Class 8: Linear Inequations (Lecture Notes)

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: