Graph the system of inequalities and indicate the region (if a...

Question

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

Graph showing two linear inequalities with shaded regions indicating the solution area where both inequalities overlap.

Coordinate plane with x and y axes labeled, grid lines from -5 to 5 on both axes, no plotted points or shading.

Accepted Answer

In this next practice problem, we're going to put together all the steps in graphing a system of inequalities. Let's get started here. We're given these two inequalities and the problem asks us to graph the system and indicate the region, if there is any, of the solutions that satisfy all of the inequations or inequalities. So basically, we're gonna find the one solution region for this system. Let's go ahead and get started here. The general idea is that for each of the inequalities, and we have two of them here, we're gonna use different colors or shading styles, but we have to graph the line by pretending there's an equal sign, and and then we have to figure out which direction to shade on either side of the line. Then when we're done with all the inequalities, we just shade the overlap of all the regions, and that's our answer. If there is none, then there is no solution. Alright? So we've got our two inequalities here. Y is greater than or equal to two x minus two in red, and In blue, we've got y is greater than three minus x. Alright. So let's get started with our first inequality here. We're going to graph a solid or dashed line for y is greater than two x minus two by basically replacing the inequality with an equal sign. So what would y equals two x minus two look like? Well, first of all, is it going to be solid or dashed? We know the rule. If it's a less than or equal to greater than or equal to, it's going to be solid. Anything else is going to be dashed. So this is a greater than or equal to symbol. So it's going to be a solid line and it's going to pass through the y intercept of negative two and it's going to have a slope of positive two. So that just means that if find your next point after y equals negative two, you're going to go up two and then over one, right? So that's the rise over the run. The next point is going to be here. Rise over run is going to be over here. Why rise of run is going to put the next point over here, so on and so forth. And then you connect all these lines these points to form your line. Alright? So that's basically what this inequality looks like, but now we have to figure out where to shade. And in order to do that, let's take a look at our next step. So we're gonna either test the points, on one side of the line by plugging the x y into the inequality. There are shortcuts to use or just go ahead and go through this step over here. Again, I like to use the point zero comma zero, or you can just use a point on the x or y axis. Zero comma zero works perfectly fine here because this inequality doesn't pass through that point. So we just go ahead and plug it in. So you remember what this looks like. We plug in zero comma zero. This just kind of becomes like a question. Is zero greater than or equal to two times zero minus two? We're just plugging in the XY into the inequality and seeing if this is a true statement or not. So in other words, is zero greater than or equal to when two times zero cancels out? Is zero greater than or equal to negative two? Is that a true statement? It is. So because it is a true statement, we're going to move on to the third step. So if the point makes that inequality true, we lightly shade the side with that point. So in other words, this point over here made this inequality true. So we're going to shade this entire area over here. I'm going to do it kind of slowly. If you have some, like, colored pencils or if you want to just use some, like, hash marks or something like that, that's perfectly fine. But basically, that is going to be this region right over here in red. Alright. You can kind of tidy it up a little bit, but that's pretty much all there is. Alright, so now we move on to the second one, the second inequality, because we're pretty much done here. So for this blue inequality, notice that it's not exactly in y in your slope intercept form. It's y is greater than three minus x. So I like to often manipulate these inequalities so that they're in y intercept form, slope intercept form because they're easier to graph. So if I sort of manipulate this, this is going to be y is greater than negative x plus three. Now it's sort of in slope intercept form, so we can go ahead and just graph it a lot easier. So this is going to be an inequality that passes through the y intercept of positive three. And then what we also can tell here is it's going to be a dashed line or a dotted line because it has a greater than symbol. All right. It's going to have a slope of negative one, which means that the slope of the rise over the run is going to be down one over to the right one, down one to the right one, down one to the right one, and then so on and so forth. And if you go ahead and just connect those points, it should look something like this. Alright, so that's going to be the inequality. Now, again, you could use your shortcuts here or you can go ahead and plug in a test point. So what I can do over here is I can just use the exact same test point I used over here with with zero comma zero to see if this inequality is true. So zero comma zero, just testing that point is zero greater than negative zero plus three. So in other words, is zero greater than positive three? That's actually not true. So because that is not a true statement, so we're going to move on to our third step. We've already done steps one and two. This just means that because our statement made the inequality false, we're going to lightly shade the side that does not include that point. In other words, what we see here is that it's going to be this shaded region over here. It's going to be basically above the line. All right. Now we can see over here that we've already kind of have step four that's already done for us because we've done this with different shading styles and colors. We see here is that the overlap of all the regions is going to be the area that is right over here in this little sort of triangle like this. This is where the red and the blue overlap to form this sort of like purple color. But basically, this is going to be you can shade this a different color if you would really like to, but this is going to be the region that satisfies all of the inequalities. That's all there is to it. Thanks for watching. Let me know if you have any questions.

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

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