We're on problem number six and I actually figured out how to cut and paste from the PDF file to my little writing board, so we actually have access to the problem. So it says: What system of inequalities best represents the graph shown below? So we want to figure out what inequalities represents this shaded area. So a good place to start is just to figure out what these dotted line graphs, what the equations of them look like. So this one, let's do the hard one first, right? This is at an angle. So whenever I look at a line, I like to think what's it's slope and what it's y-intercept, right? And then you could just write it in the form y is equal to mx plus b, where b is the y-intercept and m is the slope. So let's try to figure out the equation of this line up here, this dotted line. So what's its slope? So when I move over 1 and x, when I move over to the right, when x increases by 1, or let's say when x increases by 2, how much does y increase? Well, y increases by 2 as well. Or whatever we increase x by. If x increases by 1, then y increases by 1. So the slope is 1, right? Slope is just change in y over change in x. So the slope of this first line is 1. So m is equal to 1, for at least this first line up here. And what's it's y-intercept? The y-intercept is just what does the graph equal when x is equal to 0 or where does it intersect the y-axis? Well, it intersects the y-axis right here at y is equal to 1, so b is equal to 1. So the equation of this dotted line up here is y is equal to mx, the slope times x, we figured out the slope was 1, so that's x plus the y-intercept, which we also figured out was 1. So that's the graph of that line, right? But notice, the shaded region is below that line. And it also does not include the line, right? That's why they drew a dotted line. If it was a solid line, that means that we're including the line itself. But the shaded region does not include the line. But it's below it. Think of it this way: For any given x-- let's pick this x. Let me do it in a different color. Let me do it in blue. If I were to pick x is equal to 1, this is the point y is equal to x plus 1. That's the equation. Now, my question to you is y above y is equal to x plus 1? Is the shaded region y is greater than x plus 1, which would be that region? Or would it be y is less than x plus 1, which is this region? Well, clearly, it is y is less than x plus 1. You pick any x, x is equal to 3. This is what y would be equal to. y would be equal to 3 plus 1 if you picked this choice. But the y's that satisfy the grey area are always less than this point. So at least the grey area relative to this equation, we can write the inequality y is less than x plus 1. And we know it's not less than or equal to. It would be less than or equal to if they had filled in this line, if this was part of the area, but it's not. They drew a dotted line. So that's why we know it's less than, right? If the grey area was up here it would be greater than. You just pick an x, you say, oh, the y's less than the line, not the y is greater. But we're not done, right? Because this would be the solution set if all of this area was grey. But it's only grey up to a certain point. It's only grey up to this line down here. This line is easier. This line has no slope, right? No matter what my change in x, my change in y is 0. So what's the equation of this? The equation of this line is just slope is 0. So y is equal to 0x plus the y-intercept. Well, the y-intercept is minus 2. So the equation of this line is y is equal to minus 2. Let me do it in the purple. I'll just write it here. That equation is y is equal to minus 2, Which makes sense. For all x, y is equal to minus to 2. Now, my question to you is this grey area. For any given x, the y is greater than the line or the y is less than the line? Well, let's pick this x. When x is equal to 3, y is equal to negative 2. That's this line. Now, is the grey area-- the y's for that x-- above the line or below it? Well, it's above the line, right? So if we look at it from the point of view of this boundary, we say that y has to be greater than minus 2. That's the shaded area. y is equal to negative 2 is this dotted line's equation. The shaded area, at least relative to this line, is y is greater than negative 2. If I just had this by itself, I would shade up everything above y is equal to negative 2, right? But I want just the combination of both of these. So essentially, we have to say that y-- let me do it in a nice, vibrant color. Since the shaded area is below this line, we know that y is less than x plus 1. And since the shaded area is above this line, we know that y is greater than negative 2. Where could I write that? I'll write it right here. So which of the choices satisfy that? Let's see. Whoops, I went to the next problem. So we know that y is greater than negative 2. So these can't-- let me just cross those out. That and that can't be the answer. And then we also know that y is less than x plus 1. So we know that this is the answer. And frankly, if you just looked at the choices, you really didn't have to figure out the slope and y-intercept. They're essentially giving you the equations of the two lines, right? You say, like, oh, one is x plus 1 and one is negative 2, and I just have to figure out whether y is greater than or less than each of those lines. And you can say, OK, relative to the y equals negative 2, we're definitely greater than negative 2, so I'm one of these two choices, and relative to y is equal to x plus 1, we're definitely less than that line, so we have to be choice b. Next problem. Which point lies in the solution set for the system? So you essentially just have to say which coordinates satisfy both of these equations? And frankly, you could graph them and you could draw the area like we did in the last one, or you could just try out the numbers, right? Let's just try the numbers. That never hurts. So minus 4, 1. Let's see if it satisfies it. So remember, the first coordinate is x, the second one is y. So 2y would be 2 times minus 1 minus x. So minus minus 4. Now, is that greater than or equal to minus 6? Well, this is minus 2 plus 4. And this is what? Minus 2 plus 4 is equal to 2, which is definitely greater than or equal to minus 6. So it satisfies the first equation. Let's see if it satisfies the second equation. I'll do it in another color and I'll do it up in the black area up here. So 2 times y. So it's 2 times minus 1-- y is minus 1-- minus 3 times minus 4. And they want to know is that less than minus 6? So you get minus 2. 3 times 4 is 12, so it's plus 12. Negative times a negative is a positive, so that equals plus 10, which is not less than negative 6. So the first one does not work. That is not the answer. Let's try the second one. 3 comma 1. So y is 1. So it becomes 2 times 1, which is just 2, minus 3. Is that greater than or equal to negative 6? So you get negative 1 is definitely greater than negative 6, so this satisfies the first equation. Does it satisfy the second one? 2 times y. 2 times 1 is 2, minus 3 times 3. Is that less than minus 6? So you get 2 minus 9. Is that less than minus 6? You get minus 7 is less than minus 6. So that is true. So choice b works. You don't have to go any further. That point lies in the solution set, which just means it satisfies both of these inequalites. Next problem. Scroll down a little bit. Which system of linear equalities is represented by this graph? So now we can use some of the skills we used the last time. I don't have much space to draw all of them. So let's just to figure out the-- let's see, do they gives us-- no, they don't even give us the equations because all the equations are different, so we have to figure out the equations of these two graphs. So let's do this top one first. So let's figure out its slope. When you increase-- and you can just eyeball this one. When you increase x by 3, y increases by the same amount, right? So change in y over change in x would be 3 over 3. Whatever you change x by, y increases the same amount. So the slope of this first one at least-- let me do it in a color you can see over black-- the slope is equal to 1. And what's its y-intercept? Well, you just look at where did it intercept the y-axis? Right there. So the y-intercept is equal to minus 2. So this first equation right here is y is equal to x minus 2. Fair enough. And this grey area, is it above or below it? It's below it. You pick any x. For any x, this is y is equal to x minus 2. The grey area is for all the y's that are less than it, right? And notice, now the line is shaded in. So the line is part of the solution set. So at least relative to this, we know that the grey area is all y's that satisfy y is less than or equal to x minus 2. Less than because it's below it, and it's equal to because they actually drew in the line. But now we have to figure out the lower bound. What's the equation of this line? So let's think about it a little bit. This one's a little more interesting. If I increase x by 1, so if I'm going from this point, what happens to y? y goes down by 2, right? 1 minus 2. If I increase x by 2, y goes down by 4, right? Minus 4. So at least in this case, the slope, change in y, is minus 4 whenever change in x is positive 2. Or it could've been change in y is minus 2 whenever change in x is plus 1, right? So it equals minus 2. And that makes sense. For every 1 you go over, the slope goes down 2. And what's it's y-intercept? Let's see, y-intercept is just right there. y is equal to 3. So b is equal to 3. So this second graph is y is equal to minus 2x plus 3. And then, once again, the shaded area includes this, but we have to be greater than or equal to this graph. We have to be greater than, because if you pick any x, the y's that satisfy the shaded area are equal to the graph or greater than for any x. So there we have our inequality. So this one, this area, you could kind of say is y has to be greater than or equal to minus 2x plus 3. That's from this boundary. And we also know that y has to be less than or equal to x minus 2. Now, hopefully, that's one of the choices. If we haven't a made a careless mistake. Let me see. All right. y is greater than or equal to-- OK, so this is not the answer. Neither of these are the answers. That's not the answer. Let me cross that out. y is greater than or equal to-- we have minus 2x plus 3, right? Let me make sure I got that right. Minus 2x plus 3. Yeah, let me see. Right, so they're tricky. So the slope on that second line was minus 2x plus 3. They got the second part right. Maybe I made a careless mistake, but I don't think this is right either. Let's see if we can rearrange. So these other two, they put the x's and the y's on the same side, so let's do that. That top equation, we can rewrite it. Add 2x to both sides. You get 2x plus y is greater than or equal to 3. And the second equation, if we were to subtract x from both sides, you get y minus x is less than or equal to minus 2. And now this is interesting. So there's no minus 2 on the right-hand side here, but what can we do? If we multiply both sides of this equation by minus 2, what happens? Sorry, multiply both sides of the equation by minus 1. You get minus y plus x. And when you multiply an inequality by a negative number, multiply or divide, you switch the inequalites. So it's greater than or equal to-- and we're multiplying by negative 1-- 2. So it becomes x minus y is greater than or equal to 2. So our two equations turn into 2x plus y is greater than or equal to 3, which is this one. And we get x minus y is greater than or equal to 2, which is that one. And we're done. It's choice d. See you in the next video.