Questions related to coded inequalities are essential part of middle level competitive examinations such as Bank PO, CPO, AAO, SSC etc. Such problems are not very difficult and very easy for them who are even slightly comfortable with basis mathematics. But for those, who are not at case with maths may find it a bit difficult. This chapter would give you the basic idea of inequalities and methods to solve it in time saving way.

### What is Inequality?

As we know, 3 × 3 = 9

Now, we can say that the result of multiplication between 3 and 3 is equal to 9. Therefore, 3 × 3 = 9 is a case of equality. But when we multiply 3 × 4, we get 12 as a result of this multiplication. It does mean that

3 × 4 ≠ 9

As 3 × 4, is not equal to 9, it is a case of inequality.

When, we come to know that one thing is not equal to another; there can be only two possibilities:

(i) One thing is greater than another thing.

or

(ii) One thing is less than the another thing.

When, we denote (i) and (ii) mathematically, then we will write.

(i) One thing > another thing.

or

(ii) One thing < another thing. Hence, you can write, 3 × 4 > 9,4 × 1 < 9 (3 × 4 > 9) does mean ‘Product of 3 and 4 is greater than 9’.

(4 × 1 < 9) does mean ‘Product of 4 and 1 is less than 9’. Sometimes we come across two numbers where, we do not know the exact state of inequality between them. For example, we may have two numbers m and n and all that we know that ‘n’ is not less than m’. In such case m can be either greater than or equal to n. This situation is represented as ≥ sign. When we have to represent ‘m is less than or equal to n’ then we will use ‘≤’ sign. Let us see: m ≥ n does mean m is either greater than or equal to n. m ≤ n does mean n is either less than or equal to m. Hence, we can summarize the signs to be used in inequalities as below:

### What is chain of Inequalities?

Sometimes two or more inequalities are combined together to create a single inequality having three or more terms. Such combination is called chain of inequalities.

For example 24 > 20 and 20 > 16 can be combined as 24 > 20 > 16.

In the same way, 13 < 17; 17 < 31 and 31 < 38 may be combined as 13 <17 < 31 < 38. How to derive conclusions To derive conclusion from a combined inequality, you have to eliminate the common term. For example, (a) If we have, m > l > n

then, our conclusion is m > n

(b) When, we have, m < l < n then, our conclusion is m < n (c) When, we have ‘≥’ signs in the combined inequalities then you have to think a little bit more. Let us consider the combined inequality: m ≥ l > n

Here, m is either greater than l or equal to l.

Hence, the minimum value for m is equal to l. But l is always greater than n. Therefore, m is always greater than n.

∴ Our conclusion is m > n

(d) When, we have the following inequalities:m > l ≥ n

In this case, m is always greater than l and l is either greater than n or equal to it. When l is greater than n; m will obviously be greater than n. Even when l is equal to n; m will be greater than n as m is always greater than l.

∴ Our conclusion is m > n

(e) When, we have combined inequality, m ≥ l ≥ n

Here, m is either greater than l or equal to l.

When m is greater than l; we have m > l ≥ n, which gives the conclusion.

m > n … (A)

When m is equal to l; we have, m = l ≥ n, which gives the conclusion

m ≥ n … (B)

Combining (A) and (B), we have the final conclusion as

m ≥ n

**Examples:** In the following questions, the symbols ©, @, =, * and $ are used with the following meanings :

- P © Q means ‘P is greater than Q’;
- P @ Q means ‘P is greater than or equal to Q’;
- P = Q means ‘P is equal to Q’;
- P * Q means ‘P is smaller than Q’;
- P $ Q means ‘P is either smaller than or equal to Q’.

Now in each of following questions, assuming the given statements to be true, find which of the two conclusions I and II given below them is/are definitely true.

**Q.1. Statements:** P © T, M $ K, T = K

**Conclusions:**

I. T © M

II. T = M

- only conclusion I is true
- only conclusion II is true
- either I or II is true
- neither I nor II is true

**Solution. (c):** Given statements :

P > T | M ≤ K | T = K |

T = K, K ≥ M ⇒ T ≥ M

⇒ T > M or T = M ⇒ T © M or T = M

So, either I or II is true.

**Q. 2. Statements:** D © F, F = S, S $ M

Conclusions:

I. D © M

II. F @ M

- only conclusion I is true
- only conclusion II is true
- either I or II is true
- neither I nor II is true

**Solution. (d):** Given statements:

D > F | F = S | S ≤ M |

F = S, S ≤ M ⇒ F ≤ M

Therefore, II is not true.

Now, D > F, F ≤ M

⇒ nothing can be said about F and M.

So, I is not true.

**Q. 3. Statements:** J = V, V * N, R $ J

Conclusions:

I. R * N

II. J @ N

- only conclusion I is true
- only conclusion II is true
- either I or II is true
- neither I nor II is true

**Solution. (a): **Given statements:

J = V | V < N | R ≤ J |

R ≤ J, J = V, V < N ⇒ R < N

i.e, R * N.

So, I is true.

Now, J = V, V< N ⇒ J < N

So, J @ N

i.e., J ≥ N is not true.

Thus, II is false.

## Solved Examples

Directions: In the following questions, the symbols @, ©, %, $ and ★ are used with the following meanings illustrated.
- ‘P © Q means ‘P is not greater than Q’.
- ‘P & Q’ means ‘P is not smaller than Q’.
- ‘P % Q’ means ‘P is neither greater than nor equal to Q‘.
- ‘P $ Q’ means ‘P is neither smaller than nor equal to Q’.
- ‘P @ Q’ means ‘P is neither greater than nor smaller than Q’.
In each of the following questions assuming the given statements to be true, find out which of the three conclusions I, II and III given below them is/are definitely true. |

**Question 1.**

**Statements:** J $ D, D © K, K % R

**Conclusions:**

I. R $ J

II. R $ D

III. K $ J

- None is true
- Only I is true
- Only II is true
- Only III is true
- Only II and III are true

**Solution. (3):** J > D, D≤ K, K < R Combining all, J > D≤ K < R or R > D

So, only II is true.

**Question 2.**

**Statements:** M & K, K @ R, R % N

**Conclusions:**

I. R % M

II. R @ M

III. N $ K

- Only I is true
- Only II is true
- Only III is true
- Only either I or II is true
- Only either I or II and III are true

**Solution. (5):** M ≥ K, K = R, R < N
Combining all,
M ≥ K = R < N or R ≤ M and N > K

So, either I or II and III are true.

**Question 3. **

**Statements:** B % H, H $ J, J & M

**Conclusions:**

I. B % J

II. M % B

III. H $ M

- None is true
- Only I is true
- Only II is true
- Only III is true
- Only II and III are true

**Solution. (4):** B < H, H > J, J≥M

Combining all,

B < H > J≥M

or H > M

So, only III is true.

**Question 4. **

**Statements: **Z © K, K % E, E @ R

**Conclusions:**

I. R $ K

II. Z % E

III. R $ Z

- Only I is true
- Only I and II are true
- Only I and Ill are true
- Only II and III are true
- All I, II and III are true

**Solution. (5):** Z≤K, K < E, E = R
Combining all,
Z≤K < E = R or R > K, Z < E and R > Z

So, all are true.

**Question 5. **

**Statements:** W @ M, M © R, R $ F

**Conclusions:**

I. F % M

II. R & W

III. W % F

- None is true
- Only I is true
- Only II is true
- Only III is true
- Only I and II are true

**Solution. (3):** W = M, M ≤ R, R > F

Combining all,

W = M ≤ R > F or R≥W

So, only II is true.