Inequalities & Coded Inequalities

Theory

Coded inequalities use codes to express inequalities in the form of statements. Generally a couple of conclusions are also included in the question and the candidate is supposed to select which of the two conclusions logically follows the statements. It’s important to exercise caution when it comes to decoding the inequalities as an error could lead to getting all the five questions wrong.

Tip

An important trick in solving coded inequalities is to try and get one inequality with all the variables included in the statement. Using that inequality, the validity of the conclusions can be verified.

Solved Example

[2011 IBPS Clerk]
Directions: In the following questions, the symbols !, %, $, # and @ are used with the following meaning as illustrated below: P$Q means ‘P is not smaller than Q’
P@Q means ‘P is not greater than Q’
P ! Q means ‘P is neither smaller than nor equal to Q’
P # Q means ‘P is neither greater than nor equal to Q’
P % Q means ‘P is neither smaller than nor greater than Q’
In each of the questions, assuming the statements to be true, find which of the conclusions is definitely true. Give answer:
(a) If only conclusion 1 is true
(b) If only conclusion 2 is true
(c) If either conclusion 1 or 2 is true
(d) If neither conclusion 1 nor 2 is true
(e) If both conclusions 1 and 2 are true
Question 1) Statements F @ N, N ! R, H@ R
Conclusions 1) H ! N 2) F # R
Solution: F @ N => $$F \leq N$$
N ! R => N > R
H @ R => H $$\ leq$$ R
So, F $$\leq N > R \geq H$$
Lets check the conclusions
1) H ! N => H > N
From the statements, as N>R and $$R \geq H$$, which implies that N>H. Hence, conclusion 1 is false.
2) F # R => F < R
R < N and F $$\leq$$ N, so we cannot say if F So answer is (d)

Question 2) Statements M # T, T @ K, K $N Conclusions 1) M # N 2) K ! M Solution: M # T => M < T T @ K => T $$\leq$$ K K$ N => K $$\geq$$ N
M < T $$\leq K \geq N$$
Lets check the conclusions:
1) M # N => M < N
M < K and K $$\geq$$ N. So we cannot say if M2) K ! M => K > M
As M < T and T $$\leq$$ K, K>M. Hence, conclusion 2 is definitely true.

Solved Example

Which of the following symbols should replace the question mark (?) in the given expression in order to make the expressions P > A as well as T $$\leq$$ L definitely true? P > L ? A $$\geq$$ N = T
a) $$\leq$$
b) >
c) <
d) $$\geq$$
e)Either $$\leq$$ or <;

Solution: In order for P to be greater than A '?' should be replaced with '>' but in order for T $$\leq$$ L to be true '?' must be replace by $$\geq$$ . Hence, option D.

Solved Example

Which of the following expressions will be true if the expression R > O = A > S < T is definitely true?
a)O > T
b)S < R
c)T > A
d)S = O
e)T < R