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

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

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

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

A statement of any of the following forms: $\displaystyle i) \hspace{0.5cm} ax + b > 0 \ \ ii) \hspace{0.5cm} ax + b < 0 \ \ iii) \hspace{0.5cm} ax + b \ge 0 \text{ or } \ \ iv) \hspace{0.5cm} ax + b \le 0$ where $\displaystyle a \text{ and } b$ are real numbers  and  is called a Linear Inequation in $\displaystyle x$.

Replacement Set or Universal Set: This is a set which contains all the values of the variable $\displaystyle x$ , 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 $\displaystyle \text{If } a \le b\text{ , then } a + c \le b + c\text{ and } a - c \le b - c$ $\displaystyle \text{If } a \ge b\text{ , then } a+b \ge b+c\text{ and } a-c \ge b-c$

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

If $\displaystyle c \text{ is positive }$ , then  multiplying or dividing by $c$ does not change the inequality: $\displaystyle \displaystyle \text{If } a \ge b\text{ and } c > 0\text{ , then } ac \ge bc\text{ and } \frac{a}{c} \ge \frac{b}{c}$ $\displaystyle \displaystyle \text{If } a \le b\text{ and } c > 0\text{ , then } ac \le bc\text{ and } \frac{a}{c} \le \frac{b}{c}$

If $\displaystyle c$  is negative , then multiplying or dividing by $\displaystyle c$ inverts the inequality: $\displaystyle \displaystyle \text{If } a \ge b\text{ and } c < 0\text{ , then } ac \le bc\text{ and } \frac{a}{c} \le \frac{b}{c}$ $\displaystyle \displaystyle \text{If } a \le b\text{ and } c < 0\text{ , then } ac \ge bc\text{ and } \frac{a}{c} \ge \frac{b}{c}$

Transitive property of inequality states that for any real numbers $\displaystyle a, b, c$ : $\displaystyle \text{If } a \ge b\text{ and } b \ge c\text{ , then } a \ge c$ $\displaystyle \text{If } a \le b\text{ and } b \le c\text{ , then } a \le c$

Converse Property: The relations ≤\text{ and  } ≥ are each other’s converse. For any real numbers $\displaystyle a\text{ and } b:$ $\displaystyle \text{If } a \le b\text{ , then } b \ge a$ $\displaystyle \text{If } a \ge b\text{ , then } b \le a$

Additive inverse: The properties for the additive inverse state: For any real numbers $\displaystyle a\text{ and } b$ , negation inverts the inequality: $\displaystyle \text{If } a \le b\text{ , then } -a \ge -b$ $\displaystyle \text{If } a \ge b\text{ , then } -a \le -b$

Multiplicative Inverse: The properties for the multiplicative inverse state:

For any non-zero real numbers $\displaystyle a\text{ and } b$ that are both positive or both negative: $\displaystyle \text{If } a\le b\text{ , then } \frac{1}{a} \ge \frac{1}{b}$ $\displaystyle \text{If } a\ge b\text{ , then } \frac{1}{a} \le \frac{1}{b}$

If anyone of $\displaystyle a\text{ and } b$ is positive  and  the other is negative, then: $\displaystyle \text{If } a< b\text{ , then } \frac{1}{a} < \frac{1}{b}$ $\displaystyle \text{If } a> b\text{ , then } \frac{1}{a} > \frac{1}{b}$

For any non-zero real numbers $\displaystyle a\text{ and } b$ : $\displaystyle \text{If } 0 < a\le b\text{ , then } \frac{1}{a} \ge \frac{1}{b} > 0$ $\displaystyle \text{If } a \le b < 0\text{ , then } 0 > \frac{1}{a} \ge \frac{1}{b}$ $\displaystyle \text{If } a< 0 < b\text{ , then } \frac{1}{a} < 0 < \frac{1}{b}$ $\displaystyle \text{If } 0 > a \ge b\text{ , then } \frac{1}{a} \le \frac{1}{b} < 0$ $\displaystyle \text{If } a \ge b > 0\text{ , then } 0 < \frac{1}{a} \le \frac{1}{b}$ $\displaystyle \text{If } a> 0 > b\text{ , then } \frac{1}{a} > 0 > \frac{1}{b}$