Directed Numbers

Lesson

An* inequality* exists when one amount does not equal the other. In other words, one side of an expression is greater than the other. You can think of this like a set of unbalanced scales, where one side is heavier than the other.

So far we have compared the relative size of numbers using a number line - the further to the right a number is the larger it is. In maths, we have special symbols to indicate that one quantity is bigger or smaller than another. Let's run through them now.

The symbol > is used to tell us that the number on the left is greater than the number on the right.

For example, $3>2$3>2.

The symbol < is used to tell us that the number on the left is less than the number on the right.

For example, $2<3$2<3.

Remember!

**$>$> means "is greater than"**

**$<$< means "is less than"**

**The inequality symbol always points to the smaller number.**

Write a mathematical statement for:

- "Two is greater than one": $2>1$2>1
- "Three is greater than negative four": $3>-4$3>−4
- "Minus seven is less than minus two": $-7<-4$−7<−4

What does each inequality statement expression?

$-2>-5$−2>−5 | $-2$−2 is greater than $-5$−5 | |

$5>0$5>0 | $5$5 is greater than $0$0 | |

$-15<-6$−15<−6 | $-15$−15 is less than $-6$−6 | |

$-10<3$−10<3 | $-10$−10 is less than $3$3 |

Let's look at some examples where we use an inequality statement to compare two numbers.

Write an inequality comparing the numbers $2$2 and $-9$−9.

Think: The number $2$2 is greater than the number $-9$−9. So we want to write an inequality that says "$2$2 is greater than $-9$−9".

Do: We can do this in two ways:

1) We can use the "$>$>" symbol to state $2>-9$2>−9. This is read as "$2$2 is greater than $-9$−9".

2) We can use the "$<$<" symbol to state $-9<2$−9<2. This is read as "$-9$−9 is less than $2$2".

Sharon has made $\$86$$86 selling ice creams. Patricia has made $\$125$$125 selling lemonade.

Choose the correct number statement below that represents the relationship between Sharon and Patricia's earnings.

A) $125\le86$125≤86 B) $86>125$86>125 C) $86<125$86<125 D) $86\ge125$86≥125

**Think**: What do each of these inequalities mean?

**Do**:

A) means $125$125 is less than or equal to $86$86 - this is not true.

B) means $86$86 is greater than $125$125 - this is not true.

C) means $86$86 is less than $125$125 - this is true

D) means $86$86 is greater than or equal to $125$125 - this is not true.

$125$125 is a larger number than $86$86. In other words, $86$86 is less than $125$125. This means C) $86<125$86<125 is true.

Tobias has $778$778 songs on his mobile phone. Marge has $525$525 songs on her tablet.

Construct an inequality directly comparing these quantities.

Ray's account balance is $-\$10$−$10. Mohamad's account balance is $-\$13$−$13.

Choose the correct number statement below that represents the relationship between Ray and Mohamad's savings.

A) $-13\ge-10$−13≥−10 B) $-10>-13$−10>−13 C) $-13>-10$−13>−10 D) $-10\le-13$−10≤−13

Ray is designing a dress pattern using $\frac{3}{5}$35 of a metre of blue fabric and $1$1 metre of orange fabric.

Construct an inequality directly comparing the amount of blue and orange fabric used.

**Think**: Which is the bigger number? What side of the inequality sign do we write the bigger number on?

**Do**: There are two ways we can write this relationship. We can write $\frac{3}{5}<1$35<1 or $1>\frac{3}{5}$1>35.