In Exercises 39–45, graph each inequality. 3x - 4y 12

Question

In Exercises 39–45, graph each inequality. 3x - 4y > 12

Accepted Answer

Hey everyone in this problem. We are asked to draw the graph of the given inequality. In the rectangular coordinate system. We have two, X plus three, Y is less than or equal to 24. And when they say rectangular coordinate system here, you can see in the graphs of these potential answers, we're just talking about the regular Cartesian coordinate system. Okay. All right. Let's scroll down. Give ourselves some more room to work. We're just going to rewrite the equation. We were given her the inequality we were given which was two X plus three, Y is less than or equal to 24. Now, when we're given an inequality like this and we're asked to graph it, the first thing we wanna do is graph the equality. Okay, so we want to graph two, X plus three Y is equal to 24. And what you'll see in our inequality is that we have two, X plus three. Y is less than or equal to 24. So this equality, this is going to be included in our inequality. Okay, this is included. Alright, so if we look at this equation, we can see that this is a linear equation. This is just the equation of a line. Okay, we can write, this is why equals stuff. Um We'll go ahead and do that because that's how most people are familiar with graphing lines. Um But if you want to go right from this, if this makes more sense to you go ahead and do that. Okay, so this is going to give us three, Y is equal to 24 minus two X. Okay. And then dividing by three, Y is equal to eight minus two thirds X. Okay, so that's a line of that standard form that we're used to seeing we know how to graph that. Okay, let's find the X and Y intercepts. That's going to help us with their graph. So the X intercepts occur when y is zero. Yes, we have zero is equal to eight minus two thirds X. And this is going to give us X Is equal to 12. So we're gonna have the .120 included. And then we have our y intercept, this is when X is zero and we get Y is equal to eight. So we have the 80.8 included as well. Alright, so going over to our graph, It's 12 which is going to be handy because that's our x intercept. And then for our y axis this is eight. Alright, so we know we have an X intercept at 12. So we have the 120.12 0 which is way over here on the right hand side. Okay? And then we have the 00.8 or y intercept is eight. So we have 08 and we can see the coefficient in front of the X. Is negative. So we have a negative slope. So this makes sense. We can connect these two points through this line like this. Alright, so this is our line two X plus three, Y is equal to 24. And what you'll notice is that line has cut the plane into two. Okay we have this portion kind of on the left or below the line and then we have the portion on the right or above the line over here. What we want to figure out is what side of that plane is included in our inequality. So in our case we have two X plus three Y. Less than or equal to 24. Okay now we've done all of this work to rearrange this. Okay Using an equal sign. But what if we have it as an inequality? Okay so let's just right over this. We're gonna erase these equal signs for now and we're going to do it as an inequality. Okay and you're gonna see what happens. So we have less than or equal. Okay we move the two X. Over. This stays less than are equal. We divide by three. This stays less than or equal. Okay so we want why to be less than or equal to this line that we drew. If we want why to be less than that then we want to be below this line. This is where all the Y. Values are less than what we've drawn. Okay so we're gonna be on this side of the plane Now. Another way you can do this sometimes that can be a little bit confusing. So another way you can do this is pick a particular point. So say we choose the zero. If we were to put 00 into our original inequality, we have zero plus zero on the left hand side. That just zero we have zero is less than or equal to 24. Well that's true. That's a true statement that satisfies the inequality. So that means that point is going to be included in our inequality. Okay, so this part of the plane is the one that we shaded. Okay. Alright. So there we go. That's the graph of our inequality. And let's go back up and look at our answer choices. We know that we're on the kind of bottom or left side. So we're looking at either answer choice B or D. Now we just need to pay attention to our intercepts. Okay? We had a y intercept of eight and x intercept of 12. And so we see that we have answer choice B. That's it for this one. Thanks everyone for watching. See you in the next video

In Exercises 39–45, graph each inequality. 3x - 4y > 12

Key Concepts

Inequalities

Graphing Linear Inequalities

Slope-Intercept Form

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