Inequality

Meaning of inequality 

Inequality in the context of mathematics is a type of statement of an order or sequence relation like less than, greater than, greater than equal to, or less than equal to between any two numbers. Since equations are not always balanced on both sides, which are LHS and RHS. This is the “not equal to” situation, where the LHS≠RHS, each of the sides is either greater or less than the other. So in layman’s terms, inequality is the relationship between the equations which are not equal to each other. These equations, therefore, are called inequality where both sides remain unequal.  

• Understanding inequality

As discussed in the above part, inequality is the mathematical expressions or equations which are not equal on both sides or unlike in LHS=RHS. So in this situation, the equal to sign (=) is converted into less than (<), or more than (>), or not equal to sign(≠). There are different situations of inequality, let’s understand it below by taking expressions as variables a and b. 

1. a≠b, it means a is not equal to b.
2. a<b, it means a is less than b. 
3. a>b, it means a is greater than b. 
4. a≤b, it means a is less than or equal to b. 
5. a≥b, it means a is greater than or equal to b. 

Alert from these inequality scenarios, there are also various types of inequality. Like:

• Rational inequality 
• Polynomial inequality 
• Absolute value inequality 

Linear Inequality class 11

Inequality takes place when there is no equal situation arising between two expressions or equations. In terms of mathematics, inequality can be of algebraic form or numerical forms or in cases in combined form. So linear inequality is a type of inequality that includes a minimum of one linear algebraic equation or expression with a polynomial degree 1 or one side and another if less than me equal to

The definition of linear inequality can be :

When two linear expressions or equations are compared through inequality symbols. These inequality symbols are:

1. Not equal, ≠
2. Less than, <
3. Greater than, >
4. Greater than or equal to, ≥
5. Less than or equal to, ≤

Rules of Linear Inequality class 11

There are four types of rules of inequality which are subtraction, addition, multiplication, and division. When linear equality with the same side, LHS=RHS, then it’s equivalent inequality. All the rules for linear inequality are mentioned below which are needed to be followed while dealing with inequality.  So here are the four rules of inequality in brief:

1. Addition Rule

According to this rule, when the same number is added to both sides of the expression then it will be an equivalent inequality. Suppose a<b, then if 1 is added to both the side, the result will be the same, 1+a<b+1 And if the expression is a>b, then by adding 1 on both sides, we will get a+1>b+1. 

2. Subtraction Rule

As per this rule,  when the same number is subtracted from both sides of the expression, then it is the equivalent inequality. Suppose a<b, then if 1 is subtracted from both the side, the result will be the same, a-1b, then by subtracting 1 both the side, we will get a-1>b-1. 

3. Multiplication Rule

As per this rule,  when the same number is multiplied to both sides of the expression, then it is the equivalent inequality. Suppose a<b, then if 2 is multiplied to both the side, the result will be the same, a*2<b*2 And if the expression is a>b, then by multiplying 2 both the side, we will get a*2>b*2. 

4. Division Rule

As per this rule,  when the same number is divided to both sides of the expression, then it is the equivalent inequality. Suppose a<b, then if 2 is divided to both the side, the result will be the same, a/2<b/2 And if the expression is a>b, then by dividing 2 both the side, we will get a/2>b/2. 

Cauchy Schwartz Inequality 

The Cauchy Schwartz Inequality which is also known as Cauchy’s inequality Is used to bound the expected expressions which are a bit difficult to calculate. This method us one if the widely used inequality. It lets you to divide E[X1, X2] into an upper bound in two parts. The formula for calculating  Cauchy Schwartz Inequality is 

EXY≤ EX2EY2

where, X and Y have finite variance. 

It means for any random variable like we have taken X and Y, the expected value if they are multiplied together, E(XY)² will come as less than or equal to the expected value of squat of variables, E(X²)E(Y²). 

Application of Cauchy Schwartz Inequality

The Cauchy Schwartz Inequality has inequality with almost all the numbers of variables. This inequality is used in various other fields like complex and Classical Real Analysis, Hilbert spaces theory, Qualitative analysis, and Numerical analysis. 

Proofs of Cauchy Schwartz Inequality

Let’s suppose that E[X²]>0 and E[Y²]>0, and 

U=XEX2   and V=YEY2

Then, 2UVU2+V²

So, 

2EUV≤2EUVEU2+EV2=2

Conclusion 

Inequality is the mathematical expression in which both sides are not equal. Generally, these expressions are of none equal properties. So in general words, inequality is a relationship that tries to make comparisons between two expressions which are of non-equal category. While the linear inequality that includes a minimum of one linear algebraic equation or expression with a polynomial degree 1 or one side and another if less than me equal to 1.