Learn about writing inequalities from word problems with help from our practice examples.

If you want to test yourself, or get some practice, then try one of our graded worksheets, or our online quiz.

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An inequality is when you have a relationship between two values of expressions which are not equal to each other.

There are a few different options for different types of inequalities:

> | Greater than |

< | Less than |

≥ | Greater than or equal to |

≤ | Less than or equal to |

≠ | Not equal to |

- Greater than (>) where one expression or value is greater than another, e.g. 7 > 5
- Less than (<) where one expression or value is less than another, e.g. 9 < 2 x 6
- Greater than or equal to (≥) where one expression or value is greater than or equal to another, e.g. 20 + 4 ≥ 17
- Less than or equal to (≤) where one expression or value is less than or equal to another, e.g. 18 ≤ 9 x 2
- Not equal to (≠) where one expression or value is not equal to another, e.g. 7 ≠ 4

When writing inequalities from word problems, we have to look carefully at and understand the language being used.

Different words and phrases have different meanings when deciding on which inequality to use.

The mathematical notation is really just a shorthand way of writing the words more efficiently and clearly.

Here is a quick table showing some of the written expressions often used and which inequality they are represented by.

> | < |
---|---|

greater than | less than |

more than | smaller than |

over | under |

above | below |

≥ | ≤ |

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

a minimum of | a maximum of |

at least | at most |

> and < |
---|

strictly between |

between ... and ... (exclusive) |

≥ and ≤ |

between ... and ... |

between ... and ... (inclusive) |

from ... to ... |

Note: the word 'between' is mainly used to mean between inclusively (including end points).

However, sometimes 'between' is used to mean between exclusively (excluding end points).

To avoid ambiguity, it is good practice to include the word 'inclusive' or 'exclusive' to make it completely clear if the end points are included or not.

Expression in words; | Variable | Inequality |
---|---|---|

A maximum of 5 minutes | t | t ≤ 5 minutes |

More than $100 | m | m > $100 |

A minimum of 50mph | s | s ≥ 50 mph |

Less than 12 ounces | w | w < 12 oz |

Over 20 seconds | t | t > 20 seconds |

At least 50 points | p | p ≥ 50 points |

Not equal to three | x | x ≠ 3 |

Between 5 and 12 (inclusive) | y |
y ≥ 5 and y ≤ 12 or 5 ≤ y ≤ 12 |

Between 5 and 12 (exclusive) | z |
z > 5 and z < 12 or 5 < z < 12 |

The variable names (letters) have been chosen at random - you can use any variable name to represent any value.

Note: you need to read the word problem carefully because sometimes the inequality does not match the language used, especially when the inequality involves finding out what is left over or what remains after an amount is taken away. See Examples 2) and 7) below.

When we are writing an inequality from a word problem, we are basically translating the word problem into mathematical language and symbols.

When writing an inequality from a word problem, there are two simple steps you need to follow...

- use the language of inequalities table to help you select the right inequality

The best way to learn how to write inequalities from word problems and see how they work is to look at some ready made examples.

Here are some examples of writing inequalities from word problems.

The vocabulary which tells us about the inequality are the words: at least.

This means we need to use the ≥ symbol.

So the inequality is t ≥ 12 minutes

The vocabulary which tells us about the inequality are the words: over.

However, because he has drunk over half the bottle, it means that there is under half a bottle left.

So the symbol we need is < and the amount is ½ of 30 = 15.

So the inequality is b < 15 ounces

The vocabulary which tells us about the inequality are the words: more than.

So the symbol we need is > and the amount is 3 x 8 = 24.

So the inequality is A > 24 years old

The vocabulary which tells us about the inequality are the words: shortest.

If the shortest chapter has 11 pages, then there must be some chapters with more than 11 pages.

So the symbol we need is > and the amount is 14 x 12 = 168.

So the inequality is p > 168 pages.

These examples use two different variables and express one variable in terms of another.

The vocabulary which tells us about the inequality are the words: at least.

So the symbol we need is ≥

So the inequality is c ≥ 3f.

The vocabulary which tells us about the inequality are the words: a maximum of.

So the symbol we need is ≤

We know that there are f flights of steps and also 3 extra steps.

So the inequality is s ≤ 12f + 3.

These examples involve solving word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers.

There are also examples where the variable lies between two values.

The vocabulary which tells us about the inequality are the words: more than and less than.

However, because we are looking at what is left in the bottle, rather than what has been drunk, we need to think carefully about the inequalities!

He drinks more than one-quarter of the bottle, so there will be less than three-quarters of the bottle left, so we need the symbol <

He drinks less than one-half of the bottle, so there will be one-half or more of the bottle left, so we need the symbol ≥

Half of the bottle = ½ liters = 500 ml. 1000 - 500 = 500 ml

Quarter of the bottle = ¼ liters = 250 ml. 1000 - 250 = 750 ml

So the inequality is b ≥ 500 ml and b < 750 ml

This can be simplified to: 500 ≤ b < 750 ml

This means that he has at least 500 ml but less than 750 ml left.

The vocabulary which tells us about the inequality are the words: a minimum of.

So the symbol we need is ≥

The inequality we get from this problem is 6t + 8 ≥ 50

We are not finished yet, because this needs to be simplified and written in terms of t.

6t + 8 ≥ 50 so 6t ≥ 42

If we divide both sides of this inequality by 6, we get:

t ≥ 7

So the inequality is t ≥ 7 hours

He needs to fish for at least 7 hours to reach his target.

The vocabulary which tells us about the inequality are the words: between (inclusive).

So the symbol we need is ≤ and ≥

3 x 23 = 69 and 3 x 28 = 84

So the inequality is t ≥ 69 and t ≤ 84

This can be simplified to: 69 ≤ t ≤ 84

It will take him between 69 and 84 seconds (inclusive) to swim 3 lengths.

We have a range of different inequality worksheets which involve writing inequalities from a range of word problems..

We have split the sheets into 3 sections: A, B and C

- Section A involves basic level questions aimed at 6th grade
- Section B involves medium level questions aimed at 6th and 7th grade
- Section C involves more advanced questions aimed at 7th and 8th grade

Sheet 1 involves picking the vocabulary and relevant information from the problem and writing the inequality

Sheet 2 involves the same skills as Sheet 1, but also involves an arithmetic operation to get the inequality.

Sheet 1 involves using two variables and writing an inequality for one variable in terms of the other variable

Sheet 2 is similar to Sheet 1 but with slightly harder problems.

Sheet 1 involves using one variables and using the information to solve the inequality, usually in the form px + q > r or px + q < r, where p, q, and r are specific rational numbers

Sheet 2 involves the same skills as Sheet 1 but has compound inequalities in each question

Take a look at some more of our worksheets similar to these.

These 5th grade ratio worksheets are a great way to introduce this concept.

We have a range of part to part ratio worksheets and slightly harder problem solving worksheets.

If you are looking for some 6th grade algebra worksheets to use with your child to help them understand simple equations then try our selection of basic algebra worksheets.

There are a range of 6th grade math worksheets covering the following concepts:

- Generate the algebra - and write your own algebraic expressions;
- Calculate the algebra - work out the value of different expressions;
- Solve the algebra - find the value of the term in the equation.
- Use the distributive property to factorize and expand different expressions

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This quick quiz tests your skill at writing inequalities from a range of word problems.

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