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

Graphical Representation of Linear Inequalities

This concept involves plotting each inequality on a coordinate plane as a region defined by a boundary line and a corresponding half-plane. The boundary line is drawn by converting the inequality to an equality, and the half-plane is determined by choosing a test point. This method is essential in visualizing the solution set of a system of inequalities.

Half-Planes and Boundary Lines

Each inequality in a linear system defines a half-plane on one side of its boundary line. The boundary line represents the equality form of the inequality. Determining which side of the line satisfies the inequality is done by testing a point not on the line. This understanding allows one to correctly represent the area that satisfies each condition of the system.

Feasible Region

The feasible region is the intersection of all the half-planes created by the system of inequalities. It represents the set of all points that satisfy every inequality in the system concurrently. In many optimization and linear programming problems, including this type, identifying the feasible region is crucial as it contains the potential solutions.

Bounded vs. Unbounded Regions

Determining whether the feasible region is bounded or unbounded involves assessing if the region is contained within a finite area or extends indefinitely. A bounded region is completely enclosed by boundary lines, whereas an unbounded region extends infinitely in at least one direction. Recognizing this property is important for understanding the limitations and behavior of the solution set.

Transcript

00:01 All right, people grab those colored pencils because we're going to have some kind of fun because we can use colors.

00:05 All righty, i'm going to do one boundary at a time.

00:11 And my first boundary is x is greater than equal to zero.

00:16 Seems to be habitually our first one, doesn't it? which is going to be a vertical line going down the y -axis.

00:26 And it looks like it's going to be going that direction.

00:31 And let's go to our next.

00:33 One which is y is greater than equal to zero that sounds familiar too if you've done some of the other ones with me so we have oops we don't have that that's for sure come back here you goofy thing all right so it's gonna be a horizontal line going right there and it's greater than so it's going up that direction so let's try our third boundary which is going to be x plus y greater than equal to 2.

01:13 So if x is 0, y is 2, if y is 0, x is 2.

01:21 So at 0 2 and 2, 0, i'm going to draw that line up through here.

01:41 I could go across a little bit more, not you, you.

01:45 You.

01:47 There we go.

01:51 And we have if i put zero in for x and y zero is not greater than two, so it's going to shade this direction.

02:05 We've got the green one shading that direction and we've got the gray one shading this direction so it looks like they're kind of headed towards the same place or a common area.

02:20 So let's try the next one which is 2x plus y greater than equal to 4...

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