Much like solving equations from real world problems, there are certain keywords or phrases to look out for. When it comes to inequalities, we now have a few extra keywords and phrases to represent the different inequality symbols.

Recall from the equations section that we have the following key terms to indicate operations and quantities:

Addition | Subtraction | Multiplication | Division | Parentheses |
---|---|---|---|---|

plus | minus | times | divided by | quantity |

the sum of | the difference of | the product of | the quotient of | times the sum |

increased by | decreased by | multiplied by | half | times the difference |

total | fewer than | each/per | split | |

more than | less than | twice | equally shared | |

added to | subtracted from | double |

In addition to operations, certain key terms and phrases will indicate which inequality will be needed for the given situation:

\gt | \lt | \geq | \leq |
---|---|---|---|

greater than | less than | greater than or equal to | less than or equal to |

more than | is below | at least | at most |

over | under | minimum | maximum |

no less than | no more than |

Recall that a **solution** to an inequality is any value that makes the inequality statement true. Based on the context, some values might be calculated algebraically, but are not reasonable based on the restrictions of the scenario. For example, time and lengths generally cannot be negative, which can create restrictions on the possible values x and y can take on.

We can represent the solutions to inequalities on a number line. The inequality symbol will determine if the ray should have a filled (closed) or unfilled (open) endpoint and which direction the arrow extends.

The less than or equal to symbol \left( \leq \right) will be represented with a filled (closed) endpoint and an arrow pointing left. The inequality x \leq 2 will be graphed like this:

This graph tells us that any number that falls on the ray, including 2 is a part of the solution set.

The less than symbol \left(<\right) will be represented with an unfilled (open) endpoint and an arrow pointing left. The inequality x < 2 will be graphed like this:

This graph tells us that any number that falls on the ray, except for 2 is a part of the solution set.

The greater than or equal to symbol \left( \geq \right) will be represented with a filled (closed) circle and an arrow pointing right. The inequality x \geq 2 will be graphed like this:

This graph tells us that any number that falls on the ray, including 2 is a part of the solution set.

The greater than symbol \left(>\right) will be represented with an unfilled (open) circle and an arrow pointing right. The inequality x \gt 2 will be graphed like this:

This graph tells us that any number that falls on the ray, excluding 2 is a part of the solution set.

Consider the inequality: k \geq -25

a

Represent the inequality on a number line.

Worked Solution

b

Name 3 values that are in the solution set.

Worked Solution

Sasha is drawing a pentagon-shaped house design and wants to make sure the combined length of two sides representing the roof are at least twice as long as the sum of the remaining sides.

a

Write two different inequalities that Sasha could use to represent this situation.

Worked Solution

b

Is 4 a possible value for x?

Worked Solution

Calandra charges \$ 37.72 to style hair, as well as an additional \$ 6 per foil. Pauline would like the total cost for her styling to be no more than \$ 95.86.

a

Write an inequality that represents the number of foils Pauline could get.

Worked Solution

b

The solution for the inequality is N \leq 9.69. Determine whether N=-2 is a viable solution to the inequality in the context of the question.

Worked Solution

Given the inequality 3x + 5 - 6x > 8, create a verbal situation that can be represented by this inequality.

Worked Solution

Idea summary

The following key terms can be used to indicate operations and quantities in real-world situations:

Addition | Subtraction | Multiplication | Division | Parentheses |
---|---|---|---|---|

plus | minus | times | divided by | quantity |

the sum of | the difference of | the product of | the quotient of | times the sum |

increased by | decreased by | multiplied by | half | times the difference |

total | fewer than | each/per | split | |

more than | less than | twice | equally shared | |

added to | subtracted from | double |

Phrases that describe inequalities include

\gt | \lt | \geq | \leq |
---|---|---|---|

greater than | less than | greater than or equal to | less than or equal to |

more than | is below | at least | at most |

over | under | minimum | maximum |

no less than | no more than |

Because inequalities have infinite solutions, inequalities used to represent real-world situations often include solutions that are unreasonable in context and therefore non-viable.