In Exercises 1–26, graph each inequality. x+2y≤8

Question

Accepted Answer

Hey everyone in this problem we are asked to graph the following inequality and were given two X plus four, Y is less than or equal to six. So let's just move down and give ourselves some room to work this out, we're gonna rewrite that inequality. We were given two X plus four, Y. Is lessen or equal to six. Now when you look at this inequality you'll notice that we have an X. Term of degree one, we have a wide term of degree one and we have this constant term on the right hand side. And these are the three pieces that we have when we write a standard equation of a line, so recall that the standard of equation of a line is Y equals mx plus B. Okay we have each of these three pieces. So what I like to do when I get in inequality like this is rewrite the inequality to be kind of of the form of this standard equation of a line with the inequality instead of the equality. And that just makes it easier when we're drawing the graph to figure out which portion of the plane or which portion of our graph. We should include with our inequality shade in in which portion we don't. Okay, so I'm gonna go ahead and rewrite it. To look kind of like the standard equation of a line and you'll see when we do the graph why that helps. So we're gonna move the two X to the right hand side, we get four, Y is less than or equal to six minus two X. And then we divide by four. We get y is less than or equal to 6/4 minus 2/ X. And if we simplify those fractions, we're going to get why is less than or equal to three halves minus one half X. Okay. So it looks a little bit like this standard of equation of a line where we have Y on one side and everything else on the other side. Now, when we're asked to graph an inequality, what we want to do first is graph the equality. Okay, that's gonna be a line in this case we're gonna graft that boundary. It's gonna break the plane up into two pieces and then we can figure out which side of that line is included in our inequality that we should shade in. So if we're graphing this as an equality, we have Y is equal to three halves minus one half X. Now, in our standard equation of a line, the coefficient in front of the X is going to be our slope. Okay, so this negative a half. So in our case the slope of our line is negative a half. Ok. And that's just gonna help guide us when we do our dry, let's go ahead and find the X and Y intercepts. That's going to give us potentially two points and then we can connect those points in our graph. So if we're looking for the X intercept, this is going to occur when y is equal to zero. When y is equal to zero, we get three halves minus one half X. So you get that one half X is equal to three halves or X is equal to three. So we get the point 30. Okay. Which we can add to our graph and then we're gonna do the same for the Y intercept and that occurs when X is equal to zero and when X is equal to zero, we get that Y is equal to three halves. And so we get the 30.3 halves. So we have two points. We know our slope. So let's go ahead and draw our graph. So we have the zero and 3/2ves we can say is here, This is gonna be 3/2ves And then we have the .30. And we can say that that's about here and this is going to be three on our x axis. Alright, We have a negative slope. So it makes sense if we connect these lines or these two points, we're going to get this negative sloping line. Now, before I draw this line we have to figure out do I draw it as a solid line indicating that we're including in our inequality or do I draw it as a dotted or dashed line? Now, if we look at our inequality, we have that, why is less than or equal to Originally we had less than or equal to. So because we have that less than or equal to, we want to include the line. Okay, so we're going to draw it as a solid line. We're going to connect these two points and kind of continue our line out like this. Okay. And this is the line Y equals three halves minus one half X. Just kind of like our boundary. Now we need to figure out if we include what's kind of above this line or what's below this line and rewriting it as kind of an equation of a line helps here. Okay, so we have why is less than or equal to this line? If we want Why less than or equal to this line. That means we want why? Below it. Okay. The y values that are less than something on our graph are going to be below it. So we want everything below this line. Okay, so you can see why rewriting it as that equation of a line. Kind of helps at this point. Now, if that's confusing or you ever get stuck doing that, what you what you can do is test a point. So say you choose the zero. If you choose the zero on the left hand side of this original inequality, you're gonna have zero on the right hand side, you have six. You have zero less than or equal to six. That's a true statement. Okay. Which means that it satisfies the inequality. So that point is included. So if the zero is included, that means that this part of the plane that includes 00 is included. And then you can test a point on the other side as well if you wanted to. Okay. So that's always you can always check particular points if you get stuck or even if you just want to double check your work, that's a great way to do it. Alright? So if we go back up to our answer choices and we want to compare it to that sketch we did below and we found that we had a line with a negative slope as our boundary case. We're looking at option A or B. C and D. Have positive slope which don't look like the line we drew. And then we're just looking at what part of the plane we included and what we found was that everything below the line was going to be included in our sketch or in our inequality. And so we have option a answer A looks like the sketch we drew below. Thanks everyone for watching. I hope this video helped see you in the next one

In Exercises 1–26, graph each inequality. x+2y≤8

Key Concepts

Graphing Linear Inequalities

Slope-Intercept Form

Testing Points to Determine the Solution Region

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