An **inequality ** is a mathematical statement that compares the size of two values. Inequalities use symbols such as:

The images below show another demonstration of the inequality symbols:

We are familiar with being able to write an equation in two orders. For example, x=10 and 10=x mean the same thing.

Every inequality can be written in two ways, but we need to be careful about the symbols we use.

In the following image, there are five cats on the left and three cats on the right. We would say that five cats is greater than three cats. We can write this as an inequality 5\gt 3.

If we switch the order so that there are three cats on the left and five cats on the right, we can say that three cats is less than five cats. We can write this as an inequality 3\lt 5.

Both images mean the same thing but are stated differently. If we switch the order of an inequality, we have to change the inequality sign. This is also true with algebraic inequalities.

For example, x\gt 10 means the same thing as 10\lt x. In other words, "x is greater than ten" is the same as "ten is less than x".

For example, the expressions x \gt 5 and 5 \gt x represent different sets of numbers, while x \gt 5 and 5 \lt x represent the same set of numbers.

We can use this understanding of inequality symbols to write inequalities that represent real world situations. Let's write an inequality to represent the statement: "a student needs to score at least 75 points to pass an exam."

Let s represent the student's score. The key phrase "at least 75 points" tells us that the lowest passing score is 75. So the student will pass the exam if they score 75 points or if they score more than 75 points. If we use s to represent the score we can write the inequality s \geq 75.

Here are some common phrases and examples used for the different inequality symbols.

Inequality Symbol | Vocabulary/Representations | Example |
---|---|---|

\enspace\enspace\enspace\enspace\enspace\enspace \lt | \text{less than, fewer than, under} | \text{"The speed limit is less than 60 mph."} \\\ \text{translates to } S \lt 60 |

\enspace\enspace\enspace\enspace\enspace\enspace\gt | \text{greater than, exceeds, more than} | \text{"The temperature is greater than } 30\degree \text{C"} \\\ \text{translates to } T \gt 30 |

\enspace\enspace\enspace\enspace\enspace\enspace\leq | \text{less than or equal to, at most,}\\\ \text{ no more than, up to} | \text{"You can spend up to 50 dollars."}\\\ \text{translates to } C \leq 50 |

\enspace\enspace\enspace\enspace\enspace\enspace\geq | \text{greater than or equal to, at least,} \\\ \text{ no less than} | \text{"You need at least 8 hours of sleep."}\\\ \text{ translates to }H \geq 8 |

For the number sentence \dfrac{2}{3} \enspace ⬚ \enspace 0.3

a

Choose the mathematical symbol that makes the number sentence true.

Worked Solution

b

Write the statement in words.

Worked Solution

Write an inequality to represent each of the following situations.

a

n is greater than 9

Worked Solution

b

The weight of the package is under 5\text{ kg}. Let w be the weight of the package.

Worked Solution

c

You must be at least 18 years old to vote in the United States. Let a be the age of the voters.

Worked Solution

d

The maximum height for the ride is 120\text{ cm}. Let h be the height of the riders.

Worked Solution

Write a real-world scenario for each inequality.

a

x \leq 20

Worked Solution

b

y \geq 5

Worked Solution

c

a \lt 0

Worked Solution

d

s \gt 20

Worked Solution

A clothing store has a fitting room policy that limits the number of garments one can bring in at a time. The policy can be represented by s \leq 4, where s represents the number of garments. Which options show the number of garments you can try at once? Select all correct options.

A

1

B

2

C

3

D

4

E

5

F

6

Worked Solution

Idea summary

Inequalities are mathematical sentences where two expressions are not necessarily equal, indicated by the symbols: \lt , \gt , \leq, and \geq.

Symbol | Meaning | Example |
---|---|---|

\enspace \enspace\, \lt | \text{less than, fewer than, under} | \enspace 3\lt 6 |

\enspace \enspace\, \gt | \text{greater than, exceeds, more than} | \enspace 6\gt 3 |

\enspace \enspace\, \leq | \text{less than or equal to, at most, no more than, up to} | \enspace 4\leq 6 |

\enspace \enspace\, \geq | \text{greater than or equal to, at least, no less than} | \enspace 6\geq 5 |