Graphing Systems of Inequalities - Online Tutor, Practice Problems & Exam Prep

Graphing inequalities involves plotting corresponding lines or curves and shading regions that satisfy the inequality. For linear inequalities, a solid line indicates "greater than or equal to" or "less than or equal to," while a dashed line indicates "greater than" or "less than." Testing points helps determine which side to shade. Nonlinear inequalities, like quadratics, follow the same process. When dealing with multiple inequalities, the solution is found in the overlapping shaded regions, representing points that satisfy all inequalities simultaneously.

Linear Inequalities

Welcome back everyone. Previously, in other videos, we've seen how to graph a one-dimensional inequality, something like x≥1. It was pretty straightforward. You just find the point at x=1 on a number line, and then you would just figure out all of the numbers that satisfy this inequality. It would be everything to the right of that point. But we're going to do the exact same thing here for a 2-dimensional inequality. It's just that now our equations that we've been dealing with have multiple variables like ys and x. So the whole idea is that we're going to have to plot inequalities now on a 2-dimensional graph instead of just a 1-dimensional number line. But I'm going to show you that it's pretty straightforward, and I'm going to show you a step-by-step way to do this.

So let's go ahead and take a look here. So, if I had to graph the inequality x≥1 on a two-dimensional number line, let's see how this would work here. We've already seen what the equation x=1 looks like. Remember, on a two-dimensional graph, this will actually be a vertical line. It's kind of weird because on a one-dimensional number line, it's just a point. But remember, now we have the sort of a y-axis, so you almost could extend this up and down. You'll see that you'll get a vertical line. So, the key idea here is that to graph an inequality; first, you actually have to graph what the corresponding line is. But now we have to look at how to shade the parts of the graph that will satisfy this inequality.

So what you're going to do here is you're going to shade the side of the graph with points that make the inequality true. You remember, you're just going to figure out the points that make this inequality a true statement. And what I can do here is just pick random points. Like, let's say, for example, I pick this point, 2,0. If I take these x and y coordinates and I plug them into this inequality, we'll see that we get true statements for this. For example, the x coordinate for this is 2, so 2≥1. That is a true statement. But now we also have an infinite number of y values to consider. We can pick a point over here. This would be 4,3. When you plug it in this inequality, you'll also get a true statement. So the key difference here is that we're not just going to shade the axis like one line; we're actually going to have to shade everywhere on this graph with points that satisfy this inequality. And for this particular line, it's actually going to be all of these points that are to the right of this inequality. So it kind of makes sense. For a one-dimensional number line, you just had a line. For a two-dimensional graph, we actually have an area, a region of points that satisfy this inequality.

That's all there is to it. All the points on this side of the graph are where x>1, so that satisfies this inequality. All the points over here are less than 1. So if you had something like y<x, for example, we could graph this pretty simply. All you'd have to do is just graph the corresponding line, y=x. Now one thing I want to point out here is the symbols are different. Notice we have a less than symbol, whereas we had a greater than or equal to symbol. So, if you have equations with the less than or equal to or greater than equal to symbols, you draw a solid line. The way I like to think about this is if you see a solid bar underneath the symbol, then you draw a solid line. But if you have something like less than or greater than alone, then you draw a dashed line. So y<x is going to look like y=x, but we're going to have to draw it with a dashed line. So it's basically going to look something like this.

These examples are pretty straightforward, but sometimes you're going to get more complicated equations, something like y>2x-4. So I actually want to show you a step-by-step process of how to graph these inequalities. Let's get started here. So the first thing you want to do is you're going to graph the solid or dashed line depending on the symbol by switching the inequality symbol with an equal sign. So remember that this symbol over here means that we're going to be drawing a dashed line. And the way we graph this is by graphing y=2x-4. Well, it just looks like a line that goes through the y-intercept of negative 4 and has a slope of 2. So it's going to look something like this. Remember, we have to use a dashed line for this. So this is going to be what that graph looks like. It's kind of just sketched out. So that's the first step.

Right? So the second thing we have to do is we have to figure out which points will satisfy our inequality. With x≥1, it was pretty simple because we just highlighted everything to the right of this graph. But for this, it's going to be a little bit tricky. Right? Is it going to be this side over here? Is it going to be this side? How does it actually work out for different angles and different steepnesses? So the second step is you're actually going to basically test a point on either side of the line. And the way that you test it is you just basically Plug the x, y coordinates into the inequality, which is exactly what we did with this 2,0 over here. So the idea is that we're just going to pick a point randomly. You can just pick any point that you want. It won't matter. But it's always just best to use a point on the x or y axis because it makes math a little bit easier. So for example, if I pick this point at random, this is going to be 0,1. I'm going to test it by plugging the x and y values into this inequality. So 0,1, if I plug it into this inequality over here, what this says is that is 1 greater than 2 times 0 minus 4. Right? Because that's what this inequality says over here. So let's work this out. Is 1, in fact, greater than this inequality? Well, 2 times 0 is 0. And again, that's why using a point on the axis is easier because one of the variables will sort of cancel out. So 1, we'll see, is greater than negative 4. Is that a true statement? It is.1 is greater than negative 4, and so this is going to be a true statement. So that's the second step, sort of testing out a point. Now the third one says that if the point makes the inequality true, like exactly what we had in this case over here, then you're going to shade the side that includes that point. So shade the side with that point. So on this graph here, the line 2x-4 is like a barrier, and we're going to shade everything that includes this point over here that makes our statement true. So really what happens is the region that makes this inequality true is actually going to be all of this area over here. If you were to take a point at random anywhere inside this region, the x and y values would make this inequality a true statement. Versus if you pick something in this region over here, you'll get something that doesn't make sense. Like, 5 is greater than 7. Right? That does not make any sense. Alright.

So if your point makes the inequality false, which is not what happened here, then you're going to shade the side without that point. And that's really all there is to it. Now there's actually a cool shortcut here that I'll teach you, which is that if you can ever isolate y to one side of your inequality and get it in slope-intercept form, which is basically what we have here, then if you have something like a greater than symbol or greater than or equal to, then you can just shade everything that is above the line. So that's exactly what we had here. y>2x-4. This is the line. You would shade everything that is above this line. Whereas if you had something like a less than symbol, then you would shade everything below that line. Alright? So that's how to graph linear inequalities. Thanks for watching, and I'll see you in the next one.

Example 1

Welcome back, everyone. Let's continue our journey through systems of inequalities by taking a look at this example problem here. Here, we're told we're given these three equations, a, b, and c, and we're told to match each inequality with its graph. So we're not actually going to graph anything. We're just going to sort of be matching up equations to graphs. Let's go ahead and get started here.

So, I've got equation a, which is y>-3x+4. Then for equation b, we've got x+y<1. Equation c is -3x-y≥-4. Let's take a look at the first equation over here.

Y is greater than -3x+4. Now, in order to visualize the graph, remember, the first thing you do is just to pretend here that there is an imaginary equal sign. What does the graph of -3x+4 look like? It looks like the equation of a line with a y-intercept of 4. If you look through these two graphs, there are 2 equations with y-intercepts of 4, graphs 2 and 3. And they almost look the same, except for how the shading is in different areas, and one solid and one is a dashed line. So which one does this equation correspond to?

The next thing you should look at is the sign or the symbol that we see, because here we see a greater than symbol. Whenever you see a greater than symbol or a less than symbol without the solid bar underneath, that means that we're dealing with a dashed line, not a solid line. So that means that without a doubt, equation a corresponds to graph number 3. So this corresponds to graph a. Alright? And if you can see here, so we've got the dashed line with a negative slope of 3, and we also can see that the shaded area is everything that's above that line, which is exactly what the shortcut for lines tells us. Notice how this symbol here is a greater sign, so we're going to shade everything that's above that line.

Alright. So now let's take a look at equation b. X plus y is less than 1. So this may be hard to visualize what the graph looks like, so let's rewrite it in slope-intercept form. I could just move the x to the other side by subtracting x, and this is going to be y is less than -1x+1. So same thing over here. We've got a y-intercept of positive one. And notice how what I said in the beginning or when we were dealing with the first equation is that these two lines have y-intercepts of 4. The only one that has a y-intercept of 1 is going to be this line. So this definitely corresponds to equation b. Alright? And just to sort of double-check here, we can see that this equation tells us that we should be dealing with a dashed line because it's just a less than symbol, and that's exactly what we have. We have a dashed line. It's got a slope of negative one. And, also, what we can see is that because of the less than symbol, our shortcut tells us that we have to shade everything that's below that graph. So this is, without a doubt, the equation for the graph of b.

And that just means that by default, this second graph over here is going to be the equation for c. But let's understand why that works. So here, we've got this other sort of more complicated, negative expression. One of the things that you can do here is you can sort of tell that all of these numbers here have a negative sign. And so one thing you can do is you can actually flip all of those to positives, but then you have to flip the inequality symbol. So one way this sort of, like, simplifies is this actually just turns into 3x+y≤4. When you flip all the signs, you actually flip the symbol as well. And finally, what we can do is move the 3x to the other side, which you'll get is y is less than or equal to -3x+4. That would be in slope-intercept form. So, again, notice how we still have that positive 4, as a y-intercept. We still have the same slope of negative 3. But now what happens is that we have a less than or equal to, so this should be a solid line, and we should shade everything that's underneath that line instead of above like we did for graph number 3. Alright? So that means that this is definitely the equation for this graph over here.

That's it, folks. Let me know if you have any questions, and thanks for watching. I'll see you in the next video.

Graph the inequality $2x+3y$ < $6$.

• A.
• B.
• C.
• D.

Nonlinear Inequalities

Everyone. Welcome back. So in previous videos, we saw how to graph linear inequalities, something like y>2x-1. But now you might start to see problems in which you have nonlinear equations, things like quadratics or exponentials, radicals, and rational functions, things like that. I might think that it's an entirely new process, but I'm actually going to show you here that graphing nonlinear inequalities is exactly like graphing linear inequalities. We're actually going to follow the exact same list of steps here. So let's just jump right into this problem and see how it works. We've got y≥x2-1. This is clearly going to be a quadratic equation. Right? So we have something squared. So let's just go ahead and stick to the steps here. The first thing we do is figure out if we're dealing with a solid or dashed line. Let's look at the symbol. So we see a greater than or equal to symbol. If I see a solid line underneath, it's going to be a solid line. That's going to be my equation. Alright? So what does x2-1 look like? Well, first, you're going to have to sort of pretend that there's, like, an equal sign there. So y=x2-1 is going to be a parabola in which the vertex is going to be at 0,-1, and then it opens up like this. So what's going to happen is you're going to look something like this. I'm just going to sketch it out. It doesn't actually have to be perfect unless your professor, really wants this to be perfect. But this is going to be what our parabola looks like. Right? So that's y=x2-1. Let's take a look at the second step. The second step says we're going to test the points on either side by plugging and, actually, it doesn't need to be a line. It could be a curve or something like that. By plugging x,y, the values, into the inequality. Alright? So we can use something on the x or y axis. So we could just pick a point at random. What I'm going to do is I'm just going to pick this point over here, 0,2. That's pretty much, like, well within, sort of the upper portion of the parabola. So let's go ahead and test this out, 0,2. Remember, this is just my x and y values. So what this means here is that y≥x2-1. So I'm just going to replace them now. 2≥02-1. So does this make a true statement? Well, what you'll see here is that you get 2≥02=0, and is 2≥-1? This actually is a true statement. 2>-1. So because this is a true statement now, we can move on to step number 3. We're just going to shade the side that includes that point. If our statement is true, we include that point. So really all this is here is, remember, the curve or this equation here is kind of like a barrier. We can't shade anything that's on the opposite side of that barrier. So you might think, oh, well, I just shade everything that's above this line over here, but that's not how it works. Because if you were to pick a point, let's say, that's out here, above the equation or above the graph but still on the opposite side of this line, you'll find that this actually won't be true. So what you have to do is you're sort of, like, bounded by the shape of the graph itself. So, really, the points that make this inequality true are actually going to be everything that's just above the parabola, but still kind of, like, on the inside of it. So it's kind of, like, on the inside of the bowl. Alright? And that's really all there is to it. So that would be how to graph this inequality and shade the appropriate areas. Alright. So that's how to graph nonlinear inequalities. Let's go ahead and take a look at some practice.

Graph the inequality $y$ < $2^{x}$.

• A.
• B.
• C.
• D.

Example 2

Welcome back, everyone. So just in case you were struggling with this problem, I'm here to help. So we're going to graph this inequality,

x 2 + ( y - 1 ) 2 ≤ 9 . Let's get started here with our steps. First, we're going to have to figure out whether you're dealing with a solid or a dashed line, and we do this by looking at the equation or the the inequality symbol. Remember, if there's a bar underneath the symbol, that means it's going to be a solid line. That's exactly what we have here. So, we're going to have a solid line, not dashed, and we're going to have to graph this by switching the inequality symbol with an equal sign. So we're basically just going to have to graph the equality, x 2 + ( y - 1 ) 2 = 9 .

Alright? So what does this look like? It's a circle, and the center is at the coordinates that are inside the parentheses. Here, if we have no numbers inside the parentheses for x, that means it’s 0. So, the center is at coordinate (0, 1). Remember, when we see a minus sign, that indicates the value is actually positive.

The circle has a center at (0, 1). But remember, we actually need the radius of the circle, and that's given to us by the number 9 on the right side. So clearly, we can see that r² equals 9. And so the radius, r, is equal to the square root of 9, which is just equal to 3. We're going to have to go up, down, and to the right and left by 3 units and connect all of those points. Once I do this, it will look something like this.

Remember, we're going to use a solid line for this, but this is basically what our graph is going to look like. Now let’s take a look at the second step here, which is we’re going to have to test a point. Even if you got this far, you may not have noticed where to shade the graph. Do we always go below the circle? Do we go above it? What’s going on here?

Let’s go ahead and test the point (2, 0). So, 2 2 + ( 0 - 1 ) 2 ≤ 9 results in 4 + 1 is less than or equal to 9. That's a true statement, so we’ll shade the point including that point within the graph. But, do we have to graph everything that's below the circle?

Actually, no. Testing this out further will show that only the points within the circle make the inequality true. So, these types of inequalities are tricky because you can’t just shade everything below or above the center of the circle; you actually have to test all these points.

That’s how to graph these types of inequalities with equations of circles. Let me know if you have any questions, and thanks for watching.

Systems of Inequalities

Everyone, welcome back. So, up until now, we've seen how to graph an individual inequality, something that looks like this. The basic idea was that we would draw this inequality and test the points, and then we would shade the area, an area that makes that inequality true with those points. But now, we're actually going to see some problems that have multiple inequalities. And in this case, what we're going to do is we're just going to repeat this process over and over again, go through the same list of steps that we have for as many inequalities as we have. Now we're going to see something interesting that happens. So, for example, if I were to graph this inequality and, let's say, it just looks something like this, what we're going to see is that when we graph this inequality where our shading area is going to overlap with the shading area of the first graph. And what we're going to see here is that there's going to be this one region where all of the shadings will overlap, and this will be the solution to the system of inequalities. So the whole idea here is that to graph a system of inequalities, you're going to plot all the lines, all the curves, and shade the regions making points or, containing points that make all of the inequalities true. And to do this, we're going to use different colors and different styles of shading. You can do this however you want, basically shading each curve first. And then later on, we'll find the overlap. So let me go ahead and just break this down for you. We'll follow this list of steps here. Let's take a look at our first example. We've got \( y \leq -x + 4 \). So let's go through our steps. We're going to graph the solid or dashed curve. This is going to be a solid line because we have a solid symbol underneath the equation. And we're just going to replace it with an equal sign. So what does \( y = -x + 4 \) look like? It looks like it goes through the point, the y intercept of \( 0, 4 \), and it has a slope of negative 1. So this is going to look something like this. It's going to be a solid line, so it looks like that. Alright? So that's our equation or that's our graph. And then to test the point, we can just pick any point that's underneath this graph or really anywhere. So I'm just going to pick this point over here. This is going to be a point \( 0, 2 \). So if I test out this point \( 0, 2 \), what happens? Well, this is saying that \( 2 \) is less than or equal to \( -0+4 \). So is this actually a true statement? The \( 0 \) will go away. So is \( 2 \) less than or equal to \( 4 \)? That is actually a true statement. So we also just could have used the shortcut that because of the sign here, where we're just going to graph everything that's underneath this line, and we that would have been perfectly fine. So this all of this area here, this whole thing satisfies, the red inequality, the red equation that we have here. Alright? Now, let's do the same exact thing for the blue equation. Right? So we did all these things. For the blue equation, we're going to graph this equation, \( y > 2x + 1 \). So, in this case, we're going to have is a y intercept of 1, and it's going to have a slope of 2. But this one's going to be a dash line because of the symbol. So in other words, it's going to look something like this. So this equation is going to look something like that. So that's going to be our line. And now let's go ahead and graph the, inequality. Now what we could do or sorry, shade the area. Now in order to test the points, you actually don't have to pick a different point. You could pick the exact same point that we did over here. So what about \( 0, 2 \)? Does this satisfy this inequality? What this is saying is that \( 2 \) is greater than \( 0 + 1 \). Is that a true statement? Well, \( 2 \), in fact, is greater than 1, so this is a true statement. And therefore, we've tested a point, and we're going to shade that inequality. So in other words, for the blue equation, it's actually going to be everything that includes this point as well. So what would we see here? We're going to see that when we shade this equation, we're going to see some yellow, some blue, and some green. You also could have seen this use different sorts of styles of hatching, of, you know, crossing out. So I've seen some people do something like this, and then they'll use markings like this for the blue one. And you'll just find the ones the ones where they overlap. So there's a bunch of different ways to do it. But in this case here, the point is that in some cases or sorry, for this region, these are going to be the points

Graph the system of inequalities and indicate the region (if any) of solutions satisfying all equations.

$3x-2y>6$

$3x-2y$ < $-4$

• A.
• B.
• C.
• D.

Graph the system of inequalities and indicate the region (if any) of solutions satisfying all equations.

$x+y\le4$

$y\ge1$

$x>0$

• A.
• B.
• C.
• D.

Here’s what students ask on this topic:

How do you graph a system of linear inequalities?

To graph a system of linear inequalities, follow these steps:

1. Graph each inequality separately by first plotting the corresponding line. Use a solid line for ≤ or ≥ and a dashed line for < or >.
2. Determine which side of the line to shade by testing a point not on the line. If the point satisfies the inequality, shade that side.
3. Repeat for all inequalities in the system.
4. The solution to the system is the region where the shaded areas overlap, representing points that satisfy all inequalities simultaneously.

What is the difference between a solid line and a dashed line when graphing inequalities?

A solid line is used when the inequality includes ≤ or ≥, indicating that points on the line are part of the solution set. A dashed line is used for < or >, indicating that points on the line are not included in the solution set. This visual distinction helps identify whether boundary points are included in the region that satisfies the inequality.

How do you determine which side of the line to shade when graphing an inequality?

To determine which side of the line to shade, select a test point not on the line (commonly the origin (0,0) if it is not on the line). Substitute the coordinates of the test point into the inequality. If the inequality holds true, shade the side of the line where the test point is located. If it does not hold true, shade the opposite side.

How do you graph nonlinear inequalities like quadratics?

Graphing nonlinear inequalities, such as quadratics, involves similar steps to linear inequalities:

1. Graph the corresponding equation (e.g., y = x² - 1) using a solid or dashed line based on the inequality symbol.
2. Determine which side of the curve to shade by testing a point not on the curve.
3. If the test point satisfies the inequality, shade the region that includes the point. Otherwise, shade the opposite region.
4. The shaded region represents the solution set for the inequality.

What happens if a system of inequalities has no solution?

If a system of inequalities has no solution, it means there is no region where the shaded areas of all inequalities overlap. This can occur if the inequalities represent regions that do not intersect. In such cases, the system is considered inconsistent, and there are no points that satisfy all inequalities simultaneously.