# Properties of Inequalities

**Inequality**
tells us about the **relative size** of two values.

*(You might like to read a gentle Introduction to Inequalities first)*

## The 4 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 |

The symbol "points at" the smaller value

## Properties

Inequalities have properties ... all with special names!

Here we list each one, with examples.

Note: the values **a**, **b** and **c** we use below are Real Numbers.

## Transitive Property

When we link up inequalities in order, we can "jump over" the middle inequality.

If a < b **and** b < c, then a < c

Likewise:

If a > b **and** b > c, then a > c

### Example:

- If Alex is older than Billy and
- Billy is older than Carol,

then Alex must be older than Carol also!

## Reversal Property

We can swap **a** and **b** over, if we make sure the symbol still "points at" the smaller value.

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

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

## Law of Trichotomy

The "Law of Trichotomy" says that** only one** of the following is true:

It makes sense, right? **a** must be either **less than b** or **equal to b** or **greater than b**. It must be one of those, and only one of those.

### Example: Alex Has More Money Than Billy

We could write it like this:

a > b

So we also know that:

- Alex does
**not**have**less**money than Billy (not a<b) - Alex does
**not**have**the same amount**of money as Billy (not a=b)

(Of course!)

## Addition and Subtraction

Adding **c** to both sides of an inequality just **shifts everything along**, and the inequality stays the same.

If a < b, then a **+ c** < b **+ c **

### Example: Alex has less coins than Billy.

If both Alex and Billy get 3 more coins each, Alex will still have less coins than Billy.

Likewise:

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

So adding (or subtracting) the same value to both a and b **will not change** the inequality

## Multiplication and Division

When we multiply both a and b by a **positive number**, the inequality **stays the same**.

But when we multiply both a and b by a **negative number**, the inequality **swaps over**!

Notice that **a<b** becomes **b<a** after multiplying by (-2)

But the inequality stays the same when multiplying by +3

Here are the rules:

- If a < b, and
**c is positive**, then**ac < bc** - If a < b, and
**c is negative**, then**ac > bc**(inequality swaps over!)

A "positive" example:

Example: Alex's score of 3 is **lower than** Billy's score of 7.

a < b

If both Alex and Billy manage to **double** their scores (×2), Alex's score will still be lower than Billy's score.

2a < 2b

But when multiplying by a negative the opposite happens:

But if the scores become **minuses**, then Alex **loses 3** points and Billy **loses 7** points

So Alex has now done **better** than Billy!

−a > −b

**Why does multiplying by a negative reverse the sign?**

Well, just look at the number line!

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

So the inequality sign reverses (from < to >)

## Additive Inverse

As we just saw, putting minuses in front of a and b **changes the direction** of the inequality. This is called the "Additive Inverse":

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

This is really the same as multiplying by (-1), and that is why it changes direction.

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

But a new law says "all your money is now a **debt** you must repay with hard work"

So now Alex is worse off than Billy.

## Multiplicative Inverse

Taking the reciprocal (1/value) of both a and b **can change the direction** of the inequality.

When a and b are **both positive** or **both negative**:

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

### Example: Alex and Billy both complete a journey of 12 kilometers.

Alex runs at **6 km/h** and Billy walks at **4 km/h**.

Alex’s speed is greater than Billy’s speed

6 > 4

But Alex’s time is less than Billy’s time:

12/6 < 12/4

2 hours < 3 hours

But when either **a or b is negative ** (not both) the direction stays the same:

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

## Non-Negative Property of Squares

A square of a number is greater than or equal to zero:

a^{2} ≥ 0

### Example:

- (3)
^{2}= 9 - (−3)
^{2}= 9 - (0)
^{2}= 0

Always greater than (or equal to) zero

## Square Root Property

Taking a square root will not change the inequality *(but only when both a and b are greater than or equal to zero)*.

If a ≤ b then √a ≤ √b

(for a,b ≥ 0)

### Example: a=4, b=9

- 4 ≤ 9 so √4 ≤ √9