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Step one: Graphing two variable inequalities

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Step one: Graphing two variable inequalities

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Look at the attached video and than solve the exercise of the second step

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We're asked to graph the inequality y is less than 3x plus 5. So if you give us any x-- and let me label the axes here. So this is the x-axis. This is the y-axis. So this is saying you give me an x. So let's say we take x is equal to 1 right there. 3 times 1 plus 5. So 3 times x plus 5. So 3 times 1 is 3 plus 5 is 8. So one, two, three, four, five, six, seven, eight. This is saying that y will be less than 8. y will be less than 3 times 1 plus 5. So the y-values that satisfy this constraint for that x are going to be all of these values down here. Let me do it in a lighter color. It'll be all of these values. For x is equal to 1, it'll be all the values down here, and it would not include y is equal to 8. Y has to be less than 8. Now, if we kept doing that, we would essentially just to graph the line of y is equal to 3x plus 5, but we wouldn't include it. We would just include everything below it, just like we did right here. So we know how to graph just y is equal to 3x plus 5. Let me write it over here. So if I were to write y is equal to 3x plus 5, we would say, OK, 3 is the slope. Slope is equal to 3, and then 5 is the y-intercept. Now, I could just graph the line, but because that won't be included in the y's that satisfy this constraint, I'm going to graph it as a dotted line. So we'll start with the y-intercept of 5. So one, two, three, four, five. That's the y-intercept. And the slope is 3. So if you go over to the 1, you go up 3. So let me do that in that darker purple color. So it'll look like this. It will look like that. That point would be on it. That point would be on it. If you go back, you're going to go down by 3. So that point will be on it, that point, and that point, and I'll just connect the dots with a dotted line. That dotted line is the graph of y is equal to 3x plus 5, but we're not going to include it. So that's why I made it a dotted line because we want all of the y's that are less than that. So for any x-- so you pick an x. Let's say x is equal to negative 1. If you evaluate 3x plus 5 for that x, you'd get here. But we only care about the y's that are strictly less than that. So you don't include the line. It's everything below it. So for any x you pick, it's going to be below that line. You take the x, go up to that line and everything below it. So for all of the x's, it's going to be this entire area. Let me draw it a little bit neater than that. It's going to be this entire area that's under the line. I'll do it in this orange. It's a little bit easier to see. So this entire area under the line is y is less than 3x.

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Graphing two-variable inequalities (video) | Khan Academy

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We're asked to graph the inequality y is less than 3x plus 5. So if you give us any x-- and let me label the axes here. So this is the x-axis. This is the y-axis. So this is saying you give me an x.

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Here you find a nice worksheet : solve the first three inequalities in classroom and the others at home

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Step three: Solutions of step two worksheet

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Step three: Solutions of step two worksheet

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Step Four:Graphing systems of inequalities

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Step Four:Graphing systems of inequalities

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look at the attached video

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Graph the solution set for this system. It's a system of inequalities. We have y is greater than x minus 8, and y is less than 5 minus x. Let's graph the solution set for each of these inequalities, and then essentially where they overlap is the solution set for the system, the set of coordinates that satisfy both. So let me draw a coordinate axes here. So that is my x-axis, and then I have my y-axis. And that is my y-axis. And now let me draw the boundary line, the boundary for this first inequality. So the boundary line is going to look like y is equal to x minus 8. But it's not going to include it, because it's only greater than x minus 8. But let's just graph x minus 8. So the y-intercept here is negative 8. When x is 0, y is going to be negative 8. So just go negative 1, negative 2, 3, 4, 5, 6, 7, 8. So that is negative 8. So the point 0, negative 8 is on the line. And then it has a slope of 1. You don't see it right there, but I could write it as 1x. So the slope here is going to be 1. I could just draw a line that goes straight up, or you could even say that it'll intersect if y is equal to 0, if y were equal to 0, x would be equal to 8. So 1, 2, 3, 4, 5, 6, 7, 8. And so this is x is equal to 8. If it has a slope of 1, for every time you move to the right 1, you're going to move up 1. So the line is going to look something like this. And actually, let me not draw it as a solid line. If I did it as a solid line, that would actually be this equation right here. But we're not going to include that line. We care about the y values that are greater than that line. So what we want to do is do a dotted line to show that that's just the boundary, that we're not including that in our solution set. Let me do this in a new color. So this will be the color for that line, or for that inequality, I should say. So that is the boundary line. And this says y is greater than x minus 8. So you pick an x, and then x minus 8 would get us on the boundary line. And then y is greater than that. So it's all the y values above the line for any given x. So it'll be this region above the line right over here. And if that confuses you, I mean, in general I like to just think, oh, greater than, it's going to be above the line. If it's less than, it's going to be below a line. But if you want to make sure, you can just test on either side of this line. So you could try the point 0, 0, which should be in our solution set. And if you say, 0 is greater than 0 minus 8, or 0 is greater than negative 8, that works. So this definitely should be part of the solution set. And you could try something out here like 10 comma 0 and see that it doesn't work. Because you would have 10 minus 8, which would be 2, and then you'd have 0. And 0 is not greater than 2. So when you test something out here, you also see that it won't work. But in general, I like to just say, hey look, this is the boundary line, and we're greater than the boundary line for any given x. Now let's do this one over here. Let's do this one. The boundary line for it is going to be y is equal to 5 minus x. So the boundary line is y is equal to 5 minus x. So once again, if x is equal to 0, y is 5. So 1, 2, 3, 4, 5. And then it has a slope of negative 1. We could write this as y is equal to negative 1x plus 5. That's a little bit more traditional. So once again, y-intercept at 5. And it has a slope of negative 1. Or another way to think about it, when y is 0, x will be equal to 5. So 1, 2, 3, 4, 5. So every time we move to the right one, we go down one because we have a negative 1 slope. So it will look like this. And once again, I want to do a dotted line because we are-- so that is our dotted line. And I'm doing a dotted line because it says y is less than 5 minus x. If it was y is equal to 5 minus x, I would have included the line. If it was y is less than or equal to 5 minus x, I also would have made this line solid. But it's only less than, so for any x value, this is what 5 minus x-- 5 minus x will sit on that boundary line. But we care about the y values that are less than that, so we want everything that is below the line. And once again, you can test on either side of the line. 0, 0 should work for this second inequality right here. 0 is indeed less than 5 minus 0. 0 is less than 5. And then you could try something like 0, 10 and see that it doesn't work, because if you had 10 is less than 5 minus 0, that doesn't work. So it is everything below the line like that. And like we said, the solution set for this system are all of the x's and y's, all of the coordinates that satisfy both of them. So all of this shaded in purple satisfies the second inequality. All of this shaded in green satisfies the first inequality. So the stuff that satisfies both of them is their overlap. So it's all of this region in blue. Hopefully this isn't making it too messy. All of this region in blue where the two overlap, below the magenta dotted line on the left-hand side, and above the green magenta line. That's only where they overlap. So it's only this region over here, and you're not including the boundary lines.

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Intro to graphing systems of inequalities (video) | Khan Academy

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Graph the solution set for this system. It's a system of inequalities. We have y is greater than x minus 8, and y is less than 5 minus x. Let's graph the solution set for each of these inequalities, and then essentially where they overlap is the solution set for the system, the set of coordinates that satisfy both.

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Here you fond a nice worksheet: solve these four exercises

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Step six: Solution of step five worksheet

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Step six: Solution of step five worksheet