**Tips to solve Inequalities (Coded & Direct)**

There are two types of Inequalities

1. **Direct Inequalities:** In this type, **signs >, <, =, ≤, ≤ and ≠ are mentioned directly** in question. You just have to make it in order as per the question is asked.

2. **Coded Inequalities:** In this type, **signs >,<,=,≤,≤ and ≠ are mentioned in the coded form**. You have to find out the code of signs and solve it.

Now, let’s discuss with the help of examples.

**Examples of Direct Inequality **

**Directions (Q. 1-5):** In these questions, the relationship between different elements is shown in the statements. These statements are followed by two conclusions. Mark answer

(a) If only conclusion I follows.

(b) If only conclusion II follows.

(c) If either conclusion I or II follows.

(d) If neither conclusion I nor II follows.

(e) If both conclusions I and II follow.**1.** **Statements:** A > B ≤ C = D ≤ E, C ≥ F = G > H** Conclusions:** I. G ≤ E II. A > H**Solution:** In conclusion I, the relation is asked between G and E. So, we will try to find the relation between it and we can see that G and E are in different statements. So, first we will identify the element which is common in both statements given i.e. C.

**C = D ≤ E and C ≥ F = G**

Let’s combine these both relation: **G = F ≤ C = D ≤ E****Hence, the conclusion I follow**.

In conclusion II, the relation is asked between A and H, C is also between two statements.

**A > B ≤ C and C ≥ F = G > H**

Combining these, A > B ≤ C ≥ F = G > H.

In this, C > H but the relation between A and C cannot be defined as the relation is changed at B.**So, Conclusion II does not follow.**

Now, a question often came into student's mind that **How to approach inequality questions in Exam without taking more time?**

You should check that inequality sign should follow in the same direction like in statement 1 from C to E signs are in the same direction and In statement 2 from G to C are also following in the same direction. So conclusion 1 follows.

But in case of A and H, the direction of sign between A and C changes at B. It discontinued the relation between A and C, from C to H relation are in the same direction. So the conclusion 2 does not follow.

**2.** **Statements**: H ≥ T > S ≤ Q, T ≥ U = V > B** Conclusions:** I. V > S II. B ≤ H

**Solution:** In statements, T element is common.

For the relation between S and V, the sign changes at T itself. **So Conclusion 1 does not follow.**

For the relation between B and H, the sign does not change

i.e. **H ≥ T ≥ U = V > B** but you can see that sign between V and B is ‘>’. In conclusion H ≥ B**So, Conclusion 2 does not follow.**

**3. Statements:** F < K ≤ L, H ≥ R > K** Conclusions:** I. H > L II. R > F

**Solution:** The element common between given statements is K**For conclusion I:** K ≤ L and H ≥ R > K

combining these, H ≥ R > K ≤ L

For H to L, the relation is discontinued at K. **So it does not follow.****For conclusion II:** R > K and F < K or K > F

combining these, R > K > F**Hence this follows.**

**4. Statements:** N ≥ P > K = L, P ≤ Q < Z, T > K**Conclusions:** I. N < Q II. Z > T

**Solution:** There are three statements in statement 1 and 2 P is common, statement 1 and 3 K is common.**Conclusion I:** Relation is asked between N and Q

N ≥ P and P ≤ Q, i.e. N ≥ P ≤ Q

We can see that directions of signs are changed at P. **So it does not follow.**

**Conclusion II:** Now, we have to connect all three statements.

P > K, P ≤ Q < Z and T > K

K < P ≤ Q < Z and T > K

In this also the direction of sign changes at K. **It does not follow.**

**5.** **Statements:** P < H = O ≥ N, E ≥ H < S**Conclusions:** I. N ≤ E II. S > P

**Solution:** In statement 1 and 2, element H is common.

**Conclusion I:** In statement 1 and 2, H = O ≥ N , E ≥ H

Combining these, E ≥ H = O ≥ N**Hence E ≥ N follows.**

**Conclusion II:** In statement 1 and 2, P < H and H < S

Combining these, P < H < S **Hence S > P. **

These were examples of Direct Inequality. Now we will move on to the examples of Coded Inequalities.

**Coded Inequalities**

In these questions symbols @, #,$, *, and % are used with the different meaning as follows:**'A @ B' means 'A is not smaller than B. ****'A # B' means 'A is neither smaller than nor equal to B’. ****'A $ B' means 'A is neither greater than nor smaller than B. ****'A * B’ means 'A is not greater than B’. ****'A % B’ means 'A is neither greater than nor equal to B’. **

In each of the following questions assuming the given statements to be true, find out which of the two conclusions I and 1I given below them is/are definitely true.**Solution:** Firstly, we will decode the given relations

1. A @ B' means 'A is not smaller than B, it means A is either > or = to B. **So, @ is ≥**

2. 'A # B' means 'A is neither smaller than nor equal to B’. it means A > B. So, **# is >**

3. 'A $ B' means 'A is neither greater than nor smaller than B. it means A = B. So, **$ is =**

4. 'A * B’ means 'A is not greater than B’. it means A is either < or =. So, *** is ≤**

5. 'A % B’ means 'A is neither greater than nor equal to B’. it means A < B. So, **% is <**

**1.Statements **N @ W, W # H, H % T

**Conclusions **

I. H % N

II. T # W

A. Only conclusion I is true.

B. Only conclusion H is true.

C. Either conclusion I or conclusion II is true.

D. Neither conclusion I nor conclusion II is true.

E. Both conclusions I and II are true.

**Solution:**

In statements 1 and 2, W is common.

N@W#H or N ≥ W > H**H < N i.e H % N**

In statements 2 and 3 H is common.

W # H % T or W > H < T

Relation is discontinued at H. so **it does not follow.**

**2.Statements** F # R, H % R, L * H

**Conclusions **

I. F # L

II. R @ L

A. Only conclusion I is true.

B. Only conclusion H is true.

C. Either conclusion I or conclusion II is true.

D. Neither conclusion I nor conclusion II is true.

E. Both conclusions I and II are true.

**Solution:** @ is ≥, # is >, $ is =, * is ≤ and % is <

F > R , H < R and L ≤ H

F > R > H ≥ L**So, conclusion I follows**

Now, conclusion II, R > L i.e. R # L**So it does not follow.**

**3.Statements **J @K, K % M, M # T

**Conclusions **

I. K %T

II. K#T

A. Only conclusion I is true.

B. Only conclusion H is true.

C. Either conclusion I or conclusion II is true.

D. Neither conclusion I nor conclusion II is true.

E. Both conclusions I and II are true.

**Solution:** @ is ≥, # is >, $ is =, * is ≤ and % is <

From statement 2 and 3,

K % M # T i.e. K < M > T

Relation is discontinued at M as sign changes. **So it does not follow.**

The relation is asked between K and T. There is no need to check other statements.

But you can see that **in both conclusions, all three (>, =, <)possible relations have been given.**

So at least, **one relation has to be followed.**

The answer is **either conclusion I or conclusion II follows**.

**4.Statements **V * W, W $ H, H @ I

**Conclusions **

I. V * I

II. I * W

A. Only conclusion I is true.

B. Only conclusion H is true.

C. Either conclusion I or conclusion II is true.

D. Neither conclusion I nor conclusion II is true.

E. Both conclusions I and II are true.

**Solution:** @ is ≥, # is >, $ is =, * is ≤ and % is <

V * W $ H @ I, V ≤ W = H ≥ I

Conclusion 1: Relation is already discontinued at W and H. So,** it does not follow.**

Conclusion 2: W ≥ I, W @ I You will mark that it does not follow.

But see carefully, if it is written I ≤ W, i.e. I * W. **It follows.**

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