Graph each inequality. x

Question

Graph each inequality. x < 3 + 2y

Accepted Answer

Hello, everybody. I hope you're doing night today. Today, we're going to be looking at this question that states draw the graph for the given inequality where the given inequality is X is less than four plus three Y. Now, the first thing I'll note here is that we have a strict inequality. Now, what does this mean when we have a strict inequality is when we have less than or greater than if we were to have less than, or equal to or greater than, or equal to. Those are called non strict inequalities. Now, we have a strict inequality and therefore we will have a dashed graph and I'll show you exactly what this means once we get to actually graphing. So now we'll shift back into focusing with our inequality and I'll change, change the lesson sign with an equal sign. So I'll have X is equal to four plus three Y and I do this just for the sake of starting to solve. Now, I'm going to change this into a different form where I have the Y on one side and the X or I'll have a Y equal to an equation because let's recall really quick. The equation for a line is Y equals M X plus B where M equals slope and B equals Y intercept. And I write Y in Y I N T as Y as an abbreviation for that. So Y equals M X plus B is the equation four line. So we want to get our equation into that form. So to do that, I'm going to move the four over from the right hand side to the left hand side. And I'll have that X minus four is equal to three Y. And then if I divide both sides by three, I'll isolate the Y. So I'll have that Y is equal to one third, X, which is the same as dividing X by three minus four, divided by three. So four thirds. So our equation for our line is going to be Y equals one third, X minus four thirds. Now that minus four thirds is the same as negative four thirds. And that's going to serve as R B or R Y intercept. So, or Y intercept just from the equation on the line will be at zero comma negative four thirds. Now, for the X intercept, we'll do some quick math and I'll use the original formula for this. So I'll have that X is equal to four plus three. It was three, Y. Now, for the X intercept, we plug in zero for the Y. So I'll have three, multiplied by zero and I'll have that X is equal to four because three, multiplied by zero is zero and four plus zero is four. So our X intercept, we'll be at four comma zero. So now we have the equation for our line, our Y intercept and our X intercept. So let's go ahead and start graphing. We have a Y axis and an X axis. And I'll draw our, I'll label the, on the X axis, I'll label the points 123 and four and on the Y axis, I'll do the same thing. But with the negatives, I'll label negative one, negative two, negative three, the negative four. Now I know I'm going to have a point at four comma zero. So I'll put a point there. N R Y intercept with zero comma negative four thirds which is a bit above one or a bit above negative one. So it's, we go past negative one and put a point there and now we can connect those two points. As we said earlier, we have a strict inequality, meaning we're going to have a dashed graph. What does that mean? Well, we're going instead of using a solid line connecting our two points, we're going to use dashed lines across our graph to connect those points and we will extend our graph both to the left and to the right. And this might look a little curved because it's handwritten, but just know that this is going to be a line now because we have an inequality, we know we're going to have to shade in a region because inequalities don't have just one answer. There could be multiple answers for an inequality. So for example, for our inequality, we have X is less than four plus three Y. So it could be any X that is less than four plus three Y, not just one X. So we have to test a point. In this case, I'm going to test the origin zero comma zero. And all I have to do for that is plug in zero for my X and zero for my Y in our inequality. So I have that zero is less than four plus three multiplied by zero. So I have that zero is less than four because once again, three multiplied by zero is zero and four by zero is four. So we have that zero is less than four, which is true, meaning that the origin is part of the shaded region. And if we look at our graph, the origin is above our line, so the region that we have to shade is the region above our line. So we shave that in and just like that, we have been able to graph our inequality. We have a line. The equation for a line that has been dashed because we have a strict inequality and we shade the region above it. So I hope that was helpful. And until next time

Graph each inequality. x < 3 + 2y

Key Concepts

Inequalities

Graphing Linear Equations

Boundary Lines

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