# Graphing the Solution of the System of Linear Inequalities

This video shows examples of how to graph solutions to linear inequalities. To graph a linear inequality, graph the line as if the ‘greater than’ or ‘less than’ sign was an ‘equal’ sign. If it is ‘greater than’ or ‘less than’, you draw a dashed line. If it is ‘greater than or equal to’ or ‘less than or equal to’, then it is a solid line. Shade the side of the line that satisfies the inequality.

## Graphing Solutions to Linear Inequalities

First, we’re going to graph y is greater than 1/3x, plus 2. To graph a linear inequality, we’re going to graph the line as if this were an equals sign. Then if it’s greater than or less then, it’s a dashed line. If it’s greater than or equal to, less than or equal to, we use a solid line.

This is just greater than, we need to use a dashed line. This is in slope intercept form, y equals mx plus b. Our slope is 1/3, and our y intercept is 2. We’re going to start with our y intercept at 2, and then use our slope to find our next point.

Rise 1, run 1, 2, 3. Then we’ll draw our dashed line, connecting our two points. The final thing we need to do to graph an inequality is to shade. Where do we shade? One way to figure this out is to pick a point on either side of the line, and see if it satisfies your inequality.

If it does, then that is the side you shade. If it doesn’t, then you shade the opposite side. A good point to use is 0,0 because it’s pretty easy to solve for. y is zero, is 0 greater than -and then, again, we plug in 0 for x, one third time 0 is 0, plus 2 is 2.

Is 0 greater than 2? No, it isn’t. Which means since 0,0 does not work and is not a solution, then this is not the side we shade, so we shade the other side. Really everything above this would be shaded, but I’m not going to color the whole board.

This would be your solution to this inequality. Every point in the shaded region would make this inequality true. However, points on the line would not make the inequality true, sense it’s y is greater than, not greater than or equal to.

That’s why this line is dashed, to symbolize that the points that are on this line, the points that lie on this line, do not satisfy this inequality. Now we’re going to graph a system of inequalities. The same rules apply, but where you shade is a little bit different.

First, we’re going to graph y is greater than or equal to 2x plus 3. Since it’s greater than or equal to, we use a solid line. Again, it’s in slope intercept form. Our slope here is 2, or 2 over 1, and our y intersect is 3.

We start with 3, up 1, 2, 3 and then we use our slop rise 2, run 1. Now we can draw our solid line through these points, because again it’s a greater than or equal to. There are a couple of different ways you can do this shading.

One thing you could do is start by figuring out where you would shade for this line. Again, you can use that point 0,0. 2 times 0 is 0, 0 plus 3 is 3. Is 0 greater than or equal to three? No it isn’t.

For this line, we’d be shading on this side of the line, and we can put a little dot here just to remind ourselves, “this is the side we’ll be shading for this line.” Then you can graph your other line and see where you would shade for that one.

Where they overlap is going to be the solution. That’s one way, and I’ll show you another way when we’re done graphing the other line. This next line is y is less than -1/2 x minus 2. Less than, since it’s not less than or equal to, that means we’re going to be using a dashed line again.

Again, it’s in slope and intercept form so our slope is -1/2 and our y intercept is -2. Start with the y intercept, -2, and then use your slope to find your next point. Rise 1 -or actually, it’s, we’re going down 1 since it’s a -1, then over 2.

Again, this is a dashed line, because it’s just less than, not less than or equal to. We’re going to connect our points with our dashed line. Put our arrows on the end, to signify that that line goes on. Now that we have our two lines crossing, we’ve actually created four regions.

We have this region, this region, this one, and this one. For our first line, this region was what was shaded. Now we need to see where we’re shading for our next line. Is it above the line or below this dashed line?

Again, I’m just going to use 0,0 in order to check. -1/2 times 0 is 0, 0 minus 2 is -2, so is 0 less than -2? It is not. This is not where we’re shading for this one, we’re shading down here in this region. Where the two shaded areas overlap would be right over here.

This is your solution, since for this line we were shading on this side and for this line we were shading on this side. The two shaded areas that overlap is just this region right here. Another way you could figure this out, is just to pick a point in all four of the regions.

If the point satisfies both the equations, then that’s where you shade. If your point doesn’t satisfy both equations, then we don’t shade in that region.