● Here we get certain statements involving a sign `‘<’` (less than), `‘>’` (greater than), `‘≤’` (less than or equal) and `≥` (greater than or equal) which are known as inequalities.

● we will study linear inequalities in one and two variables. The study of inequalities is very useful in solving problems in the field of science, mathematics, statistics, optimisation problems, economics, psychology, etc.

`\color{purple}ul(✓✓) \color{purple} \mathbf("DEFINITION ALERT") `
● Two real numbers or two algebraic expressions related by the symbol `‘<’, \ \ ‘>’, \ \ ‘≤’ `or `‘≥’` form an inequality.

`\color{green}✍️\color{green}\mathbf("Numerical inequalities")`
● `3 < 5; 7 > 5` are the examples of numerical inequalities while

`\color{green}✍️\color{green}\mathbf("Literal inequalities")`
● `x < 5; y > 2; x ≥ 3, y ≤ 4` are the examples of literal inequalities.

`\color{green}✍️\color{green}\mathbf("Double inequalities")`
● `3 < 5 < 7` (read as `5` is greater than `3` and less than `7)`
● ` 3 < x < 5` (read as `x` is greater than or equal to `3` and less than `5)` and `2 < y < 4` are the examples of double inequalities.

`\color{green}✍️\color{green}\mathbf("Linear inequalities in one Variable")`
These are linear inequalities in one Variable `x` when `a≠ 0`
● `ax + b < 0`
● `ax + b > 0`
● `ax + b ≤ 0`
● `ax + b ≥ 0`

`\color{green}✍️\color{green}\mathbf("Linear inequalities in two variables")`
These are linear inequalities in two variables `x` and `y` when `a ≠ 0, b ≠ 0.`
● `ax + by < c`
● `ax + by > c`
● `ax + by ≤ c`
● `ax + by ≥ c`

`\color{green}✍️\color{green}\mathbf("Quadratic inequalities in one variable")`
These are quadratic inequalities in one variable `x` when `a ≠ 0.`
● `ax^2 + bx + c ≤ 0 `
● `ax^2 + bx + c > 0`

`\color{purple}ul(✓✓) \color{purple} \mathbf("DEFINITION ALERT") `
● Any solution of an inequality in one variable is a value of the variable which makes it a true statement.

`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)` we followed the following rules:
`"Rule 1: "` Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality.

`"Rule 2:"` Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed.

`\color{red} \ox \color{red} \mathbf(COMMON \ CONFUSION) :` in `"Rule 2,"` the sign of inequality is reversed (i.e., ‘<‘ becomes ‘>’, ≤’ becomes ‘≥’ and so on) whenever we multiply (or divide) both sides of an inequality by a negative number.

It is evident from the facts that
` \ \ \ \ \ \ 3 > 2` while `– 3 < – 2,`

` \ \ \ \ \ \– 8 < – 7` while `(– 8) (– 2) > (– 7) (– 2) ,` i.e., `16 > 14.`