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# Inequality

In inequality problems, each problem will have a statement followed by conclusions. We can easily score 4-5 marks from this topic in few minutes. Inequality is common topic for all competitive exams.

So, let’s learn this topic.

__RULES:__

__“>” (more than) & “≥” (more than equal to) symbol explanation:__

If the conclusion contains “>” (more than) symbol:

- It will satisfies if, “>” (more than) symbol present in the statement between objects.
- It will also follow “= (equal to)”, “≥ (more than equal to)” symbols but “> (more than)” symbol must be present at least in statement between the objects as per the conclusion.

If the conclusion contains “≥” (more than equal to) symbol:

- It will satisfies if, “≥” (more than equal to) symbol present in the statement between objects.
- It will also follow “= (equal to)” symbol, but “≥ (more than equal to)” symbol must be present at least in statement between the objects as per the conclusion.

Note: if the statement contains “≥ (more than equal to)” symbol, and conclusion has “= (equal to)” & “>” (more than) symbol, in this case either option will come into existence.

__“<” (less than) & “≤” (less than equal to) symbol explanation:__

If the conclusion contains “<” (less than) symbol:

- It will satisfies if, “<” (less than) symbol present in the statement between objects.
- It will also follow “= (equal to)”, “≤ (less than equal to)” symbols but “< (less than)” symbol must be present at least in statement between the objects as per the conclusion.

If the conclusion contains “≤” (less than equal to) symbol:

- It will satisfies if, “≤” (less than equal to) symbol present in the statement between objects.
- It will also follow “= (equal to)” symbol, but “≤ (less than equal to)” symbol must be present at least in statement between the objects as per the conclusion.

Note: if the statement contains “≤ (less than equal to)” symbol, and conclusion has “= (equal to)” & “<” (less than) symbol, in this case either option will come into existence.

__Important Points :-__

__Example 1: __

**Statement:**

P < Q > R < S > T

**Conclusion:**

**Explanation:** Sign used in statement as compared to conclusions are opposite, so the conclusion will be false.

__Example 2: __

**Statement:**

P ≥ Q ≤ R ≥ S ≤ T

**Conclusion:**

**Explanation:** Sign used in statement as compared to conclusions are opposite, so the conclusion will be false.

__Example 3: __

**Statement:**

P ≤ Q ≤ R ≤ S ≤ T

**Conclusion:**

**Explanation:** Sign used in statement as compared to conclusions are SAME, so the conclusion will be TRUE.

**Important conclusion Based on Statement **

S.No |
Statement |
Conclusion |

1. | A > B > C | A > C |

2. | A > B ≥ C | |

3. | A ≥ B > C | |

4. | A = B > C | |

5. | A > B = C | |

6. | A <B < C | A < C |

7. | A < B ≤ C | |

8. | A ≤ B< C | |

9. | A = B < C | |

10. | A < B = C | |

11. | A ≥ B ≥ C | A≥C(Either A>C or A=C) |

12. | A = B ≥ C | |

13. | A ≥ B = C | |

14. | A ≤ B ≤ C | A ≤ C (Either A < C or A = C) |

15. | A = B ≤ C | |

16. | A ≤ B = C | |

17. | A < B > C | Either 1 or 2 follows if any of the following cases (a, b, c and d) are given as they form a complementary pair.
a) 1. A > C 2. A ≤ C b) 1. A ≥ C 2. A < C c) 1. A < C 2. A ≥ C d) 1. A ≤ C 2. A > C |

18. | A ≤ B> C | |

19. | A < B≥ C | |

20. | A > B < C | |

21. | A > B ≤ C | |

22. | A ≥ B < C |

*Example of Coded Inequality in Reasoning*

*Example of Coded Inequality in Reasoning*

**Directions:** In the following questions, the symbols δ, @, ©, % and ⋆ are used with the following meaning as illustrated below.

‘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 equal to B’.

‘A δ B’ means ‘A is not greater than B’.

‘A @ B’ means ‘A is neither greater than nor smaller than B’.

Now in each of the following questions assuming the given statements to be true, find which of the four conclusions I, II, II and IV given below them is / are definitely true and give your answer accordingly.

**Statements:
**P δ T, T @ R, R © O, O % K

**Conclusions:
**I. R @ P

II. R % P

III. K ⋆ T

IV. O δ T

1) Only either I or II is true

2) Only III and IV are true

3) Only either I or II and III are true

4) Only either I or II and IV are true

5) Only either I or II and III and IV are true

Follow the steps given below to simplify the process.

*Steps Involved in Solving Coded Inequality in Reasoning*

*Steps Involved in Solving Coded Inequality in Reasoning*

**Step 1:** **Make Decoding Table.**

The easiest method is to first make a table as shown below.

**NOTE:** Elements used in question are A and B so we have added A and B in table.

**TIP:** Sometimes, to make questions more complicated, reverse relations may be given as:

‘A * B’ means ‘B is not smaller than A’.

So here we will write B in the first row and A in the last row.

Step 2:**Add Symbols to Table**

**Step 3:** Start decoding symbols one by one. Then add decoded operator into the table.

Here symbols are:

© → not smaller than → means greater than or equal to → **‘≥’
**% → neither smaller than nor equal to → means greater than →

**‘>’**

⋆ → neither greater than nor equal to → means smaller than →

**‘<’**

δ → not greater than → means smaller than or equal to →

**‘≤’**

@ → neither greater than nor smaller than → means equal to →

**‘=’**

So our decoding table becomes:

We will now use this decoding table to solve the actual questions.

Step 4: Decode Statements using Decoding Table.

**Statements:** P δ T, T @ R, R © O, O % K

**Decoded statements:** P ≤ T, T = R, R ≥ O, O > K

**
Step 5: Combine Decoded Statements
**Combined statement will be: P ≤ T = R ≥ O > K

Step 6:**Conclude Individually**

Look at conclusions one by one, decode each conclusion using the Decoding Table. Then check whether the conclusion follows or not.

__Conclusion I__: R @ P → R = P

Now from the combined statement we get, P ≤ T = R.

According to priority level we get, P ≤ R.

Thus R = P is false.

__Conclusion II__*:* R % P → R > P

From the combined statement we get, P ≤ T = R.

Thus again we get P ≤ R.

So R > P is false.

But we know from the combined statement that P ≤ R. Hence either conclusion I or II has to be true as they form complementary pair.

### Practice Set for Inequality

**Directions (1 – 5)**: In the following questions, the symbols %, @, #, $ and & are used with the following meaning as illustrated below:

‘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 not greater than Q’.

‘P$Q’ means ‘P is not smaller than Q’.

‘P&Q’ means ‘P is neither smaller than nor greater than Q’.

Now in each of the following questions, assuming the given statements to be true, find which of the four conclusions I, II, and III given below them is/are definitely true and give your answer accordingly.

- Statements: Q%C, C$W, W&D, D@X

Conclusions:

I. C%D

II. W@Q

III. X%W

A) Only I

B) Only II

C) Both II and IIID)AllI, II and III

E) Only I and II - Statements: S&W, W#Q, Q%X, X$V

Conclusions:

I. Q&S

II. X%W

III. V@Q

A) Only II

B) Only I

C)Only I and III

D) Only IIIE) None of these

- Statements: A@W, W$E, E#S, S&J

Conclusions:

I. A$E

II. J&W

III. A@E

A) Either I or IIIB) Only I

C) Only II and III

D) All I, II and III

E) Either I or III and II - Statements: A#B, X%B, R$X, R@S

Conclusions:

I. R%B

II. R$B

III. A@S

A)Both I and II

B) Both I and IIIC) Both II and III

D) Either I or II and III

E) Only I - Statements: X$D, D%W, W&F, F@A

Conclusions:

I. A@W

II. X%F

III. D&A

A) Only III

B)Both I and II

C) All I, II and III

D) Both II and III

E) None of these

**Direction (6-10):**Relationship between different elements is shown in the statements. Find if the conclusions also follow or not. - Statements: W > A ≤ G, S ≤ A, I ≤ B = S

Conclusions:

G ≥ I

G = I

A) only I followsB) only II follows

C) either I or II follows

D) neither I nor II follow

E) both I and II follow - Statements: Q< W ≤ B, C> B < K, W ≥ A

Conclusions:

I. K > W

II. C> A

A) only I follows

B) only II follows

C) either I or II follows

D) neither I nor II follow

E) both I and II follow

- Statements: A > E ≥ F, P = E, S > A

Conclusions:

S> P

F < S

A) only I follows

B) only II follows

C) either I or II follows

D) neither I nor II follow

E) both I and II follow

- Statements: S ≥ W > K, W > F < A, P > S

Conclusions:

I. P >A

II. S> F

A) only I follows

B) only II followsC) either I or II follows

D) neither I nor II follow

E) both I and II follow - Statements: R ≥ X ≥ S, S< C, C > W

Conclusions:

W> X

X ≥ W

A) only I follows

B) only II follows

C) either I or II followsD) neither I nor II follow

E) both I and II follow

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