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 Solution Method

This method involves plotting the boundary lines of the inequalities on a coordinate plane, determining which side of each line satisfies the corresponding inequality, and then identifying the common region shared by all the inequalities. This visual approach aids in understanding the structure and limits of the solution set.

Feasible Region

The feasible region is the set of all points that satisfy every inequality in a system simultaneously. It is derived by finding the intersection of the half-planes and represents the potential solutions that meet all the given constraints.

Bounded and Unbounded Regions

A bounded region is completely enclosed and lies within a finite area, while an unbounded region extends infinitely in at least one direction. Recognizing whether a feasible region is bounded or unbounded is essential, especially in optimization problems and in understanding the constraints of a system.

Linear Inequalities

These are mathematical expressions that relate linear functions with inequality signs instead of equality. They establish regions of the coordinate plane that satisfy the inequality rather than a single line, forming the basis for determining solution sets graphically.

Half-Planes

Each linear inequality divides the coordinate plane into two halves: one where the inequality holds true and one where it does not. These regions, called half-planes, are fundamental in understanding how multiple inequalities overlap to form a solution set.

Transcript

00:01 So the problem that we're resolving asks us to find the solution set for the system of inequalities that is made of the equations for x plus 3y is less than or equal to 18.

00:13 And 2x plus y is less than or equal to 8.

00:16 Also, we have the restrictions that x should be greater than 0 and that the y must be greater than 0, sorry, both of them.

00:27 So the first step would be to take these two equations right over here.

00:31 And solve for them, since these are just straight lines that go either horizontally or vertically.

00:39 So for this system of equations, we're going to take 4x plus 3y and set them equal to 18.

00:51 And as we know, this will be just a line that will define the region that will satisfy this inequality.

01:00 And the other equation is 2x plus y is equal to 8.

01:06 And the reason to do this is because we're going to add them by multiplying this lower part of the equation by minus 3 and the upper part times 1.

01:21 And then we're going to be adding them together.

01:24 So the first one will stay the same as 4x plus 3y is equal to 18.

01:34 The bottom one though, it's going to be minus 6x, sorry, minus 3y.

01:44 Is equal to minus 24, negative 24.

01:52 So as you can see, these two terms, the y terms, are going to be canceling out since they add up to zero.

01:59 Then we'll have negative 2x equal to negative 6.

02:06 We'll divide both sides by negative 2 to obtain the x value in which these two lines intersect, and that it's going to be just 3.

02:20 Next up, we're going to substitute this value of 3 into one right here.

02:28 So we'll have for x is equal to three, for simplicity, i'm just going to substitute it back in this equation over here.

02:37 It's going to be 2x, 2 times 3, which is the value of x plus y is less than or equal to 8.

02:47 Sorry, it's equal to 8.

02:52 So in this case, we'll have 6 plus y is equal to 8.

03:00 So we know that by subtracting six on both sides, we're going to get that y is equal to two.

03:07 So the intersection, these two lines, it's at the point three, oh, it's at the point three, comma two, these two lines intercept.

03:27 And this is also one of the vertices, also a vertex of the solution set, since this is where these two will be bounded, each other.

03:42 Finally, i'm going to take two points to make a sketch of the graph.

03:48 So for the i'm going to use in red to identify it, we're going to substitute y is equal to zero and x is equal to zero.

04:03 And this will give us two points.

04:04 So we know that it exists on a line, the y intercept, sorry, the x intercept and the y intercept for every line.

04:13 So this will help us to make a sketch, an accurate sketch of the graph.

04:19 So for the first one, we'll have 4x is equal to 18, since 3 times 0 is 0, then we'll have 18 over 4.

04:37 And this is rough, and this is 4 .5.

04:44 So we'll have the point 4 .5, comma 0.

04:50 For x is equal to 0, we'll have the y intercept.

04:55 So in this case, we'll have 3y, is equal to 18 and this is y is equal to 6.

05:06 So in this case we'll have the point 6, sorry, 0 .6, 0 .6.

05:14 And this will be the points that we'll be using to graph this first inequality right here.

05:19 Next, i'm going to use green and do the same as we did with this first inequality.

05:27 So for y is equal to 0 and x is equal to 0.

05:33 And remember that you can choose any point that you want for x, but these two points are just the easiest ones to obtain.

05:41 So it's just a matter of how do you want to do the problem.

05:46 So it's 2x plus y is equal to 8.

05:50 For y is equal to 0, we'll have 2x is equal to 8 and have that x is going to be equal to 4.

05:59 So we'll have the point 4, 0.

06:05 When x is equal to 0, we'll have that the y, it's just going to be equal to 8.

06:10 So we'll have the point 0 .8.

06:16 Next, we're going to be doing a sketch of this inequalities.

06:20 For this reason, i'm going to change pages.

06:23 So i have summarized the information that we have already obtained in this corner right here.

06:29 So we have that the inequality for x plus 3y is less than equal to 18, and we'll be using the points 4 .5 .0 and 0 .6 to graph it.

06:38 The second inequality, 2x plus y equals less than or equal to 8.

06:44 We'll be using the points 4 .0 and 0 .8.

06:48 And finally, we'll have the point, the line x is greater than 0 and y is greater than 0.

06:56 So i'm going to identify this graph, the line where this inequality is red, and this one is blue.

07:07 Sorry, green.

07:08 So for the first one, we have the point 4 .5 .0.

07:11 And it's just here, 4 .5 .0.

07:16 Then we have the point 0 .6, so it's over here...

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