# 1. Properties of Inequalities

The expression `a < b` is read as

ais less thanb

while the expression `a > b` is read as

ais greater thanb.

The `<` and `>` signs define what is known as the **sense** of the
inequality (indicated by the direction of the sign).

Two inequalities are said to have

(a) the

same senseif the signs of inequality point in the same direction; and(b) the

opposite senseif the signs of inequality point in the opposite direction.

### Example 1

The inequalities `x + 3 > 2` and `x + 1 > 0` have the same sense.

So do the inequalities `3x - 1 < 4` and `x^2- 1 < 3`.

### Example 2

The inequalities

`x - 4 < 0` and `x > - 4`

have the **opposite** sense as do the following 2 inequalities:

`2x + 4 > 1` and `3x^2- 7 < 1`.

## Solution of an Inequality

The **solution** of an inequality consists of all the
values of the variable that make the inequality a true
statement.

**Conditional inequalities** are those which are true for
some, but not all, values of the variable.

**Absolute inequalities** are those which are true for all
values of the variable.

A solution of an inequality consists of only real numbers as
the terms *"***less than or greater than***"* are not defined for
complex numbers.

### Example 3

The inequality `x + 1 > 0` is true for all values of *x* greater than `-1`.

Hence the solution of the inequality is written as `x
> -1` and so this is a **conditional** inequality.

### Example 4

The inequality `x^2+ 1 > 0`
is true for all values of *x* and hence is an **absolute** inequality.

## Graphical Representation of Inequalities

### Example 5

(a) To show `x > 2` graphically, we use an open circle at `2` on the number line and a line to the right of this point, with an arrow pointing to the right:

The open circle shows that the point is not part of the indicated solution.

(b) To show `x ≤1` graphically, we use a solid circle at 1 on the number line and a line to the left of this point, with an arrow pointing to the left:

The solid circle shows that the point is part of the indicated solution.

(c) To indicate ` −2 < x ≤ 4` graphically, we draw a bold line between the 2 values, an open circle at `−2` (since it is not included) and a closed circle at `4` (since it is included).

We now examine some of the key properties of inequalities.

## Property 1 - Adding or Subtracting a Number

The

senseof an inequality is not changed when the same number is added or subtracted from both sides of the inequality.

### Example 6

Using the inequality:

`9 > 6`

adding `4` to both sides gives

`9 + 4 > 6 + 4`

i.e. `13 > 10` which is still true

subtracting `12` from each side of the original gives

`9 - 12 > 6 − 12 `

i.e. `-3 > -6` which is still true

## Property 2 - Multiplying by a Positive Number

The

senseof the inequality is not changed if both sides are multiplied or divided by the same positive number.

### Example 7

Using the inequality:

`8 < 15`

Multiplying both sides by `2` gives

`8 × 2 < 15 × 2`

i.e. `16 < 30` which is still true

Dividing both sides of the original by `2` gives

`8/2 < 15/2`

i.e. `4 < 7.5` which is still true

## Property 3 - Multiplying by a Negative Number

The sense of the inequality is

reversedif both sides are multiplied or divided by the same negative number.

### Example 8

We start with the inequality `4 > −2`.

Multiplying both sides by `-3` gives

`4 × −3 > -2 × −3`

`-12 > 6` which is

not true

Hence the correct solution should be

`4 > −2`

`4 × −3 < −2 × −3`

`−12 < 6` (Note the change in the sign used)

Similarly dividing both sides of the original inequality by ` −2` gives

`4 > −2`

`4 ÷ −2 < −2 ÷ −2`

`-2 < 1` (Note the change in the sign used)

## Property 4 - *n*-th Power

If both sides of an inequality are positive and

is a positive integer, then the inequality formed by then-th power orn-th root of both sides have thensame senseas the given inequality.

### Example 9

Using the inequality:

`9 > 6`

Squaring both sides gives

`9^2> 6^2`

i.e. `81 > 36` which is still true

Taking square root of each side gives

`sqrt(9)>sqrt(6)`

i.e. `3 > 2.45` which is still true

[**Note:** `sqrt(9)` does not equal `±3`. By convention, we take the positive square root only. See the discussion at √16 - how many answers?]

## Exercise

Graph the given inequality on the number line:

`1 < x ≤ 4`

Answer

We need to have an **open** circle for `1`, since it is not included, but a **closed** circle for `4`, since it is included.