1. Properties of Inequalities

The expression `a < b` is read as

a is less than b

while the expression `a > b` is read as

a is greater than b.

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 sense if the signs of inequality point in the same direction; and

(b) the opposite sense if 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:

Graph of x greater than 2

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:

graph of x less than or equal to 1

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).

graph of x greater than -2 and less than or equal to 4

We now examine some of the key properties of inequalities.

Property 1 - Adding or Subtracting a Number

The sense of 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 sense of 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 reversed if 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 n is a positive integer, then the inequality formed by the n-th power or n-th root of both sides have the same sense as 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`

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