Graph the inequality 2 x + 3 y 2x+3y 2 x + 3 y < 6 6 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 don't 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. And 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.

Table of contents

Skip topic navigation

0. Fundamental Concepts of Algebra3h 32m
1. Equations and Inequalities3h 27m
2. Graphs1h 43m
3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
19. Conic Sections2h 36m
20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

18. Systems of Equations and Matrices

Graphing Systems of Inequalities

Precalculus

18. Systems of Equations and Matrices

Graphing Systems of Inequalities

Previous problemNext problem

Struggling with Precalculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

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

option a

option b

option c

option d

Previous problemNext problem

Verified step by step guidance

Start by rewriting the inequality 2x + 3y < 6 in slope-intercept form (y = mx + b). To do this, solve for y.

Subtract 2x from both sides to get 3y < -2x + 6.

Divide every term by 3 to isolate y, resulting in y < -\frac{2}{3}x + 2.

The inequality y < -\frac{2}{3}x + 2 represents a line with a slope of -\frac{2}{3} and a y-intercept of 2. The line itself is dashed because the inequality is strict (<), meaning points on the line are not included.

Shade the region below the line y = -\frac{2}{3}x + 2, as the inequality y < -\frac{2}{3}x + 2 indicates that the solution set includes all points below the line.