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About this Lesson
 Type: Video Tutorial
 Length: 9:04
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 97 MB
 Posted: 08/18/2009
This lesson is part of the following series:
College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Systems of Equations (33 lessons, $44.55)
Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra.
About this Author
 Thinkwell
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Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
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Systems of Equations
Systems of Inequalities
Graphing Linear and Nonlinear Inequalities
So, let’s see how you would actually go about graphing inequalities. So, again, it’s just a visual picture of what’s going on algebraically. So, how about graphing the following, 2x  3y and let’s say that’s greater than or equal to 1. Well, how would you graph this kind of inequality? Well, the first thing I would do would be to consider these sort of auxiliary object going on in the background which would be what I would have if I had an equal sign there. So, I’d first do a warm up problem and the warm up problem I would consider would be 2x  3y = 1. What’s the graph of that? Well, that’s some sort of line. If you want to see exactly what that is I could write this maybe in slope intercept form. So, I’ll solve for y.
If I were to solve for y I would bring this 3y to this side and bring the 1 over here and I would see 2x 1 = 3y. If I divide both sides by 3, I write the y over here though, y equals, I would see 2/3x  1/3. Just divided both sides by 3 and then just wrote the reverse order. So, I can see exactly what’s going on here. I see that y is going to be equal to 2/3x  1/3. So, the y intercept is 1/3 and the slope is 2/3. What I’m going to just do here is record the original question and then write down this auxiliary thing, y = 2/3x  1/3.
This here actually represents sort of the boundary and this is going to be one of the regions. Now, let’s actually see what this would look like graphically. Well, we have this line and we know the line is this. So, it’s y intercept is 1/3. So, I wrote a line and it’s intercept is 1/3. So, 1 is here, so 1/3 would be around here and the slope is what? It’s 2/3, so it’s three over and two up. It would just like that. So a thick line, but notice that the origin is above the line. I hope you can see that. Now, that’s not what we’re asked to find. What we were asked to find of course was this inequality thing. But I know that this is going to cut the world up into two pieces and one of those pieces, either the upper piece here or the lower piece here, is going to actually represent all the solutions to this.
How do I figure out which is the solution? Well, all I do is pick a point in either region and see if it satisfies the original inequality. So, for example, I could pick (0,0), there’s the origin, and ask does (0,0), x = 0, y = 0 satisfy this? Well, let’s see. If you plug in (0,0) I’d see 0 1. Is that a truism? No. Zero is not greater than or equal to one. So, in fact, the upper region must be the wrong region. We must be looking at the lower region. So, all I’m going to do is shade in the lower region. So, I would do this like so.
So, this inequality is, the graph of that, is this line and all the things here in green. In fact, notice that since this is a greater than or equal to I actually include that boundary right there. I actually include that boundary. That’s part of it, because that’s what this equals. Neat, so there’s a neat one to sort of take a look at. Let’s try another one. Sort of get in the mood for this. Let’s look at the following. Let’s take a look at x + y < 3.
Well, what would I look at first? I’d first look at sort of a warm up auxiliary thing, where this equals. If I put an equals there I’d see x + y = 3. If I solve it for y, I would see y = x + 3. So, that’s sort of a good warm up to look at. What’s the graph of the warm up. Well, the graph of the warm up has a y intercept of 3 and a slope of 1. So, it looks just like this. That’s actually a pretty easy one to do. There you have it. The slope is negative one, one down, one over, one down, one over and it goes through three. So, there’s the line. Now, I’m looking at, the actual thing I’m looking at is x + y > 3.
So, I’m either going to be in this region up here or that region down there. Which one? Just pick a point. In fact, you could pick any point. I usually pick the origin, when I can. Of course I couldn’t if the line went right through it, but if the line doesn’t go through it I pick that because it’s so easy. Plug in (0,0) and see if it’s a truism. If I plug in (0,0) in here I see 0 + 0 is 0 and that is less than 3. So, this is the region of excitement and so I can color it. Of course you want to color the whole region there.
Now, the question is do I want to include the line or not? Well, since there’s strict inequality, in fact, I don’t want to include the line. So, I actually remove the line. So it’s anything up to the line, but not including the line. So, it actually sort of looks like this. So, there’s the region right there. I’m going to put my hands here so you don’t see the fact that I ran out of stuff there. So, the region is all the stuff that’s sort of below the invisible line. There’s the graph of that. So, again, it’s pretty easy. You just graph the equality part and then ask yourself what side or what part do you want.
Well, let’s try some that actually are lines just to see that in fact there’s no difference whether you have a line or not. Suppose we look at y < x². What would I do first? Well, first I would graph the analogous thing with an equal sign. So, I’d graph y = x². That’s a standard happy face parabola right here. Notice that I have strict inequality. So, I’m not going to actually want to include that curve. So, I’m going to draw that curve dotted. So, I’m going to draw that curve dotted. So, I’m going to put in a dotted standard parabola. It’s dotted because I don’t want to include it.
Now, I’ve got to figure out whethernow notice the two regions are a little bit weird. There’s sort of this region here and there’s this region up here and the question is which one should I shade in when I pick a point. Well, you might say, “Gee, I pick the origin.” I can’t pick the origin today because this curve actually passes through it. I’ve got to pick a point that’s in one of the regions. So, why don’t I pick this point right here that’s (0,1). So, I’ll let x be 0 and y be 1 and I’ll ask, “Does that satisfy that?” If I put a zero in for x and a one in for y, is that true? Is 1 < 0? No it’s not. So, this region must not be the region that I want. So, in fact, I shade the outside region and so I’d shade this. Here’s how I do it by the way. I shade everything. It just keeps going. I shade everything out here and this actually is unshaded. So, there’s the graph of y < x².
Let’s try one last one. How about looking at the following, y x². Well, same picture but now, notice that I have a possibility for equality. So, when I actually graph it I can use a solid graph now. Still passes through the origin. Now, I pick a point. I’ll pick the same point. I’ll pick (0,1). If I put in 0 and 1, is this a truism? Is 1 0? Well, now it is true. So, now I shade in this upper part here. This is the graph of this inequality. Notice the solid line because I’m allowed to have equals. Here there’s a dotted line because I’m not allowed to have equals and I shaded in the right region. This curve, notice that each curve will cut the world up into two pieces and I just shade the appropriate piece by picking a point, plugging in, see if it satisfies it. Up next we’ll take a look at more of these shady issues.
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