# Inequalities

While discussing Equations and Inequations in Polynomials, we have seen what Inequations are. However while discussing Inequalities here, our focus will be to discuss this concept in isolation with Equations, for real numbers only.

# Tools of Inequality

â€˜>â€™ means â€˜greater thanâ€™â€˜<â€™ means â€˜less thanâ€™
â€˜â‰¥â€™ means â€˜greater than or equal toâ€™
â€˜â‰¤â€™ means â€˜less than or equal toâ€™

If â€˜
When â€˜
Similarly When â€˜
The two signs â€˜>â€™ and â€˜<â€™ are called the signs of inequalities.

*N*â€™ is any real number, then value of â€˜*N*â€™ will be either positive or negative or zero.*N*â€™ is positive, we write*N*> 0; which is read â€˜*N*is greater than zeroâ€™.*N*â€™ is negative, we write*N*< 0; which is read as â€˜N is less than zeroâ€™. If â€˜*N*â€™ is zero, we write*N*= 0 and in this case, â€˜*N*â€™ is neither positive nor negative.

# Standard Definitions

â€‹For any set of real numbers*M*and

*N*,

*M*is said to be greater than*N*when*M*â€“*N*is positive*M*>*N*when*M*â€“*N*> 0 and*M*is said to be less than*N*when*M*â€“*N*is negative.*M*<*N*when*M*â€“*N*< 0.

# Number Line

The number line is used to represent the set of real numbers. Below is the brief representation of the number line:

# Basics of Inequalities

It is quite pertinent here to understand some of the very basic properties related to inequalities. These properties should be seen as the building block of the concepts of Inequalities. Assume all the numbers used here are real numbers.

For any two real numbers

*M*and*N*, either*M*>*N*or*M*<*N*or*M*=*N*.If

*M*>*N*, then*N*<*M*.- If
*M*>*N*and*N*>*P*, then*M*>*P*. - If
*M*<*N*and*N*<*P*, then*M*<*P*. - If
*M*>*N*, then*M*Â±*c*>*b*Â±*c*. - If
*M*>*N*and*P*> 0 then*MP*>*NP*. - If
*M*>*N*and*P*< 0, then*MP*<*NP*. - If
*M*<*N*and*P*> 0, then*MP*<*NP*. - If
*M*<*N*and*P*< 0, then*MP*>*NP*. - If
*M*>*N*and*P*>*Q*, then*M*+ P >*N*+*Q*. - If
*M*<*N*and*P*<*Q*, then*M*+*P*<*N*+*Q*. - However if
*M*>*N*and*P*<*Q*, Or,*M*<*N*and*P*>*Q*then we cannot comment about the inequality between (*M*+*P*) and (*N*+*Q*). - If
*M*>*N*and*P*>*Q*, then we cannot infer the inequality sign between (*M*â€“*N*) and (*P*â€“*Q*). Depending on the values of*M*,*N*,*P*and*Q*, it is possible to have (*M*â€“*N*) > (*P*â€“*Q*), (*M*â€“*N*) = (*P*â€“*Q*) or (*M*â€“*N*) < (*P*â€“*Q*) - The square of any real number is always greater than or equal to 0.
- The square of any NON-ZERO real number is always greater than 0.
- If
*N*> 0, then â€“*N*< 0 and if*M*>*N*, then â€“*M*< â€“*N*. - If
*M*and N are positive numbers and*M*>*N*, then- 1/
*M*< 1/*N* *M*/*P*>*N*/*P*if*P*> 0 and*M*/*P*<*N*/*P*if*P*< 0

- 1/
- For any two positive numbers
*M*and*N**If**M*>*N*then*M*^{2}>*N*^{2}.*If**M*^{2}>*N*^{2}, then*M*>*N*.*If**M*>*N*, then for any positive value of*n*,*M*>^{n}*N*.^{n} **For two positive numbers M and N****If***M*/*N*< 1 then*M*<*N**M*/*N*= 1 then*M*=*N**If**M*/*N*> 1 then*M*>*N***Relationship between a number and its square root***N*=*N*, for*N*= 0 or*N*= 1*N*>*N*, for 0 <*N*< 1*N*<*N*, for*N*> 1- Let
*A*,*G*and*H*be the Arithmetic Mean, Geometric Mean and Harmonic Mean of n positive real numbers, then*A*â‰¥*G*â‰¥*H*. Equality occurs only when all the numbers are equal. - If the sum of two positive quantities is given, their product is greatest when they are equal; and if the product of two positive quantities is given, their sum is least when they are equal.
- For any positive number, the sum of the number and its reciprocal is always greater than or equal to 2,
*x*+ where*x*> 0.*x*= 1.

# Cauchy-Schwarz Inequality

If

*a*_{1},*a*_{2}, â€¦â€¦.*a**and*_{n}*b*_{1}*b*_{2}â€¦*b**are 2*_{n}*n*real numbers, then (*a*_{1}*b*_{1}+*a*_{2}*b*_{2}+ â€¦.. +*a*_{n}*b**)*_{n}^{2}__<__

With the equality holding if and only if

Despite all the points given above, however, we should not let the LOGIC die. Most of the questions asked in CAT can be solved by using options and we wonâ€™t be in need of using any concept of Inequalities. But this should not be seen as a case in support of not-going through concepts.

Example-1

If (CAT 05)

- 0 <
*R*__<__0.1 - 0.1 <
*R*__<__0.5 - 0.5 <
*R*__<__1.0 *R*> 10

Solution

As, 30

^{65}â€“ 29^{65}> 30^{64}+ 29^{64}Or, 30

^{64}(30 â€“ 1) > 29^{64}(29 + 1)Or, 30

^{64}Ã— 29 > 29^{64}Ã— 30Or, 30

^{63 }> 29^{63}Hence option (4)

Example-2

If 13

*x*+ 1 < 2*z*, and*z*+ 3 = 5*y*^{2}, then(CAT 03)1.

*x*is necessarily less than*y*.2.

*x*is necessarily greater then*y*.3.

*x*is necessarily equal to*y*.4. None of the above is necessarily true.

Solution

We have

13

and

From (1) and (2) we get

13

â‡’ 13

â‡’ 13

â‡’
If

i.e.

If
i.e.

13

*x*+ 1 < 2*z*.... (1)and

*z*+ 3 = 5*y*^{2}.... (2)From (1) and (2) we get

13

*x*+ 1 < 2 (5*y*^{2}â€“ 3)â‡’ 13

*x*+ 1 < 10*y*^{2}â€“ 6â‡’ 13

*x*< 10*y*^{2}â€“ 7â‡’

*x*<*y*= 1 then, we get*x*< 0.230i.e.

*y*>*x*.If

*y*= 2. Then, we get*x*< 2.538.*x*>*y*. This is not possible. Hence correct answer is (4).Example-3

If |

*b*| â‰¥ 1 and*x*= â€“ |*a*|*b*, then which one of the following is necessarily true?(CAT 03)- a â€“ xb < 0
- a â€“ xb > 0
*a*â€“*xb*> 0*a*â€“*xb*__<__0

Solution

Lets start assuming the values of

*a*and*b*.Assume

*b*= 2 and*a*= 1/2. Then*x*= â€“1.Thus options 1 and 4 have been eliminated.

Assuming
Putting the values in options 1 and 2, we get

*a*= â€“1 and*b*= â€“1, we get*x*= 1.**Ans.(2)**