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

step 1 All right, number 50. Get your colors out, people. Here we go. So, oh, guess what? The first one

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

Key Concepts

Bounded and Unbounded Regions

A region is bounded if it is completely enclosed by the constraint lines, meaning that there exists a finite area that contains the entire feasible region. In contrast, an unbounded region extends infinitely in at least one direction and does not have a finite area. Determining whether the solution set is bounded or unbounded is crucial in understanding the limits of the possible solutions.

Feasible Region

The feasible region is the set of all points that satisfy all the inequalities in the system. It is formed by the intersection of all the individual half-planes and represents all possible solutions to the problem.

Systems of Linear Inequalities

A system of linear inequalities consists of more than one inequality involving linear expressions. Each inequality represents a half-plane in the coordinate system and the solution to the system is the intersection of these half-planes, including any restrictions such as non-negativity of variables.

Graphical Representation

Graphical methods involve plotting the boundary lines of the inequalities on a coordinate plane and shading the appropriate side for each one. This visual approach allows one to see the region where all conditions are simultaneously satisfied, making it easier to understand the relationships between the constraints.

Transcript

00:01 All right, number 50.

00:02 Get your colors out, people.

00:04 Here we go.

00:08 So, oh, guess what? the first one is x is greater than equal to zero.

00:15 And it is a, say it with me, vertical line down the y -axis, and it is going to be shading that direction.

00:26 The second one, no big surprise to all of us who've been doing several of these now, is that y is greater than equal to zero.

00:36 But what's still a surprise to me is why this just randomly changes colors on me.

00:41 I don't know.

00:43 I guess it's just doing that to break the boredom.

00:49 And this is a horizontal line across the x -axis.

00:58 And our next boundary is x plus 2, sorry, x plus y, greater than equal to 2.

01:17 So if x is 0, y is 2, if y is 0, x is 2, so 0 to 0 to 0.

01:28 Did you notice i said those backwards from how i did them? just making you pay attention.

01:34 Here you go.

01:35 So we're going to go across here.

01:43 We do want to check our shading is 0 greater than 2.

01:49 New.

01:50 So we don't shade 0, we're going to go that direction.

01:55 Your next one is x plus y less than equal to 8.

02:04 So if x is 0, y is 8.

02:10 If y is 0, x is 8.

02:13 And 0 is less than equal to 8.

02:18 So we're going to shade towards 0 .0.

02:20 So let's do 8.

02:23 And 08.

02:26 Looks like i've got parallel line.

02:28 Again don't i? did you notice that? remember if the a's and the b's both match then you've got parallel lines which means those lines are going to cross but the shaded could and it looks like the shading does so it's going to go this direction.

02:51 Now those look exactly like the nuns on number 49 don't they but the odd thing here is this one's new x plus 2y greater than equal to 1 so if y is 0 then x is 1 then x is 1 and i put a 0 in for x then i get a 1 half for y which i don't want to do that so i'm using my slope is negative 1 over 2 so i'm going to do 1 .2 so i'm going to do 1.

03:38 I'm going to go down one over two to get my slope on the other one...

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