Properties of Inequalities

Inequality tells you about the relative size of two values.

Here we look at the properties of these Inequalities:

Symbol	Words	Example

>	greater than	x+3 > 2
<	less than	7x < 28
≥	greater than or equal to	5 ≥ x-1
≤	less than or equal to	2y+1 ≤ 7

Example: Alex plays in the under 15s soccer. How old is Alex?

We don't know exactly how old Alex is, because it doesn't say "equals"

But we do know "less than 15", so we can write:

Age < 15

Properties

Inequalites have several properties, all with special names! Here we list each one, with an example. (Note: the values "a", "b" and "c" are Real Numbers)

Trichotomy Property

The "Trichotomy Property" says that only one of the following can be true:

a < b
a = b
a > b

It makes sense, right? a can't be be both less than b and greater than b at the same time!

Transitivity Property

If a > b and b > c; then a > c
If a < b and b < c; then a < c

Example: if Alex is older than 12, and Billy is older than Alex, then Billy must be older than 12 also!

Reversal Property

If a > b then b < a
If a < b then b > a

Example: Alex is older than Billy, so Billy is younger than Alex

Addition and subtraction

If a > b, then a + c > b + c and a − c > b − c
If a < b, then a + c < b + c and a − c < b − c

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

Multiplication and division

If c is positive and a < b, then a × c < b × c
If c is negative and a < b, then a × c > b × c

Example: Alex has a lower score than Billy. If both Alex and Billy manage to double their scores (×2), Alex will still have a lower score than Billy.

But if you multiply by a negtaive the opposite happens

Example: Alex has a lower score than Billy. But if the scores are minuses (they take away points from their Totals), then Alex has done better than Billy!

Why does multiplying by a negative reverse the sign?

Well, just look at the number line!

For example, from 3 to 7 is an increase, but from -3 to -7 is a decrease.


-7 < -3	7 > 3

See how the inequality sign reverses (from < to >) ?

Additive inverse

If a < b then -a > -b
If a > b then -a < -b

Example: Alex has more money than Billy, and so Alex is ahead.

But suddenly money is bad, so Alex ends up behind Billy!

Multiplicative inverse

If a < b then 1/a > 1/b
If a > b then 1/a < 1/b

Example: Alex has run faster than Billy, and so Alex is ahead.

Then the rules change! It is not about how fast but how slow, so Billy is ahead.