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 the boundary line of a linear inequality on the coordinate plane, and then shading the region corresponding to the inequality. The boundary is typically plotted by converting the inequality to an equation, and the appropriate half-plane is determined by testing a point that is not on the line. This graphical method provides a visual means of identifying all the points that satisfy the inequality.

Half-Planes

A half-plane is one of the two regions into which a straight line divides the plane. When dealing with linear inequalities, each inequality represents a half-plane, which includes all the points on one side of the corresponding boundary line. The boundary line can be either included (if the inequality is non-strict, like ? or ?) or excluded (if the inequality is strict, like < or >) from the solution set.

Intersection of Systems of Inequalities

In a system of linear inequalities, the solution set is the intersection of the half-planes represented by each inequality. This intersection is the set of all points that simultaneously satisfy all the inequalities, which can be found by graphically overlaying the individual half-planes. The overall solution region is the area where the shaded regions of all the inequalities overlap.

Bounded and Unbounded Regions

A region in the coordinate plane is termed bounded if it is enclosed within a finite area, meaning that it can be contained within a circle of finite radius, while an unbounded region extends indefinitely in some direction. In the context of linear inequalities, whether the solution set is bounded or unbounded is determined by the configuration of the boundary lines and the overlapping half-planes.

Transcript

00:01 So here we have x is greater than or equal to 0, y is greater than or equal to 0, x plus 2y is greater than or equal to 1, and x plus 2y is less than or equal to 10.

00:16 And this is our system.

00:19 So the first thing we will do is look for our equations of the lines, and this will give us x is equal to 0.

00:27 This one will give us y is equal to 0.

00:32 And this, i must subtract this x to the other side, and this will give us y is greater than or equal to negative x plus 1.

00:42 Dividing by 2 to both sides will give us y is greater than or equal to negative 1 half x plus 1 half.

00:50 And our line is going to be y is equal to negative 1 half x plus 1 half.

00:56 And we want to do this because we want the lines to be of the form, y is equal to mx plus b.

01:06 And next we are going to do the same thing here.

01:09 2 .y is less than or equal to negative x plus 10.

01:14 Dividing by 2 to both sides will give us y is less than or equal to negative 1 half x plus 5.

01:23 And our line is y is equal to negative one half x plus five and with this we can start grapping first we will have our x is equal to zero then our y is equal to zero then our y is equal to one half negative one half x plus one half so this is our so it has a y intercept at one half.

02:28 And i'm just going to copy and paste the same line since it has the same slope of our yellow line...

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