Determine graphically the solution set for each system of inequalities and indicate whether the

Determine graphically the solution set for each system of...

Key Concepts

Bounded vs. Unbounded Regions

Bounded regions are those completely enclosed by boundary lines and have a finite area, while unbounded regions extend infinitely in at least one direction. Examining the constraints of the system helps determine whether the overall solution set is confined or extends indefinitely.

Systems of Inequalities

Systems of inequalities consist of multiple inequalities that must be satisfied simultaneously. The solution set to the system is the intersection of the solution sets of the individual inequalities, representing the common area where all conditions overlap.

Boundary Lines

Boundary lines are obtained by replacing the inequality sign with an equality, creating a line that delineates the regions defined by the inequality. They serve as the edges of the solution region and help in identifying on which side of the line the inequality holds.

Linear Inequalities

Linear inequalities involve expressions of the first degree and represent regions in the coordinate plane where a particular condition holds true rather than being equal. They are crucial for understanding how different parts of a graph correspond to the inequality's solution set.

Half-Planes

Half-planes are the two regions created when a line divides the coordinate plane. For a given linear inequality, one of these regions represents the set of points that satisfy the inequality. Determining the correct half-plane is often done via test points.

Transcript

00:01 So we will be solving this system of inequalities that is made of the inequality 2x plus 3y is greater than 12, and 3x minus y is less than 21.

00:11 And we're also going to be graphing the solution to the system of inequalities with its vertices and determining whether or not it's bounded or unbounded.

00:20 Well, to start off, we're going to treat these two equations separately to find two points from which we can draw their respective graphs.

00:28 So first we're going to take 2x plus 3y is equal to 12 and we're going to choose the points where y is equal to 0 and where x is equal to 0 and find the corresponding point.

00:43 So when y is equal to 0, we only have the equation 2x is equal to 12 and x, so x divided both sides by 2 is equal to 6.

00:56 So we have the point 6 comma 0.

01:01 When x is equal to 0, we, we're going to 0, we're going to 2, we're going to 6.

01:03 We have 3y is equal to 12, divided both sides by 3.

01:10 We have y is equal to 4.

01:11 So we have the point 0 .4.

01:16 The next equation is 3x minus y is equal, sorry, equal to 21.

01:29 Now we're going to treat this differently.

01:32 We will be using the point where y is equal to 0, and we're also will be using the point where x is equal to eight.

01:42 These are just numbers that i will get nice points that are just integer numbers.

01:47 So equal to zero, we have 3x is equal to 21, divided in both sides by three, we have x equal to 7.

01:59 So we'll have the point 7 .0.

02:04 Now, when x is equal to 8, 3 times 8 minus y, is equal to 21, we'll have the point 24 minus y, it's what's 21.

02:21 Substructing 24 on both sides will have minus y is equal to minus three.

02:27 And dividing both sides by one, it's just going to be three.

02:30 So we'll have the point 8, 3.

02:33 And with these four points, we can easily graph both of these inequalities.

02:38 Now, we're going to find the vertex or vertices of this system of inequalities.

02:44 And to do so, we're just going to solve for the system of inequalities to find the solution set.

02:50 So the solution set will just be 2x plus 3y, is equal to 12.

02:59 Sorry, the solution set will be the point or points in which these two graphs intercept.

03:09 And those will correspond to the vertices.

03:13 So to solve the system of inequalities, we're going to multiply this side by 3 and this side by 1, to get rid of y to find the x coordinate.

03:28 So doing that, we'll have the equation 2x plus 3y is equal to 12th.

03:37 And we're going to have 9x minus 3y.

03:42 Sorry, that's not a y.

03:46 It's equal to well.

03:48 21 times 3, it's just going to be 63.

03:54 Adding these two together, we're going to have 11x is equal to 5.

04:02 Dividing both sides by 11, we'll have 75 divided by 11, which is roughly 6.

04:16 Now with this number, we can substitute it back into one of the two equations to find out which of this, which will be the corresponding y coordinate.

04:28 Well, we're going to substitute it back into the first one, so it's going to be 2x, which is 2 times 6 .8, plus, 3y is equal to 12.

04:44 So this number right here is just going to be 6.

04:47 2, 2 times 6 is 12, 1 is 13 .6 plus 3y, it's equal to 12.

04:58 3y is equal to negative 1 .6.

05:03 Doing the math, you'll get around negative 0 .5.

05:08 So our vertex, so this is, it's 0 .6 .8 comma negative 0 .0.

05:28 So i'll have summarized of this information into this slide.

05:33 We'll have the inequality 2x plus 3 y is greater than 12...