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ZingPath: Systems of Linear Equations and Inequalities

Writing a System of Linear Inequalities that Corresponds to a Graph

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Systems of Linear Equations and Inequalities

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Writing a System of Linear Inequalities that Corresponds to a Graph

Algebra-1

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Students write a system of linear inequalities from their graphs.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Write a system of linear inequalities in two variables that corresponds to a given graph.

Everything You'll Have Covered

Recall that a linear inequality is an inequality involving linear functions. 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. So all of the inequalities in the first row are associated with the linear equation y = mx + b, those in the second row with y = b, and those in the third row with x = a.

Graphically, we can represent a linear inequality by a half-plane, which involves a boundary line. The boundary line is precisely the linear equation associated with the inequality, drawn as either a dotted or a solid line. 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).

For example, in Figure 1, the linear inequality is represented on the coordinate plane. This is an inclusive inequality since it can be interpreted as or meaning we wish to include the equality. Graphically, we represent an inclusive inequality by representing the boundary line with a solid line. A strict inequality, such as would be represented graphically with a dashed or dotted boundary line. Finally, our graph should include the points (x, y) which satisfy the inequality We can determine these points by taking a point on one side of the line and testing its coordinates in our inequality. If the inequality is then a true statement, we shade the half-plane including that point; otherwise, we shade the half-plane that does not include the point. In this example, we can use the origin (0, 0) as a test point. Notice that it is not true that and so we shade the half-plane that does not include the origin.

In order to graph a linear inequality, we can follow the following steps:

  • Graph the boundary line.
  • Determine if the boundary line should be dotted or solid (that is, check whether the inequality is strict or inclusive, respectively).
  • Choose a test point not on the boundary line.
  • Use the test point to determine which half-plane should be shaded.

The use of the test point can be bypassed and last three steps can be summarized with the following for non-vertical boundary lines:

  • If the inequality is of the form y < mx + b then the region below the line is shaded and the boundary line is dashed.
  • If the inequality is of the form then the region below the line is shaded and the boundary line is solid.
  • If the inequality is of the form y > mx + b , then the region above the line is shaded and the boundary line is dashed.
  • If the inequality is of the form then the region above the line is shaded and the boundary line is solid.

Recall that a system of linear inequalities is a set of linear inequalities in the same variables. Consider the shaded triangle in Figure 2. We can use the method described above to find each linear inequality associated with the boundary lines for this region. In this case, our system is:

Tutorial Details

Approximate Time 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.
Course Algebra-1
Type of Tutorial Skills Application
Key Vocabulary graph of linear inequality, linear inequality in two variables, systems of linear inequalities