Properties of Inequalities

“If real numbers or the algebraic expressions are related by means of the symbols “>”, “<”, “≥”, “≤”, then the relation is referred to as an inequality.”

Open Sentence – The inequality is stated to be an open sentence if it has the handiest variable.

Example -:  y > 6 ( y is greater than 6)

                     X < 9 ( x is less than 9 )

Double Inequalities – The inequality is known to be a double inequality if the statement represents the double relation of the expressions or the numbers.

 Example -: 5 < x < 9  ( x is greater than 5 and less than 9 )

                         8 < y < 9 ( y is greater than 8 and less than 9)

Inequality Symbols –

The most familiar inequality sign is the “now not identical sign (≠)”. But to evaluate the values at the inequalities, the following symbols are used.

• Strict Inequality -: 

The strict inequality symbols are less than symbol (<) and greater than symbol (>). These two symbols are referred to as strict inequalities because it suggests the numbers are strictly extra than or less than each other.

For example,

7 < 9 ( 7 is strictly less than 9)

20 >8 (20 is strictly greater than 8)

• Slack Inequality -: 

The slack inequalities are less than or same to symbol (≤) and greater than or equal to a symbol (≥). The slack inequalities constitute the relation between two inequalities that are not strict.

For example,

X ≥ 18 ( x is greater than or equal to 18)

Y ≤ 6 (Y is less than or equal to 6)

Properties of Inequalities

Transitive Property –

The relation among the three numbers has described the usage of the transitive property.

If x, y, and z are the 3 numbers, then

If x ≥ y and y ≥ z, then x ≥ z

Similarly,

If x ≤ y, and y ≤ z, then x ≤ z

Within the above-mentioned example, if one relation is described by means of strict inequality, then the end result ought to also be in strict inequality.

For example,

If p ≥ q , and q > r, then p > r.

Addition and Subtraction Property

The addition and subtraction assets of inequalities state that adding or subtracting the equal constant on both sides of inequalities are equivalent to each other.

Let “n” be constant,

If a ≤ b, then a +n ≤ b + n

If a ≥ b, then a +n ≥ b + n

Similarly, for the subtraction 

If a ≤ b, then a -n ≤ b-n

If a ≥ b, then a –n ≥ b-n

Multiplication and Division Property-

If a positive constant number is multiplied or divided by both facets of inequality, the inequality stays equal. but, if inequality is multiplied or divided through the negative constant number, the inequality expression will get reversed.

Let “n” be a positive constant,

If x ≤ y, then x n ≤ y n (if m>0)

If x ≥ y, then x n ≥ y n (if m>0

The below condition holds true for the division operation, 

Let “n” be a negative constant number,

If x ≤ y, then x n ≥ y n (if m<0)

If x ≥ y, then x n ≤ y n (if m<0)

Non-negative property of squares-

When the square of a number is done, it will always be greater than or even equal to zero. 

For example-  62=36

                       (-6)2=36

                        02=0

Property of square root-

The inequality will not be changed if the square root is done provided that the numbers are greater than or equal to zero. 

For example –              x≤y,  hence x ≤ √y   where x, y 0

Law of Trichotomy-

This law of trichotomy tells that in the below-mentioned equations only one of them is true. 

     x y   

We can say that x is less than y or x is equal to y or x is greater than y. It should be one of them. 

Conclusion –

In arithmetic, a declaration of an order relationship—greater than, more than, or identical to, less than, or much less than or equal to—between numbers or algebraic expressions. We have understood what are inequalities in mathematics and how are they important in mathematics to solve any inequalities-related problems. We have even tried explaining the inequalities symbol and how to use them in solving the problems. We have gone through the properties of inequalities and explained them with examples to understand them better. We have explained the transitive property, addition and subtraction property, multiplication and division property, non-negative property of squares and the last one is property of square root.