Rules of Inequalities
There are various rules in inequalities to help us relate to and solve various different inequalities. Some of these rules are discussed as follows:
Rule 1
If a, b, and c are three numbers then, inequality between these numbers follows transitive property.
- If a > b and b > c, then a > c
- If a < b and b < c, then a < c
- If a ≥ b and b ≥ c, then a ≥ c
- If a ≤ b and b ≤ c, then a ≤ c
Rule 2
If the LHS and RHS of the expressions are exchanged, then the inequality reverses. It is called converse property.
- If a > b, then b < a
- If a < b, then b > a
- If a ≥ b, then b ≤ a
- If a ≤ b, then b ≥ a
Rule 3
If the same constant k is added or subtracted from both sides of the inequality, then both sides of the inequality are equal.
- If a > b, then a + k > b + k
- If a > b, then a – k > b – k
Similarly, for other inequalities.
- If a < b, then a + k < b + k
- If a < b, then a – k < b – k
- If a ≤ b, then a + k ≤ b + k
- If a ≤ b, then a – k ≤ b – k
- If a ≥ b, then a + k ≥ b + k
- If a ≥ b, then a – k ≥ b – k
The direction of the inequality does not change after adding or subtracting a constant.
Rule 4
If k is a positive constant that is multiplied or divided by both sides of the inequality, then there is no change in the direction of the inequality.
- If a > b, then ak > bk
- If a < b, then ak < bk
- If a ≤ b, then ak ≤ bk
- If a ≥ b, then ak ≥ bk
If k is a negative constant that is multiplied or divided by both sides of the inequality, then the direction of inequality gets reversed.
- If a > b, then ak < bk
- If a > b, then ak < bk
- If a ≥ b, then ak ≤ bk
- If a ≤ b, then ak ≥ bk
Rule 5
The square of any number is always greater than or equal to zero.
- a2 ≥ 0
Rule 6
Taking square roots on both sides of the inequality does not change the direction of the inequality.
- If a > b, then √a > √b
- If a < b, then √a < √b
- If a ≥ b, then √a ≥ √b
- If a ≤ b, then √a ≤ √b
Inequalities
Inequalities are the expressions which define the relation between two values which are not equal. i.e., one side can be greater or smaller than the other. Inequalities are mathematical expressions in which both sides are not equal. They are used to compare two values or expressions. It is a mathematical expression used to compare the relative size or order of two objects or values.
They are fundamental in solving problems in mathematics, economics, engineering, and various other fields.
Inequalities
In this article, we will learn about Inequalities including their symbols, rules/properties, types, and their graphical representations and others in detail.