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Math Foundations

Students graph linear inequalities in two variables on the coordinate plane.

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

- Graph a linear inequality in two variables using a test point.
- Graph a linear inequality in two variables without using a test point.

Graphing linear equations into variables compared to linear inequalities in two variables

You are probably familiar with graphing a linear equation in two variables. A linear equation is graphed by plotting two or more points on a coordinate plane that make the equation true. If the points satisfy the equation, they follow a straight line.

Consider the linear equation y = 2x + 2, shown on the right.

The ordered pairs graphed on the coordinate plane are:

(-3, -4), (-2, -2), (0, 2), (1, 4), (-4, -6)

All the numbers on the line satisfy the linear equation y = 2x + 2.

Graphing linear inequalities is similar to graphing linear equations, except that the line (called the boundary line for inequalities) may or may not be included in the solution set. Plus, a half-plane is included.

The linear equation is changed to an inequality by changing the equal sign to , resulting in the inequality

The related boundary line is y = 2x + 2. We know the graph of this line. To find which half-plane to shade, identify a test point. (0, 0) is a good point to use, since the calculation will be easier. Substituting the values in the inequality, we find that:

The test point makes the linear inequality false. So, the half-plane that does not include the test point is shaded. If the test point had made the inequality true, the other half-plane would be in the solution set.

Steps to graph a linear equation

1. Identify the boundary line by graphing two ordered pairs.

2. If there is a ? or ? inequality, we use a solid line to graph the boundary line. If there is a < or > inequality, we use a dashed line to graph the boundary line.

3. Choose a test point to determine which half-plane is part of the solution set for the inequality. If the test point is part of the solution set, that half-plane satisfies the inequality. If the test point is not part of the solution set, the other half-plane satisfies the inequality.

4. Shade the half-plane which satisfies the inequality.

Graphing a linear inequality without using a test point

You can determine the half-plane that satisfies the linear inequality without using a test point. Isolate y and then apply the following rules to shade the half-plane:

Horizontal and vertical lines

y > n: horizontal dashed boundary line; shaded above the line

y < n: horizontal dashed boundary; shaded below the line

x > n: vertical dashed boundary line; shaded on the right side of the line

x < n: vertical dashed boundary line; shaded on the left side of the line.

If there is a ? or ? inequality, we use a solid boundary line.

Approximate Time | 30 Minutes |

Pre-requisite Concepts | Students should be familiar with these concepts: coordinate plane, definition of an inequality and a linear inequality in two variables, how to graph a linear equation in two variables, and solving linear equations in two variables. |

Course | Math Foundations |

Type of Tutorial | Skills Application |

Key Vocabulary | inequality, inequality symbol, linear inequality |