Learn about CA Algebra I: Graphing Inequalities Video

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Learn about CA Algebra I: Graphing Inequalities

We’re on problem 21: Stan’s solution to an equation is shown below, alright and I haven’t looked at that yet but let see what they asks us. Which statement about Stan’s solution is true? So let just work through and see if you got it right or if you made a mistake. So let see, he did; let’s just think about how we would do it. So if we have n+8(n+20)=110. So the first thing you want to do is you want to distribute this 8(n+20) right? You don’t want to multiply this whole and plus it. Remember you’re doing order of operations. So this multiplication first, so let’s does that. So you get the n plus and you just distribute this eight. So its 8n+160=110 right, 8(N)+8(20) is 160. And already I see the discrepancy between ours and his. We have n there, we have plus 8n there but where he has a 20 we have 160. He didn’t distribute the eight, 8(n+20) is 8n+160. So that’s wrong, so he made a mistake in step one. Unfortunately I need to get all that other work he had to do just because of this, so it all going to be wrong because of his mistake. Problem 22: When is this statement true? The opposite of a number is less than the original number. I mean you know, if I have a negative number, the opposite of a negative is positive. So it’s not going to be less but if I have a positive number then the opposite of it is negative which is less than it, because a negative is less than a positive. Or if we say that X is greater than zero, then (-X) is going to be less than X. If we say X is positive then definitely you know (-3) is less than (+3). So the statement is true for positive numbers. So let us see. It’s definitely not never drew, I mean you know we showed X case where it’s true. If I have (+3), (-3) is less than that. Statement B, the statement is always well no, if we start with (-3) the opposite is going to be larger then its opposite is going to be (+3). Indeed, the statement is true for negative numbers. No its not, I mean you know (-3) the opposite is three and that’s larger than (-3). Problem 23: What is the Y intercept of the graph? Okay let me see, they say so 23 they say at 4x+2y=12, so if you were to graph this line not that I'm going to you know. The Y intercept is when it intersects the Y axis. So its, that point there. It’s what this Y coordinate is and when the X coordinates, it’s when X is equal to zero, right. So it’s going to be zero comma sum Y intercept. So the easiest thing to do is when X is equal to zero what is Y? So when X is equal to zero. This four times zero is equal to zero, so that just becomes zero. So you get 2y=12, y=6 and that’s choice C and all they want to know is of this line when X=0, what's Y? That’s what the Y intercept is. Next problem, okay they’ve drawn us a graph. They say, let me copy and paste this one. Okay, which inequality is shown on the graph below? So first of all, let’s figure out the equation of this line right now. So what's its Y intercept first of all? So when X is equal to zero, Y=(-1). So if we want to know the; if we put it in slope Y intercept form. The equation of really any line is Y=MX+B; where M is the slope and B is the Y intercept, right. Here when X is zero Y is (-1). So the Y intercept is (-1), so B is (-1). So we know that Y=MX-1 and now we just have to figure out the slope. And the slope is just changing Y for given change in X, right. So if we say change in Y for given change in X is equal to; let see when I increase X by two, I increase Y by one. So change in Y is one when the changing X is two. Now you could say when you increase X by four, I'm going to four spot. I increase Y by two. So you could have done that, you could have said change in Y over change in X is 2/4 which is also equal to ½. But either way the equation, we now know the slope. The slope is change in Y over change in X. so the equation of the line is Y=1/2x-1 that’s the equation of this line. Now what they want to know the equation of the inequality is. So first of all, they’re including, you know they have this gray area here and they put this line in bold. So that means that were including the line. If they’ve drawn a dotted line here that means were not including the line but since their including the line is going to be there greater than or less than or equal to that equal to is because were including the line. But lets think about, is it; so our choices are Y is greater than equal to 1/2x-1 or Y is less than or equal to 1/2x-1. Now think about for any given X, right. Let say when X=2, when X=2 if you put it into this equation you get Y=0 and that’s this point on the line. Now is the gray area all Y values greater than zero or are Y value is less than zero. Well clearly its all Y values is greater than zero, right. It’s all the Y values above the Y value dictated by this equation. So its going to be the equation of this inequality or this area is Y is greater than 1/2x-1 and this is very easy to way to eyeball is okay. They’re including the lines, so I'm going to have this equal dense. This going to be either be greater than or equal to or less than equal to and since this is the area above the line right, above the line is going to be greater than our equal to and that is choice D. Next problem, problem 25, this is another one where I think it makes sense to copy and paste it. It’s a big one. Okay which best represents the graph of Y=2x(-2). So the slope is 2, its positive two and its Y intercept is (-2), so the Y intercept is easy to eyeball. So the Y intercept is here. The Y intercept is (-2) that’s right. Here its (-2) as well and both of these are Y intercept is plus two. So neither of these are going to bright, it intersects the Y axis to the positive 2 but we know that the Y intercept is (-2). So we know our choices are one first the top 2. It’s either A or C and which of these has a positive slope. It’s a positive slope of two. When choice A, when I go to the right by two, I go up by four, right. So change in Y over change in X, I go up by four. When I go up in X by two, I go up in Y by four. When I go up in X by two, so the slope is two. So this has a positive slope of two. So that’s choice A but this for seeing the right choice, he doesn’t work or figuring out the slope. Think about this way. Let me start at random point here. If I increase my X values by, if I increase X value by two. What's happening to my Y value? Is it increasing by two or is it decreasing by two? Well its decreasing by four, it’s decreasing at twice the rate. It’s decreasing by four. So in this case change in Y over change in X. whenever X is positive two, we increase by two here. What happen to Y? It went down four, so here the slope is (-2) and you can look at that. You could kind of eyeball it because it’s going to from the top left to the bottom right. So its going to be negative number but this where to draw these arrows and say okay when it increase X, Y is decreasing. So that’s going to be a negative slope. Lets do it, so the answer was A. Next problem, okay so they’re giving us another one of this where their shaded a graph. These are good. Thank god I have the cut and paste feature. So they ask us which inequality that the shaded region of the graph represents? So once again, it will be good just to figure out which of; what the equation of this line is. We can immediately say, okay the Y intercept is two and what's its slope? When X increases by one, what happens to Y? Y is going down by three, so the Y intercept is equal to two, we figure that out, B=2. The slope is equal to change in Y over change in X, when X increases by one, the slope; sorry the change in Y is (-3). It goes down by three. So the slope is equal to (-3). So the equation of this line is Y=(-3x)+2, MX+V and if I look at the choices already that it looks a little bit different than that, but we’ll get to that. But let’s see what the inequality is. So first of all, it’s going to be either greater than or equal or less than our equal and we say the, or equal because they filled in the line. So let’s think about it, for any; if we pick any X. Say when X=2 then according to this equation, according to this line Y would be equal to (-4). Now are we including the Y’s that are greater than that value as well or we including the Y’s that are less than that value is or were including was or less than that point for given X. So the shaded region is Y is less than or equal to (-3x)+2 and you can eyeball this. You kind of say, oh you know this is below the graph. So it’s less than our equal, not equal it’s because they’ve shaded in the line, if there was a dotted line or just be less than. Once again, this isn't the exact format that they have here. They put all the X’s in the Y’s on the same side that we can do that. Let just add three X to both sides of this equation and you get 3x+Y is less than or equal to. When you have 3x on the right hand that’s becomes zero. So less than or equal to two and that is 3x+Y is less than or equal to (+2,) and that is choice A. And I'm all out of time, see you in the next video.

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We’re on problem 21: Stan’s solution to an equation is shown below, alright and I haven’t looked at that yet but let see what they asks us. Which statement about Stan’s solution is true? So let just work through and see if you got it right or if you made a mistake. So let see, he did; let’s just think about how we would do it. So if we have n+8(n+20)=110. So the first thing you want to do is you want to distribute this 8(n+20) right? You don’t want to multiply this whole and plus it. Remember you’re doing order of operations. So this multiplication first, so let’s does that. So you get the n plus and you just distribute this eight. So its 8n+160=110 right, 8(N)+8(20) is 160. And already I see the discrepancy between ours and his. We have n there, we have plus 8n there but where he has a 20 we have 160. He didn’t distribute the eight, 8(n+20) is 8n+160. So that’s wrong, so he made a mistake in step one. Unfortunately I need to get all that other work he had to do just because of this, so it all going to be wrong because of his mistake.

Problem 22: When is this statement true? The opposite of a number is less than the original number. I mean you know, if I have a negative number, the opposite of a negative is positive. So it’s not going to be less but if I have a positive number then the opposite of it is negative which is less than it, because a negative is less than a positive. Or if we say that X is greater than zero, then (-X) is going to be less than X. If we say X is positive then definitely you know (-3) is less than (+3). So the statement is true for positive numbers. So let us see. It’s definitely not never drew, I mean you know we showed X case where it’s true. If I have (+3), (-3) is less than that. Statement B, the statement is always well no, if we start with (-3) the opposite is going to be larger then its opposite is going to be (+3). Indeed, the statement is true for negative numbers. No its not, I mean you know (-3) the opposite is three and that’s larger than (-3).

Problem 23: What is the Y intercept of the graph? Okay let me see, they say so 23 they say at 4x+2y=12, so if you were to graph this line not that I'm going to you know. The Y intercept is when it intersects the Y axis. So its, that point there. It’s what this Y coordinate is and when the X coordinates, it’s when X is equal to zero, right. So it’s going to be zero comma sum Y intercept. So the easiest thing to do is when X is equal to zero what is Y? So when X is equal to zero. This four times zero is equal to zero, so that just becomes zero. So you get 2y=12, y=6 and that’s choice C and all they want to know is of this line when X=0, what's Y? That’s what the Y intercept is.

Next problem, okay they’ve drawn us a graph. They say, let me copy and paste this one. Okay, which inequality is shown on the graph below? So first of all, let’s figure out the equation of this line right now. So what's its Y intercept first of all? So when X is equal to zero, Y=(-1). So if we want to know the; if we put it in slope Y intercept form. The equation of really any line is Y=MX+B; where M is the slope and B is the Y intercept, right. Here when X is zero Y is (-1). So the Y intercept is (-1), so B is (-1). So we know that Y=MX-1 and now we just have to figure out the slope. And the slope is just changing Y for given change in X, right. So if we say change in Y for given change in X is equal to; let see when I increase X by two, I increase Y by one. So change in Y is one when the changing X is two.

Now you could say when you increase X by four, I'm going to four spot. I increase Y by two. So you could have done that, you could have said change in Y over change in X is 2/4 which is also equal to ½. But either way the equation, we now know the slope. The slope is change in Y over change in X. so the equation of the line is Y=1/2x-1 that’s the equation of this line. Now what they want to know the equation of the inequality is. So first of all, they’re including, you know they have this gray area here and they put this line in bold. So that means that were including the line. If they’ve drawn a dotted line here that means were not including the line but since their including the line is going to be there greater than or less than or equal to that equal to is because were including the line. But lets think about, is it; so our choices are Y is greater than equal to 1/2x-1 or Y is less than or equal to 1/2x-1.

Now think about for any given X, right. Let say when X=2, when X=2 if you put it into this equation you get Y=0 and that’s this point on the line. Now is the gray area all Y values greater than zero or are Y value is less than zero. Well clearly its all Y values is greater than zero, right. It’s all the Y values above the Y value dictated by this equation. So its going to be the equation of this inequality or this area is Y is greater than 1/2x-1 and this is very easy to way to eyeball is okay. They’re including the lines, so I'm going to have this equal dense. This going to be either be greater than or equal to or less than equal to and since this is the area above the line right, above the line is going to be greater than our equal to and that is choice D.

Next problem, problem 25, this is another one where I think it makes sense to copy and paste it. It’s a big one. Okay which best represents the graph of Y=2x(-2). So the slope is 2, its positive two and its Y intercept is (-2), so the Y intercept is easy to eyeball. So the Y intercept is here. The Y intercept is (-2) that’s right. Here its (-2) as well and both of these are Y intercept is plus two. So neither of these are going to bright, it intersects the Y axis to the positive 2 but we know that the Y intercept is (-2). So we know our choices are one first the top 2. It’s either A or C and which of these has a positive slope. It’s a positive slope of two. When choice A, when I go to the right by two, I go up by four, right. So change in Y over change in X, I go up by four. When I go up in X by two, I go up in Y by four. When I go up in X by two, so the slope is two. So this has a positive slope of two. So that’s choice A but this for seeing the right choice, he doesn’t work or figuring out the slope.

Think about this way. Let me start at random point here. If I increase my X values by, if I increase X value by two. What's happening to my Y value? Is it increasing by two or is it decreasing by two? Well its decreasing by four, it’s decreasing at twice the rate. It’s decreasing by four. So in this case change in Y over change in X. whenever X is positive two, we increase by two here. What happen to Y? It went down four, so here the slope is (-2) and you can look at that. You could kind of eyeball it because it’s going to from the top left to the bottom right. So its going to be negative number but this where to draw these arrows and say okay when it increase X, Y is decreasing. So that’s going to be a negative slope. Lets do it, so the answer was A.

Next problem, okay so they’re giving us another one of this where their shaded a graph. These are good. Thank god I have the cut and paste feature. So they ask us which inequality that the shaded region of the graph represents? So once again, it will be good just to figure out which of; what the equation of this line is. We can immediately say, okay the Y intercept is two and what's its slope? When X increases by one, what happens to Y? Y is going down by three, so the Y intercept is equal to two, we figure that out, B=2. The slope is equal to change in Y over change in X, when X increases by one, the slope; sorry the change in Y is (-3). It goes down by three. So the slope is equal to (-3). So the equation of this line is Y=(-3x)+2, MX+V and if I look at the choices already that it looks a little bit different than that, but we’ll get to that.

But let’s see what the inequality is. So first of all, it’s going to be either greater than or equal or less than our equal and we say the, or equal because they filled in the line. So let’s think about it, for any; if we pick any X. Say when X=2 then according to this equation, according to this line Y would be equal to (-4). Now are we including the Y’s that are greater than that value as well or we including the Y’s that are less than that value is or were including was or less than that point for given X. So the shaded region is Y is less than or equal to (-3x)+2 and you can eyeball this. You kind of say, oh you know this is below the graph. So it’s less than our equal, not equal it’s because they’ve shaded in the line, if there was a dotted line or just be less than. Once again, this isn't the exact format that they have here. They put all the X’s in the Y’s on the same side that we can do that.

Let just add three X to both sides of this equation and you get 3x+Y is less than or equal to. When you have 3x on the right hand that’s becomes zero. So less than or equal to two and that is 3x+Y is less than or equal to (+2,) and that is choice A. And I'm all out of time, see you in the next video.

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