Section 3.1 Inequalities and their Solution Sets

3.1.1 Introduction

If two numbers are related by one of the comparators

\leq

<

\geq

, or

>

, a statement is generated that can be true or false depending on the numbers:

$a < b$ (reads: " $a$ is strictly less than $b$ " or simply " $a$ is less than $b$ ") is true if the number $a$ is less than and not equal to $b$ .
$a \leq b$ (reads: " $a$ is less than $b$ ") is true if $a$ is less than or equal to $b$ .
$a > b$ (reads: " $a$ is strictly greater than $b$ " or simply " $a$ is greater than $b$ ") is true if the number $a$ is greater and not equal to $b$ .
$a \geq b$ (reads: " $a$ is greater than $b$ ") is true if the number $a$ is greater than or equal to $b$ .

The comparators describe how the given values are related to each other on the number line:

a < b

means that

a

is to the left of

b

on the number line.

The statements

2 < 4

- 12 \leq 2

4 > 1

, and

3 \geq 3

are true, but the statements

2 < \sqrt{2}

and

3 > 3

are false.

On the number line, the number

2

is to the left of the number

4

, thus

2 < 4

Here,

a < b

means the same as

b > a

, likewise

a \leq b

means the same as

b \geq a

. But it should be noted that the opposite of the statement

a < b

is the statement

a \geq b

and not

a > b

. If terms with a variable occur in an inequality, the problem is to find the number range of the variable such that the inequality is true.