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Introduction to a System of Inequalities

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Today we're going to talk about systems of inequalities. A system of inequality is just the set of two or more inequalities that have the same variables and are considered at the same time. So on a graph, the solution of a set of inequalities is the intersection of shaded regions or half planes. And this region is bounded by the corresponding equations of the inequalities in the systems of inequalities.

And the points in this bounded region are the set of x and y values that satisfy the inequalities in the system. So we'll do some examples graphing systems of inequalities and show what their solution looks like. So for my first example, I've got a system of inequalities, x is greater than or equal to 4, and y is less than or equal to 1/4x minus 5.

I'm going to show how to create its graph. So the first inequality, x is greater than or equal to 4, is going to be a half plane, or shaded region, that is bounded by the equation x is equal to 4. And the equation x is equal to 4 is a vertical line that goes through 4 on the x-axis. So I'm going to start by plotting a point at 4 on the x-axis and then drawing a vertical line through it.

Now I notice that my inequality symbol is greater than or equal to instead of strictly greater than. So I know that I'm going to use a solid line. And I also noticed that because the inequality symbol is greater than or equal to instead of less than or equal to, I'm going to shade the half plane that is to the right of 4. These numbers are greater than or equal to 4.

For my second inequality, I notice that the equation y equals to 1/4x minus 5 is in slope intercept form. So to start my graph, I'm going to start at my y-intercept of negative 5. And then I'm going to use my slope of 1/4 to get another point. Up one and over four, and now I can connect my points. I again know that because it's a less than or equal to instead of strictly less than inequality sign, I'm going to use a solid line to connect my points.

And then I know that because, again, it's less than or equal to instead of greater than or equal to, I'm going to shade below the line. Now the solution set to this system of inequalities is going to be the overlap of all of the shaded regions for the inequalities in the system. So that will be this region here.

And now any point, xy, that's inside of this shaded region, is going to be a solution to the system of inequalities. And if we were to take that point in the values for x and y and evaluate them into our inequalities, those values would yield true statements for each and every inequality.

So here's my second example. I've got two inequalities that represent a system, y is greater than or equal to 3, and y is less than or equal to negative 2. And I've already graphed both of these inequalities onto this graph. The first inequality, y is greater than 3, represents the region that is above 3 on the y-axis. And the inequality y is less than or negative 2 is represented by this region, which shows the values of y that are less than negative 2, less than or equal to negative 2.

And so these two inequalities, when graphed, do not have any overlapping shaded regions, which means that there is no solution to this system of inequalities. Again, because they have no regions that intersect or that overlap.

So let's go over our key points from today. As usual, make sure you have them in your notes so you can refer to them later. A system of linear inequalities can be used to represent a situation with several boundaries. The solution to a system of linear inequalities fits within all boundaries defined by the system. This is shown on the graph as the region that is overlapped by all shaded region. In any point, xy within this region represents a solution to the system.

So I hope that these key points and examples helped you understand a little bit more about systems of inequalities. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.