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
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 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
does not change the inequality:
If is negative , then multiplying or dividing by
inverts the inequality:
Transitive property of inequality states that for any real numbers :
Converse Property: The relations ≤\text{ 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 :