On one side lie all the solutions to the inequality. Define Solutions to Systems of Linear Inequalities Learning Outcomes Identify solutions to systems of linear inequalities Graph a System of Two Inequalities Remember from the module on graphing that the graph of a single linear inequality splits the coordinate plane into two regions.

System of Linear Inequalities Two or more linear inequalities grouped together form a system of linear inequalities.

Graphing two-variable linear inequalities Video transcript We're asked to determine the solution set of this system, and we actually have three inequalities right here. This area up here satisfies the last one and the first one.

So the solution set for this second inequality is going to be all of the area below the line. We will verify algebraically whether a point is a solution to a linear equation or inequality. This is a quick way of determining whether the given point is a solution for the system although we will graph this system as well.

So if we were to graph 2x minus 5, and something already might jump out at you that these two are parallel to each other. The boundary line divides the coordinate plane in half. Luckily, graphs can provide a shortcut. That's the graph of y is equal to 2x plus 1. Now we need to shade the regions that represent the inequalities. The points M and N are plotted within the bounded region. The colored area, the area on the plane that contains all possible solutions to an inequality, is called the bounded region. You can check a couple of points to determine which side of the boundary line to shade. A good place to start is just to graph the solution sets for each of these inequalities and then see where they overlap. The general steps are outlined below: Graph each inequality as a line and determine whether it will be solid or dashed. And this is only less than, strictly less than, so we're not going to actually include the line. Solve the inequality If you missed this problem, review Figure. In contrast, points M and A both lie outside the shared bounded region. This is a quick way of determining whether the given point is a solution for the system although we will graph this system as well.Slope is 2 again. You can check a couple of points to determine which side of the boundary line to shade.

So before we even get to this last inequality, in order for there to be something that satisfies both of these inequalities, it has to be in both of their solution sets.

To create a system of inequalities, we need to graph two or more inequalities together. That's the graph of y is equal to 2x plus 1.

And that's the region of the x, y coordinate plane that will satisfy all of them. A system of two linear inequalities is shown below. You can check a couple of points to determine which side of the boundary line to shade. In the following video, we show another example of determining whether a point is in the solution of a system of linear inequalities. Determine whether the ordered pair is a solution to the system. If you doubt that, try substituting the x and y coordinates of Points A and B into the inequality; you will see that they work. In the following video examples, we show how to graph a system of linear inequalities and define the solution region.
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Systems of Linear Inequalities