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: