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
- is less than or equal to
- is greater than or equal to
- is not equal to
A linear inequality actually looks exactly like a linear equation , with the inequality sign replacing the equality sign .
Two-dimensional linear inequalities are expressions in 2 variables of the form: where the inequalities may either be strict or not.
A statement of any of the following forms: where and are real numbers and is called a Linear Inequation in .
Replacement Set or Universal Set: This is a set which contains all the values of the variable , 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
Multiplication and Division Property: The property states that for any real numbers,
If , then multiplying or dividing by c does not change the inequality:
If , then multiplying or dividing by inverts the inequality:
Transitive property of inequality states that for any real numbers :
Converse Property: The relations ≤ and ≥ are each other’s converse. For any real numbers
Additive inverse: The properties for the additive inverse state: For any real numbers , negation inverts the inequality:
Multiplicative Inverse: The properties for the multiplicative inverse state:
For any non-zero real numbers that are both positive or both negative:
If anyone of is positive and the other is negative, then:
For any non-zero real numbers :