Introduction to inequalities,symbol and rules of inequality for Class 11 ,CBSE Board, IITJEE maths and other exams

What is inequalities
In mathematics, an inequality is a relation that holds between two values when they are different
Solving linear inequalities is very similar to solving linear equations, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative

Symbols used in inequalities

The symbol < means less than. The symbol > means greater than.
The symbol < with a bar underneath means less than or equal to. Usually this is written as $\leq$
The symbol > with a bar underneath means greater than or equal to. Usually this is written as $\geq$
The symbol $\neq$ means the quatities on left and right side are not equal

Examples
1)a < b means a is less then b or b is greater a
2) $a \leq b$ means a is less then or equal to b
3) a > b means a is greater than b
4) $a \geq b$ means a is greater or equal to b
Things are which are safe to do in inequality which does not change in direction
1) addition of same number on both sides
a > b
=>a+c > b +c
2) Substraction of same number on both sides
a > b
=>a-c > b-c
3) Multipication/Division by same positive number on both sides
a > b
if c is positive number then
ac > bc
or
a/c > b/c

Things which changes the direction of the inequality

1) swapping the left and right sides
2) Multiplication/Division by negative number on both sides
3) Dont multiple by variable whose values you dont know as you dont know the nature of the variable

Concept Of Number line

A number line is a horizontal line that has points which correspond to numbers. The points are spaced according to the value of the number they correspond to; in a number line containing only whole numbers or integers, the points are equally spaced.

It is very useful in solving problem related to inequalities and also representing it
Suppose x >2(1/ 3), this can represent this on number line like that

Linear Inequation in One Variable:

A equation of the form
ax+b> 0
or
$ax+b \geq 0$
or
ax+b< 0
or
$ax+b \leq 0$
are called the linear equation in One Variable
Example:
x-2 < 0
or 3x +10 > 0
or $10x-17 \geq 0$

Linear Inequation in Two Variable:

A equation of the form
ax+by> c
or
$ax+by \geq c$
or
ax+by< c
or
$ax+by \leq c$
are called the linear equation in two Variable
Example:
x-2y < 0
or 3x +10y > 0
or $10x-17y \geq 0$