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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

‘>’ means ‘greater than’
 
‘<’ means ‘less than’
 
‘≥’ means ‘greater than or equal to’
 
‘≤’ means ‘less than or equal to’
 
If ‘N’ is any real number, then value of ‘N’ will be either positive or negative or zero.
 
When ‘N’ is positive, we write N > 0; which is read ‘N is greater than zero’.
 
Similarly When ‘N’ is negative, we write N < 0; which is read as ‘N is less than zero’. If ‘N’ is zero, we write N = 0 and in this case, ‘N’ is neither positive nor negative.
 
The two signs ‘>’ and ‘<’ are called the signs of inequalities.

Standard Definitions

​For any set of real numbers M and N,
  1. M is said to be greater than N when M – N is positive
     
    ⇒ M > N when M – N > 0 and
     
    As we can see, 10 is greater then 5 because 10 – 5 = 5 and 5 is greater than zero.
  2. M is said to be less than N when M – N is negative.
     
    ⇒ M < N when M – N < 0.
     
    As we can see, –10 is less than –5 because –10 –(–5) = –5 and –5 is less than zero.
     
    However, in case of numbers inequalities can be understood through Number Line also.

Number Line

The number line is used to represent the set of real numbers. Below is the brief representation of the number line:
Description: 12512.png

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 ± c > b ± 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 – N) and (P – Q). Depending on the values of MNP and Q, it is possible to have (M – N) > (P – Q), (M – N) = (P – Q) or (M – N) < (P – 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 –N < 0 and if M > N, then –M < –N.
  • If M and N are positive numbers and M > N, then
    1. 1/M < 1/N
    2. M/P > N/P if P > 0 and
    3. M/P < N/P if P < 0
  • 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 nM n > N n.
  • For two positive numbers M and N
     
    If M/N < 1 then M < N
     
    If M/N = 1 then M = N
     
    If M/N > 1 then M > N
  • Relationship between a number and its square root
     
    Let N be a natural number.
     
    N = N, for N = 0 or N = 1
     
    N > N , for 0 < N < 1
     
    N < N , for N > 1
  • Let AG and H be the Arithmetic Mean, Geometric Mean and Harmonic Mean of n positive real numbers, then A ≥ G ≥ 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,
     
    i.e., x + Description: 10794.png where x > 0.
     
    The equality in this relationship will occur only when x = 1.

Cauchy-Schwarz Inequality

If a1a2, ……. an and b1 b2 … bn are 2n real numbers, then (a1b1 + a2b2 + ….. + anbn)2
< Description: 10800.png
 
With the equality holding if and only if
Description: 10807.png
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 won’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 Description: 10813.png(CAT 05)
  1. 0 < R < 0.1
  2. 0.1 < R < 0.5
  3. 0.5 < R 1.0
  4. R > 10
Solution
Description: 10819.png
As, 3065 – 2965 > 3064 + 2964
Or, 3064 (30 – 1) > 2964 (29 + 1)
Or, 3064 × 29 > 2964 × 30
Or, 3063 > 2963
 
Hence option (4)
 
 
Example-2
If 13x + 1 < 2z, and z + 3 = 5y2, 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
13x + 1 < 2z.... (1)
and z + 3 = 5y2.... (2)
From (1) and (2) we get
13x + 1 < 2 (5y2 – 3)
⇒ 13x + 1 < 10y2 – 6
⇒ 13x < 10y2 – 7
⇒ x < Description: 10828.png
 
If y = 1 then, we get x < 0.230
i.e. y > x.
If y = 2. Then, we get x < 2.538.
 
i.e. x > y. This is not possible. Hence correct answer is (4).
 
 
Example-3
If |b| ≥ 1 and x = – |ab, then which one of the following is necessarily true?(CAT 03)
  1. a – xb < 0
  2. a – xb > 0
  3. a – xb > 0
  4. a – xb < 0
Solution
Lets start assuming the values of a and b.
 
Assume b = 2 and a = 1/2. Then x = –1.
 
Thus options 1 and 4 have been eliminated.
 
Assuming a = –1 and b = –1, we get x = 1.
 
Putting the values in options 1 and 2, we get Ans.(2)
 





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