Graph each inequality. x + 2y ≤ 6

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Hello, today we're gonna be drawing the graph for the given inequality. So what we are given is two X plus three Y is less than or equal to 12. Now, currently, our inequality matches the form of the equation of a line in standard form. See if we were to replace the inequality sign with an equal sign, we can make the equation two X plus three Y is equal to 12. In this case here, in order to get the equation of the line, we will want to solve for our Y variable. And since our inequality matches this form, we can use the same methods to solve for Y in our given inequality. So let's go ahead and solve for Y. So we can rewrite our inequality in the form of a line. So the first thing installing for Y is going to subtract two X to both sides of the inequality. That is going to leave us with three Y is less than or equal to negative two X plus 12. Then in order to solve for Y, we're gonna want to divide every term into inequality by three that is going to leave us with Y less than or equal to negative 2/3 X plus four. Now, if we were to rewrite this using an equal sign, our inequality matches the form of the equation of a line because we get Y is equal to negative 2/3 X plus four. Now, in this case here, our Y intercept is going to be four and our slope is going to be negative 2/3. So let's go ahead and graph for inequality by first plotting this information. So we know that the Y intercept or the starting point of the line is going to be at zero comma four. And we're going to have a negative line because we have a negative slope, we have negative 2/3, which means that the next point in the line is gonna be two units below the Y intercept and three units to the right. So roughly about here and this next point is gonna be located at three comma two. Now we can go ahead and connect the two dots. But before we do that, we need to ask ourselves for the inequality, do we want to include the points that are along the line? Well, in this case here, because the inequality uses the less than or equal to sign that means that we want to include the points that lie along the line of the graph. So with that being said, we can connect these two dots by using a solid line. And this is going to be the rough shape of the equation of the inequality. Now, we need to ask ourselves, do we want to include the solution set that is below the line or above the line? Well, in order to confirm that we can pick a testing point either below or above the line. And if it satisfies the inequality, then that is going to be the region that we shade the easiest testing point to pick here is going to be the origin zero comma zero. And if we were to plug in zero comma zero into our inequality, what we get is zero less than or equal to negative 2/3 times zero plus four, anything times zero is going to be zero. So this is going to cancel out leaving us with zero less than or equal to positive four. Now this is a true statement. And since our testing point satisfies the inequality, that means that we're going to shade the region that is below the line. And with that being said, this is going to be the graph for the given inequality. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Graph each inequality. x + 2y ≤ 6

Key Concepts

Inequalities

Graphing Linear Inequalities

Slope-Intercept Form

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