says, graph the line Y equals X plus three on the grid below in dashed form. So let me just write this up here really quick. Y equals X plus three. What I want you to do is compare this and this. They're nearly the same, right? With one big difference, this is Y is greater than, this is Y is equal to. All right, I'm going to go back to blue. So let's graph Y equals X plus three. We're going to do that quickly, right? The Y intercept is three. And the slope is one, right? So my Y intercept is three, and then I'll go up one to the right one. Up one to the right one. Up one to the right one. Et cetera. Go down one into the left one as well. But we should be able to graph those points very quickly. Just checks and then really quick. Now it's kind of hoping I'd have a dashed line option there, but I don't. We're going to talk in a moment about why we dash this, but for right now, just to follow along with me, draw a dashed line through those points. The second part of it really explains for why their dashed, why are points on that, why are points that lie on this line not? Part of the solution set. Now, by the way, I just want to point out something. This point right here is right here. It lies on that line. And we said that, no, it's not part of it, right? And it's not part of it for a very, very clear reason. All right? Points on this line. Are where Y equals X plus three not where Y is greater than X plus three. Seattle one points where Y is equal to X plus three, I want all the points where the Y coordinate is greater than X plus three. Now let's take a look at something really quick. It says plot the three points from part a and use them to help you shade in the proper region of the plane that represents the solution set to the inequality. Well, I already plotted the .1 comma four. Oops, let me keep that in red. All right, I think I'm going to make two 7 green. All right, where's two 7? Two, one, two, three, four, 5, 6, 7. Now remember, that's this guy. That was yes. And actually, let me take black. We'll go zero, one. And remember that was nose. That was a no. This was a no. And the green was a yes. Now, as we consider the inequality, why is greater than X plus three? It makes sense that yes, would be on this side of the line. Because every Y coordinate or every point I should say that's over in this region is one where the Y coordinate is greater than the line Y equals X plus three. So every single point on this side of the line. Solves that inequality. All right, we're going to test that in just a second letter D but every point that is above that line, Y is greater than X plus three solves it. Every point below it doesn't. And all the points on the line also don't, and that's why the line is dashed. You see, because there's going to be this boundary between the points that are shaded and the points that are unshaded. Sometimes the boundary is going to be included. We'll talk about one of those in a little bit. Sometimes the boundary won't be included, like in this case. All right? Let me go back to blue. Letter D says, choose a fourth point that lies in the region you shaded and show that it is the solution set of the inequality. All right. Now, I'd like you to do this for letter D and then we'll come back and I'll choose one of my own. But pause the video now and see if you can do letter D. All right, let's go through it. I could pick any point, but I think I'm going to go with this one, and that's what, negative two, one, two, three, four, 5, 6, 7. So I'm going to choose the point negative two comma 7. Now my claim is it's going to make the inequality true. Let's take a look. I put 7 and for Y greater than negative two plus three. Negative two plus three is one, 7 is greater than one. That is certainly true. And so it definitely falls in the solution set. All right. That's basically it. We're going to get more practice with this. Think about this more and we've got a little bit more to do in terms of exercise one. But I need to clear out this text so that we can move on to the next page. So pause the video now if you need to. All right, here we go. Let's finish up this problem. Okay, so what we see in exercise one now is we see those points that we plotted. I had actually used negative two comma 6 in an earlier demonstration of one that worked. So letter E says the .10 comma 12 can not be drawn on the graph grid above, right? So ten comma 12 is somewhere out here. All right? So it's difficult to tell whether it falls in the shaded region or not. Is ten 12 part of the solution set of the seine quality, show how you arrive at your answer. All right? So why don't you go ahead and figure out whether that point is part of the solution set? All right, let's go through it. It's not that difficult. In fact, it probably is almost harder to do this with a graph than without a graph. Because without a graph, we simply have to check the truth of the statement. So I'm going to put in 12 for Y, and for X, I'm going to find that 12 is greater than 13. That is clearly false and so no, it is not in shaded region. That's it. Very, very easy to tell whether or not a point in the coordinate plane falls in the solution set of an inequality. Just as it's pretty easy to tell whether a point falls on the equation of a line or any other curve, same thing here. Does it make it true? That's your only question. All right, I'm going to clear this out. And then we're going to keep going on. All right, it is gone. Now, some of these inequalities can get absolutely terrible. So what I thought we would do is we would pick about the worst case scenario and go through it. Because really, most of this isn't about graphing the inequality, it's about getting ready to graph the inequality. So let's take a look at this thing. We want to graph three X minus two Y is greater than or equal to two. And keep in mind what that means. I want to shade in all the points on this graph. That when I plug them into this inequality, make it true. So let me walk you through the easiest way of doing this. The first thing a asks us to do is to rearrange the left hand side of this inequality using the commutative property of addition. In other words, I'm going to take this and rewrite it like this. All right, negative two Y plus three X is equal to two. All I used was that commutative property of addition. To flip flop those. Now let her be solved the inequality, not plot it, but solve the inequality for Y by applying the properties of inequality that we used in unit two. All right, so it's been a little while, but one of the properties of inequality that we had said that we could add or subtract anything we wanted from both sides of the inequality, that's the addition property of inequality, and it doesn't change anything. Now I'm going to use the commutative property to rewrite this as negative three X plus two. I will then divide both sides of the inequality by negative two, but now we have the most important property of inequality. When we divide or multiply by a negative, we must change the inequality symbol or the inequality will become false. Now I'm also going to distribute the division by negative two. So right here I'll have negative three X divided by negative two. And two divided by negative two, so finally, I'm going to make that negative and negative into a positive. And I'm going to do two divided by negative two and make it into negative one. Now, this inequality, and this inequality are equivalent. They have exactly the same solution set. So letter C says shave the solution set of the inequality on the graph below. So here's what you do. All right, you take this inequality. And you plot the equation that it's based on. All right, we're going to do that very quick. We have a Y intercept of negative one and a slope of three halves, meaning we're going to go to the right two and up three. And then we're going to go to the right two. And up three. And perhaps we'll go to down three into the left two. All right. Now, we're going to connect these points with a solid line. I know that doesn't look much like a line, but hopefully you've got a better flat edge than I do. The reason that we're going to draw that with a solid line and not dashed, so it's a solid line. Because this time, the equality is included. The equality is included. Okay? So in the last exercise, we had just a greater than. Here we have a less than or equal to. Now the key is where should we shade? Well, we could pick a point on either side and test it. But because it's less than we should shade below. Now you might say to yourself, well, wait a second, mister wyler wait a second, that says greater than. Ah, but the inequality that we're graphing right now, and it's a little bit hard to see, so let me kind of rewrite it. We transformed that inequality into Y is less than or equal to three halves X minus one. So there it is. And we can really kind of show that we did the right shading and letter D it says pick a point in the shaded region and show that it's a solution to the original inequality. We don't want to test it in this one. We want to test it in that one. So let me go with my color red, and let me go pick this point. Okay? That's one, two, three, four, 5, comma two. Let's see a 5 two works in the original inequality. Three X minus two Y is greater than or equal to two. So where there's an X, I'm going to put in a 5, where there's a Y I'm going to put in two, three times 5 is 15, two times two is four, 15 minus four is 11. And 11 is greater than or equal to two. And that's true. All right, that's it. You know? So the basic procedure for graphing inequalities is number one, rearrange until you sort of get it into Y equals MX plus B form. Watch out for dividing or multiplying by negatives. That will flip the inequality sign. Then plot the line that the inequality is based on, if the inequality includes the equals, the line is solid. If it doesn't include the equals the line is dashed. Then shade above the line if the inequality is greater, shade below the line if the inequality is less than. If you have any question about which way you should shade, pick a point on either side of the line and test it in the inequality. If it makes the inequality true, shade on that side of the line, if it makes it false. Don't. That's essentially it. So I'm going to clear this out, copy down whatever you need to. All right, here it goes. Let's do just a couple more that are a little bit weird. All right? So finally, we've got some that just involve X and Y all right, so exercise three says, shade the solution set for each of the following inequalities in the XY planes provided. First, state in your own words, the XY pairs that the inequality is describing. So think about this for a minute, right? X is less than four. X is less than four. What does that mean? So it means to shade all points. Whose X coordinate. I guess coordinates are less than four, right? So for instance, like this one would be shaded, right? It's X coordinate is less than four. This one would be shaded. Because this X coordinate is less than four. This one wouldn't be shaded. Because it's X coordinate is not less than four. Now, how do we actually graph it? Well, we graph it just the way that we were talking. We turned it into a line, X equals four. Now remember, that's a vertical line. Right, that's a vertical line. And we have to draw dashed. Why do we have to draw a dashed? Well, we have to draw dash because it's less than, not less than or equal to. And now where do we shade? Well, we shade over here. That's not very good. There we go. A little bit better. We shade over here because we're looking for all the points whose X coordinates are less than four. And when we're talking about vertical, that's to the left. Likewise, when we have Y is greater than or equal to negative two, right? We want to shade. All points whose Y coordinates are greater than negative two. So for instance, if I had three comma four, that would work because four is greater than negative two. If I had zero comma negative one, that would work. Negative one is greater than negative two. Think about that for a minute. If I had a comma 5, that would get shaded because that is greater than negative two. If I had 7 comma negative three, that wouldn't be shaded because negative three is not greater than negative two. So again, just like before what we want to do is we want to think about Y equals negative two first. And that, as we saw not too long ago, is a horizontal line, right? In fact, it's the horizontal line right there. Now why is it solid? It's solid because the equals is included. Now I want all the points that are above it, right? Because it's greater than. So it's all the points in here. That's it. So actually, shading any qualities is not too hard, right? We shade the boundary of the inequality, which is the equality, so that the boundary of greater than or less than is equal to. Sometimes that boundary is colored in in the case of a an inequality that includes the equals. Sometimes the line is dashed in what's called a strict inequality, strict meaning that it's only greater than or only less than no equals. Then which half of the plane we shade on, all depends. On whether it's greater than or less than. All right, I'm going to clear out the screen so pause the video if you need to. Here we go. Okay, let's finish up the lesson. So today, we saw how to graph inequalities. Again, it's as simple as this. Graph the equation that the inequality is based on, make it dashed if the equals isn't there, make it solid if the equals is, test points if you need to to figure out which side you shade on or use common sense in terms of greater than being shading above, less than shading below. For now, let me thank you for joining me for another common core algebra one lesson by email instruction. My name is Kirk weider, and until next time, keep thinking. And keep solving problems.

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