Graph the inequality 2 x + 3 y < 6 2x+3y<6

Welcome back, everyone. So in this practice problem, we're going to graph this inequality that's shown to us, 2x plus 3y is less than 6. So this is clearly a linear inequality. Let's go ahead and just take a look at our steps here. So the first thing we have to do is just figure out whether we're dealing with a solid or a dashed line, and then we're gonna have to graph that line. And remember, we do this by switching this inequality symbol and kind of just pretending that there's an equal sign here. So let's go ahead and do this. I'm actually just gonna write this down here because 2x plus 3y is less than 6 may not be super easy to graph right now because it's in standard form. So one thing we can do here is we can just sort of or switch it into slope intercept form. So to do that, remember, I just have to isolate for y. So I'm gonna subtract 2 x from both sides to basically bring it over. I'm gonna subtract 2 x, and then I have that 3 y is less than negative 2 x plus 6. Now, I just divide by 3 on both sides. Now, unfortunately, this leads to a little bit of a fraction, but that's okay. So we're just gonna have to divide 3 by have to divide every one of the terms by 3. What you'll see here is that we're gonna get y is less than negative 2 thirds x plus 6. So we see that this is kind of like in in slope intercept form, y equals mx plus b. And actually, I'm sorry, this shouldn't be 6 because 6 divided by 3 ends up being 2. So this actually ends up being a little bit easier to graph because all we're gonna have to do here is take a look at the y intercept of 2. We know that we're gonna have a slope of negative 2 thirds, which means we go down 2 and to the right 3. So we go down 2 and to the right 3, and that's basically gonna be our line. And we just connect this, like this. So I just have to connect these points kind of like that. Alright? Now remember, are we dealing with a solid or dashed line? Well, remember, that just comes down to the symbol. And because we just have a less than symbol, not a less than or equal to symbol, we're going to use a dash line over here. All right. So this is going to be a dash line that goes through these points. All right. So this is going to be the equation of our line. That's what the graph looks like. Now we're going to move on to the second step over here, which is testing a point. So we're going to test a point just to make sure on either side of the line. We're just going to basically plug the x y values into the inequality symbol. It's always best to use a point that's on the x or y axis because it makes the math a little bit easier. So one thing I can do here is I'm just gonna test this point over here, which is 0 comma negative one. So 0 comma negative one. So So let's go ahead and do that here. If I plug in the point 0 comma negative 1 into this inequality, what do I get? Well, I'm gonna get the 2 times 0. So I'm just gonna use the the point 0 comma negative 1. I'm gonna get 2 times 0, which works out to 0, plus 3 times negative one. Is that less than 6? Alright? Well, clearly, what we can see here is that this works out to negative 3, and negative 3 clearly is less than 6. That is a true statement. Alright? So now that moves on to the 3rd step. We move on to the 3rd step, which says that if the point makes the inequality true, which is what we had here, then we're gonna shade the side with that includes that point with that point. Alright? So we have that as less than. So, I mean, in other words, it's just gonna be all of this area that's over here. That's gonna be the those are gonna be the the points that satisfy our inequality. You also could have just used the shortcut that for linear equations, remember, once you get it into slope intercept form like this, if it's less than, then it's going to be shading below that line. You also totally could have used that. Alright? So these are the numbers or the points that would satisfy your inequality. Thanks for watching, and I'll see you in the next one.

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Multiple Choice

Graph the inequality $2 x plus 3 y is less than 6$

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Verified step by step guidance

Start with the inequality: \$2x + 3y < 6$.

Rewrite the inequality in slope-intercept form ($y = mx + b$) by isolating $y$: subtract \$2x$ from both sides to get $3y < -2x + 6$, then divide every term by 3 to get $y < -\frac{2}{3}x + 2$.

Graph the boundary line $y = -\frac{2}{3}x + 2$. Since the inequality is strictly less than (<), use a dashed line to indicate that points on the line are not included in the solution.

Determine which side of the line to shade by testing a point not on the line, such as the origin $(0,0)$. Substitute into the inequality: \$2(0) + 3(0) < 6$ simplifies to $0 < 6$, which is true, so shade the side of the line that contains the origin.

The shaded region represents all points $(x,y)$ that satisfy the inequality \$2x + 3y < 6$. The graph with the dashed line and shading below the line is the correct representation.

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