# 4.4 2D Inequality and Absolute Value Graphs

# Graphing a 2D Inequality

To graph an inequality, borrow the strategy used to draw a line graph in 2D. To do this, replace the inequality with an equal sign.

Example 4.4.1

Consider the following inequalities:

All can be changed to by replacing the inequality sign with =.

It is then possible to create a data table using the new equation.

Create a data table of values for the equation

0 | 6 |

2 | 3 |

4 | 0 |

6 | −3 |

Using these values, plot the data points on a graph.

Once the data points are plotted, draw a line that connects them all. The type of line drawn depends on the original inequality that was replaced.

If the inequality had ≤ or ≥, then draw a solid line to represent data points that are on the line.

If the inequality had < or >, then draw a dashed line instead to indicate that some data points are excluded.

If the inequality is either or , then draw its graph using a solid line and solid dots.

If the inequality is either or , then draw its graph using a dashed line and hollow dots.

There remains only one step to complete this graph: finding which side of the line makes the inequality true and shading it. The easiest way to do this is to choose the data point .

It is evident that, for and , the data point is true for the inequality. In this case, shade the side of the line that contains the data point .

It is also clear that, for and , the data point is false for the inequality. In this case, shade the side of the line that does not contain the data point .

# Graphing an Absolute Value Function

To graph an absolute value function, first create a data table using the absolute value part of the equation.

The data point that is started with is the one that makes the absolute value equal to 0 (this is the -value of the vertex). Calculating the value of this point is quite simple.

For example, for , the value makes the absolute value equal to 0.

Examples of others are:

The graph of an absolute value equation will be a V-shape that opens upward for any positive absolute function and opens downward for any negative absolute value function.

Example 4.4.2

Plot the graph of

The data point that gives the -value of the vertex is in which This is the first value.

For which yields

Now pick -values that are larger and less than −2 to get three data points on both sides of the vertex,

1 | 0 |

0 | −1 |

−1 | −2 |

−2 | −3 |

−3 | −2 |

−4 | −1 |

−5 | 0 |

Once there are three data points on either side of the vertex, plot and connect them in a line. The graph is complete.

Example 4.4.3

Plot the graph of

The data point that gives the -value of the vertex is in which This is the first value.

For which yields

Now pick -values that are larger and less than 2 to get three data points on both sides of the vertex,

5 | −2 |

4 | −1 |

3 | 0 |

2 | 1 |

1 | 0 |

0 | −1 |

−1 | −2 |

Once there are three data points on either side of the vertex, plot and connect them in a line. The graph is complete.

# Questions

For questions 1 to 8, graph each linear inequality.

For questions 9 to 16, graph each absolute value equation.