In this section we look at how to solve linear inequalities and illustrate their solutions using a number line.

When using a number line, a small solid circle is used for ≤ or ≥ and a hollow circle is used for > or <.

For example,

*x* ≥ 5

Here the solid circle means that the value 5 is included.

*x* < 7

Here the hollow circle means that the value 7 is not included.

When solving linear inequalities we use the same techniques as those used for solving linear equations. The important exception to this is that when *multiplying or dividing by a negative number*, you must *reverse the direction of the inequality*. However, in practice, it is best to try to avoid doing this.

Solve the inequality *x* + 6 > 3 and illustrate the solution on a number line.

x + 6 | > 3 | |

x | > 3 – 6 | Subtracting 6 from both sides of the inequality |

x | > –3 |

This can be illustrated as shown below:

Solve the inequality 3*x* + 7 ≥ 19 and illustrate the solution on a number line.

3x + 7 | ≥ 19 | |

3x | ≥ 12 | Subtracting 7 from both sides |

x | ≥ 4 | Dividing both sides by 3 |

This can now be shown on a number line.

Illustrate the solution to the inequality 12 – 3*x* ≥ 6.

Because this inequality contains the term ' –3*x* ', first add 3*x* to both sides to remove the – sign.

12 – 3x | ≥ 6 | |

12 | ≥ 6 + 3x | Adding 3x to both sides |

6 | ≥ 3x | Subtracting 6 from both sides |

2 | ≥ x | Dividing both sides by 3 |

or x | ≤ 2 |

This is illustrated below.

Solve the equation –7 < 5*x* + 3 ≤ 23.

In an inequality of this type you must apply the same operation to each of the 3 parts.

–7 < | 5x + 3 | ≤ 23 | |

–10 < | 5x | ≤ 20 | Subtracting 3 from both sides |

–2 < | x | ≤ 4 | Dividing both sides by 5 |

This can then be illustrated as below.