Linear Inequation: Linear inequality is an inequality which involves a linear function and it contains one of the symbols of inequality.

• $<$ is less than
• $>$ is greater than
• $\leq$ is less than or equal to
• $\geq$ is greater than or equal to
• $\neq$ is not equal to

A linear inequality actually looks exactly like a linear equation $(y = mx + c)$, with the inequality sign replacing the equality sign $(e.g. y > mx + c)$.

Two-dimensional linear inequalities are expressions in 2 variables of the form:  $ax + by < c \ \ or\ \ ax + by \le c$  where the inequalities may either be strict or not.

A statement of any of the following forms:  $i) ax + b > 0 \ \ ii) ax + b < 0 \ \ iii) ax + b \ge 0 \ \ or \ \ iv) ax + b \le 0$  where $a$    and  $b$   are real numbers and  is called a Linear Inequation in $x$.

Replacement Set or Universal Set: This is a set which contains all the values of the variable $x$  , which would satisfy the inequation.

Solution Set: This is a subset of the replacement set which satisfy the given inequation.

Properties of Inequalities

Addition and Subtraction Property: A common constant c may be added to or subtracted from both sides of an inequality. For any real numbers

$If \ a \le b, \ then \ a + c \le b + c \ and \ a - c \le b - c$

$If \ a \ge b, \ then \ a+b \ge b+c \ and \ a-c \ge b-c$

Multiplication and Division Property: The property states that for any real numbers, $a, b \ and \ non-zero \ c:$

If $\ c \ is \ positive$  , then multiplying or dividing by c does not change the inequality:

$If a \ge b \ and \ c > 0, \ then \ ac \ge bc \ and\ \frac{a}{c} \ge\frac{b}{c}$

$If \ a \le b \ and \ c > 0, \ then \ ac \le bc \ and \ \frac{a}{c} \le\frac{b}{c}$

If  $\ c \ \ is \ \ negative$  , then multiplying or dividing by $c$  inverts the inequality:

$If a \ge b \ and \ c < 0, \ then \ ac \le bc \ and \ \frac{a}{c} \le \frac{b}{c}$

$If \ a \le b \ and \ c < 0, \ then \ ac \ge bc \ and \ \frac{a}{c} \ge \frac{b}{c}$

Transitive property of inequality states that for any real numbers $\ a, \ b, \ c$  :

$If \ a \ge b \ and \ b \ge c, \ then \ a \ge c$

$If \ a \le b \ and \ b \le c, \ then \ a \le c$

Converse Property: The relations ≤ and ≥ are each other’s converse. For any real numbers $\ a \ and \ b:$

$If \ a \le b, \ then \ b \ge a$

$If \ a \ge b, \ then \ b \le a$

Additive inverse: The properties for the additive inverse state: For any real numbers   $\ a \ and \ b$  , negation inverts the inequality:

$If a \le b, then -a \ge -b$

$If a \ge b, then -a \le -b$

Multiplicative Inverse: The properties for the multiplicative inverse state:

For any non-zero real numbers $\ a \ and \ b$  that are both positive or both negative:

$If a\le b, \ then \ \frac{1}{a} \ge \frac{1}{b}$

$If a\ge b, \ then \ \frac{1}{a} \le \frac{1}{b}$

If anyone of $\ a \ and \ b$   is positive and the other is negative, then:

$If a< b, \ then \ \frac{1}{a} < \frac{1}{b}$

$If a> b, \ then \ \frac{1}{a} > \frac{1}{b}$

For any non-zero real numbers $\ a \ and \ b$  :

$If 0 < a\le b, \ then \ \frac{1}{a} \ge \frac{1}{b} > 0$

$If a \le b < 0, \ then \ 0 >\frac{1}{a} \ge \frac{1}{b}$

$If a< 0 < b, \ then \ \frac{1}{a} < 0 <\frac{1}{b}$

$If 0 > a \ge b, \ then \ \frac{1}{a} \le \frac{1}{b} < 0$

$If a \ge b > 0, \ then \ 0 <\frac{1}{a} \le \frac{1}{b}$

$If a> 0 > b, \ then \ \frac{1}{a} > 0 >\frac{1}{b}$