Graph the solution region for each system. and indicate whether each solution region is bounded

Graph the solution region for each system. and indicate...

Key Concepts

Corner Points (Vertices)

Corner points, or vertices, are where the boundary lines of the inequalities intersect. They are significant because they often represent key values in optimization problems, especially those involving linear programming where optimum solutions occur at these vertices.

Feasible Region

The feasible region, also known as the solution region, is the common area where all the inequalities in a system overlap. This region represents all the possible solutions that simultaneously satisfy every constraint imposed by the inequalities.

Bounded and Unbounded Regions

A region is considered bounded if it is contained within a finite area; conversely, an unbounded region extends infinitely in at least one direction. Determining if a feasible region is bounded is crucial in understanding the limits of possible solutions.

Graphical Representation of Inequalities

Graphing linear inequalities requires drawing the boundary line (or line of equality) and then shading the correct half-plane that satisfies the inequality. This visual process helps in identifying regions that meet all given conditions in a system.

Linear Inequalities

Linear inequalities involve algebraic expressions with a 'greater than', 'less than', or equal component and are represented graphically as half-planes. Understanding these inequalities is fundamental for illustrating constraints and solution sets in a coordinate plane.

Systems of Inequalities

A system of inequalities is a collection of two or more inequalities that must be simultaneously satisfied. The overall solution is the intersection of their individual solution regions, which is pivotal in problem-solving scenarios like optimization and feasibility studies.

Transcript

00:01 Okay, for this problem, you need to take a system of linear inequalities and plot them and find the solution region.

00:07 Okay, so we're going to start off with our first inequality, which is x plus 2y, is greater than or equal to 8.

00:14 This is in standard form, and so it's easiest to use intercepts to graph this.

00:20 Of course, if your instructor allows you to use the calculator for this, or desmos, then you can certainly do that as well.

00:26 So if x is equal to zero, that would lead me with simply 2y, and we're going to put an equal sign here because we're just interested in the line, not the shaded part, which is what the inequality is.

00:36 And that leads me with y equals 4.

00:38 And so that's a intercept of 0, 4.

00:42 Now if i set y equal to 0, i would get x equals 8.

00:47 And so that intercept would be 8 comma 0.

00:50 So that's our first line.

00:51 Let's plot those points.

00:54 04.

00:55 1, 2, 3, 4, 5, 6, 7, 8.

00:58 There's 8 .0.

01:00 We're going to connect those with a line.

01:03 Okay? now, the shading is going to be above that line because here's your greater than right here.

01:09 Okay, that has a greater than.

01:10 So it's going to be above the line for the shading, but we'll get to that in just a little bit.

01:13 Let's go to the next one.

01:15 I'll use red for this one.

01:16 And we've got 2x plus y is greater than or equal to 10.

01:23 So we're going to set x equal to 0.

01:24 That leaves me with simply y equals 10 so the intercept is 0 .10.

01:30 Y equals 0, that leads me with 2x equals 10.

01:34 So x equals 5, which is then 5.

01:38 Let's plot that.

01:40 0, 10, so here's 4, 5, 6, 7, 8, 9, 10, and then 5 ,0, 1, 2, 3, 4.

01:50 So there's that connection.

01:55 Okay? now, the last two are just the x and y axis, because we have x is greater than or equal to zero, and y is greater than or equal to zero.

02:05 And all that means is that it has to be to the right of the y axis and above the x -axis, because they're both in the positive regions, which leaves me in quadrant one.

02:16 And so now we look back at these two...

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