# How do you write a system of linear inequalities for the shaded region

When the graphs of a system of two linear equations are parallel to each other, we found that there was no solution to the system. In addition, the half-plane involves a shaded portion of the plane either above or below the boundary line or to the left or right of a vertical boundary line.

To create a system of inequalities, you need to graph two or more inequalities together. Systems of inequalities can have no solutions when boundary lines are parallel.

## How to solve a system of inequalities without graphing

Determine which side of each boundary line represents solutions to the inequality by testing a point on each side. In this case, our system is: Tutorial Details 20 Minutes Pre-requisite Concepts Students should be able to write the equation of a line from its graph and vice versa graph a line from its equation , and define and graph a system of linear inequalities. We will verify algebraically whether a point is a solution to a linear equation or inequality. A linear inequality on the plane can have one of the following forms: Notice that all of these linear inequalities have linear equations, which can be associated with them if we replace the inequality with an equality. The general steps are outlined below: Graph each inequality as a line and determine whether it will be solid or dashed. 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. You can verify whether a point is a solution to a system of linear inequalities in the same way you verify whether a point is a solution to a system of equations. Summary Solutions to systems of linear inequalities are entire regions of points. The boundary line divides the coordinate plane in half. In contrast, points M and A both lie outside the solution region purple. Here is a graph of the system in the example above. We can use the method described above to find each linear inequality associated with the boundary lines for this region. Use the test point to determine which half-plane should be shaded. Show Solution Check the point with each of the inequalities. In this case, it is shown as a dashed line as the points on the line do not satisfy the inequality.

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. Determine Whether an Ordered Pair is a Solution to a System of Linear Inequalities On the graph above, you can see that the points B and N are solutions for the system because their coordinates will make both inequalities true statements.

A strict inequality, such as would be represented graphically with a dashed or dotted boundary line. A linear inequality on the plane can have one of the following forms: Notice that all of these linear inequalities have linear equations, which can be associated with them if we replace the inequality with an equality.

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