Introduction to Inequalities
Inequality is a relation that builds a comparison that is non-equal amongst 2 numerals or mathematical expressions. It is mostly used in the comparison of 2 numbers present on the number line by their dimensions. There are various notations used to denote varied kinds of inequalities, namely
a] The expression a < b indicates that a is less than b.
b] The expression a > b indicates that a is greater than b.
The value of a is not equal to b in either of the above cases. They are termed strict inequalities. It means that a is rigidly less than or strictly more than b. The equivalence is not included. In contrast to the concept of strict inequalities, there exist 2 kinds of inequality relations that aren’t rigid in nature.
a] The expression a ≤ b indicates that the value of a is less than or equal to the value of b (equivalently, not more than b)
b] The expression a ≥ b indicates that the value of a is greater than or equal to the value of b (equivalently, not less than b)
The relation “not greater than” can be denoted as a ≯ b, and similarly, for the relation “not less than” can be denoted as a ≮ b.
The expression a ≠ b denotes that the value of a is not equal to the value of b and is occasionally taken as one of the rigid forms of inequalities.
Properties of Inequalities
Inequalities are controlled by the following properties. These properties are valid even if the inequalities that are non-strict are restored by their corresponding rigid inequalities.
The relations greater than or equal to and lesser than or equal to are one another’s converse. It simply means that for any two real numbers a and b:
a ≤ b and b ≥ a are identical.
The transitive property of inequality for any three real numbers a, b and c is
If a ≤ b, b ≤ c then a ≤ c.
If a < b, b ≤ c, then a < c.
c] Addition and subtraction
A constant that is common can be added or subtracted on both sides of an inequality. For any three real numbers a, b and c
If a ≤ b, then a + c ≤ b + c and a – c ≤ b – c.
In other words, the inequality relation is conserved under the operation addition (or subtraction) and the numbers that are real are an ordered category under addition.
d] Multiplication and division
If x < y and a > 0, then ax < ay.
If x < y and a < 0, then ax > ay.
The characteristics that deal with multiplication and division for any three real numbers a, b and non-zero c
If a ≤ (less than or equal to) b and c > 0, then ac ≤ (less than or equal to) bc and a / c ≤ (less than or equal to) b / c.
If a ≤ (less than or equal to) b and c < 0, then ac ≥ (greater than or equal to) bc and a / c ≥ (greater than or equal to) b / c.
In other words, the inequality relation is conserved under the operations division and multiplication with the constant that is positive but is reversed when an integer that is negative is included. In general, it is applicable to an ordered field.
e] Additive inverse
For any two real numbers a and b, the property of additive inverse is given by
If a ≤ (less than or equal to) b, then – a ≥ (greater than or equal to) – b.
f] Multiplicative inverse
In the case of both numerals being positive, then the inequality relation between the inverses of multiplication is contrasting with that amongst the original numbers.
A note on inequalities is written above. For more information on various mathematical concepts such as quadratic inequalities, equations, quadratic equations, logarithmic functions, trigonometry etc., please refer to BYJU’S website. It consists of a detailed explanation that assists the students in exam preparation.