Inequalities
1.0 Definition
1.0 Definition
An inequality says that two values (say $a$ and $b$) are not equal, which is represented by $a \ne b$. There are other special symbols that show in what way things are not equal.
$a < b$ means that $a$ is less than $b$.
$a > b$ means that $a$ is greater than $b$.
$a \leqslant b$ means that $a$ is less than or equal to $b$.
$a \geqslant b$ means that $a$ is greater than or equal to $b$.
- In each case sign open towards the larger number. For example, $2 < 5$ ($2$ is less than $5$).
There are two senses of an inequality: $<$ and $>$ which is explained with the help of number line.
On number line, $a<b$ means $a$ falls to the left of $b$ as shown in figure.
| Symbols | Meaning |
| $R$ | Set of real numbers |
| ${R^ + }$ | Set of positive real numbers |
| ${R^ - }$ | Set of negative real numbers |
| $N$ | Set of natural numbers |
| $Z$ | Set of integers |
| $W$ | Set of whole numbers |
| $Q$ | Set of rational numbers |
Closed Interval
$$1 \leqslant x \leqslant 2$$
It means $x \in [1,2]$ i.e., $1$ and $2$ both are included in the value of $x$.
Open Interval
$$1 < x < 2$$
It means $x \in (1,2)$ i.e., $1$ and $2$ both are not included in the value of $x$.
Semi-closed/Semi-open Interval
$$1 < x \leqslant 2$$
It means $x \in (1,2]$ i.e., $1$ is not included but $2$ is included in the value of $x$.
