In Exercises 46–55, graph the solution set of each system of ineq...

Question

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.

This

is a piecewise function. Refer to the textbook.

Accepted Answer

Hey everyone in this problem, we are asked to consider the given system of inequalities and indicate if it has a solution by graphing it. Okay, so we have two X plus Y is greater than or equal to four and three X minus Y is greater than or equal to nine. Alright, so let's go down and give ourselves some space to work and we're just going to rewrite those inequalities we were given so we can refer to them. So we had two X plus Y Is greater than or equal to four And we had three, X -Y is greater than or equal to nine. Alright, so the first thing we're gonna do is we're gonna graph these and we're gonna graph the equality first. Okay, so put in an equal sign and then graph and what you'll see is that these are linear equations when we do that. We have a Y and an X. And most people are most familiar with graphing linear equations. When we have y equals something. Okay, that standard equation of a line. So let's go ahead and rewrite our inequalities to be why? On one side, everything else on the other. Okay, so we do that with the first one we're going to get why is greater than or equal to 4 -2 x. And on this one. Okay, the wire's gonna move to the right hand side, Which means that we have y less than or equal to three X -9. Okay. Just be careful which way that inequality goes when you're moving things around. Alright, so we have our two inequalities here and again, when we're graphing, we're gonna start with graphing the equality. Okay, so we're going to graph Y is equal to four minus two X. And we're going to graph Y is equal to three X minus nine. Okay? And then we're gonna figure out which side of that line. Okay? The line is going to cut the plane into two, and we're gonna figure out which side of that plane is included in our inequality. So this first inequality we're going to do in blue and the second will do in red cages, so we can see the difference. All right, let's just go ahead and get our X. And y intercepts. That's gonna help us graph these lines. Okay? So for the first one the x intercept is going to occur when y is zero. Okay? We get zero is equal to four minus two, X. Which gives us X equals two. Okay, So we have the 20.20. And then our y intercept is going to be when X is zero. Okay? And we get y is equal to four. So we have the zero. Similarly for the other equation, the x intercept when y is zero, we get zero is equal to three X minus nine, so X is equal to three. Yes, we have the 30.30 and for the y intercept we have y, we have X is equal to zero. We get y is equal to negative nine. So we have the 90.0 negative nine. Alright, so let's go ahead on the right hand side here and draw this out. 78,910. 12345 nine, negative X. And our Y axis. Alright, so let's start with the blue. Okay, we're gonna start with the line. Y equals four minus two X. Okay, we have the 20.20. Who's going to be here? And we have a .04 which is going to be here. Okay. And this has a negative slope. Okay, we have a negative coefficient in front of the X. So this makes sense. We connect these lines and we get this line here with negative slope. Alright, this is the line y is equal to 4 -2. Now we need to figure out which side of the plane is included in our inequality. What we have is why is greater than or equal to this? For -2x. So if Y is greater than or equal to we want to take all the plane above this line. Okay. So we're gonna take the plane on the side of the one. This is all gonna be included. Okay and if you get confused about greater than less than where we should be including the plane, what you can do is choose a particular point. Okay, so you could choose the zero. If you put that into the inequality you get zero is greater than or equal to four? That's not true. So that means that the zero is not included in our inequality. Okay, that means that this side of the plane isn't included and we instead include the other side of the plane. Okay. Alright. So let's do the other one now. So we have the .30, which is going to be here. We have the 0.0 negative nine which is going to be here. We have a positive slope. So this makes sense. We have this line connecting those two points like this. Okay. And this is the line why is equal to three X minus nine? And again we have to figure out which part of the plane we include in this case we have why less than or equal to three X minus nine. Okay. So we want to be below this line. So we're gonna take everything here Below this line. Okay. Alright. And again, if we were confused and we didn't know which to include, we can choose a point. Okay, so choose 00. You get zero is less than or equal to negative nine. We know that's not true. So we know 00 is not included. All right. Now, if we're considering the system of these inequalities. Okay. We want to find points that satisfy both. Okay, so if we're looking at it graphically we want to find points that are red and blue. Okay, so those are the points that satisfy both of those inequalities. So what we see is that between this red line in this blue line? Okay. We have both red and blue. So this we're gonna draw in green, get a little bit messy right now, but we're gonna draw in green and this is gonna be the portion that is going to be a solution to that system. Okay, it solves both of those inequalities. Alright, so we have this empty spot here on the left hand side and then this overlap on the right hand side with these triangles at the top and bottom. We're just one of the inequalities holds true. So if we go back up to our answer choices, we see that our diagram looks like the one in a Okay, we have this empty space on the left hand side, the overlap on the right hand side and these two triangles where just one inequality holds true on the top and bottom. So the answer here is going to be a thanks everyone for watching. I hope this video helped see you in the next one

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In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.
This
is a piecewise function. Refer to the textbook.