Understand the Concept of Inequality

In Reasoning Part, another important Topic is Inequality Problems. It is a common topic for all competitive exams. We can expect 5 or 6 questions from Inequality. It is one of the topic, where you can get full marks very easily. I am going to explain a very simple method and i assure you that it’s possible to solve 5 questions in just two minutes.

In Inequality, at first they give some directions for the Problems. And in each problem, there will be statements followed by 4 conclusions. Our job is to find the conclusions which are true.

There are two types of models in Inequality.

1) Coded Inequality.

2) Direct Inequality.

In Coded Inequality, symbols like $, #, %, &, @ are used. In Direct Inequality, we use symbols like <, >, =, ≤, and ≥ . Before learning shortcuts, lets understand what these symbols mean. Most of us learn these symbols in lower classes and they all are familiar ones.

< – Less than

≤ – Less than or Equal to

> – Greater than 

≥ – Greater than or Equal to

= – Equal to

I hope everybody know this. Let’s go to Coded Inequality.

Directions 

P $ Q means P is not smaller 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 greater than nor smaller than Q

P © Q means P is not greater than Q

Now our first work is to find what does the symbol means indirectly?

Lets solve it one by one.

P is not smaller than Q. So P can be either equal to Q or Greater than Q. Symbol is ≥.

P is neither smaller than nor equal to Q. So P is Greater than Q. Symbol is >.

P is neither greater than nor equal to Q. So P is lesser than Q. Symbol is <.

P is neither greater than nor smaller than Q. So P is equal to Q. Symbol is =.

P is not greater than Q. So P can be either equal to or lesser than Q. Symbol is ≤.

Now form a small table before solving this kind of problems.

Now we decoded the symbols and we clearly know what is the meaning of each symbol. Make sure that you form a table as above. The symbols >, ≥, and = should be in one row and the symbols <, ≤, and = should be in another row. Don’t memorize the table. Usually Symbols change. You have to decode it correctly and form a table.

Note:

———-> Forward Direction. 

<———- Reverse Direction.

 In Row 1, First Priority should go to >. Next ≥. Last =. Similarly,

In Row 2, First Priority should go to <. Next ≤. Last =. 

Just remember these lines. You will understand it while solving problems.

Never make a mistake in forming this table. If you make a mistake then you loose all the 5 or 6 marks.

We will try to solve some problems based on the above directions.

Example 1

Statements: N ∂ B, B $ W, W # H, H © M

Conclusions: (1) M @ W.  (2)  H @ N.  (3) W ∂ N.

Step 1 – Make a single statement. 

N ∂ B $ W # H © M

Step 2 – Analyze the conclusions one by one. Always compare with the Modified Statement. 

1. M @ W

• If you draw a line from M to W, you will get a Reverse Line.
• Between M and W two symbols are there. One is © and other is #.  Both are in row 2. Highest Priority is #. Since the letters form a reverse line, we should note the symbol which is exactly above #. The symbol above to # is @. So M @ W is TRUE. 

2. H @ N

• If you draw a line from H to N, you will get a Reverse line.
• Between H and N, the symbols are #, $, and ∂.
• If you check these symbols with the table, # is in Row 2 and $ is in Row 1. So Conclusion 2 is FALSE. 

3. W @ N

• If you draw a line from W to N, you will get a Reverse line.
• Between W and N, the symbols are $ and ∂. Between $ and ∂, the higher priority goes to $.
• The conclusion formed a reverse line. So we should note the symbol which is opposite to $. That is ©. But given conclusion is W @ N. So it is FALSE

So Conclusion one alone Follows

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Example 2

Statements – R © D, D $ J, J # M, M @ K

Conclusions – 1. K # J.   2. D @ M   3. R # M.   4. D @ K

Step 1 – Modify the statement.

R © D $ J # M @ K

Table

Step – 2 – Analyze the conclusion one by one.

1. K # J

• Between K and J – Reverse Line. Symbols are – @ and #.
• In table @ and # are in different rows.
• Conclusion 1 – FALSE

2. D @ M

• Between D and M – Forward Line. Symbols are $ and #.
• In table $ and # are in different rows.
• Conclusion 2 – False.

3. R # M

• Forward line. Symbols are ©, $, and #.
• $ and # are in different rows.
• Conclusion 3 – False.

4. D @ K

• Forward Line. Symbols are $, # and @.
• Symbols are in different rows.
• Conclusion 4 – False.

NOTE: When more than one conclusion in false, check for merging concept. If the characters are same and both the statements are false, and while merging, if we get a meaningful symbol, then the statements can be merged.

In the above problem, all the characters of the statements are different. So we can not merge it. So None is True.

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Example 3

Statements: M $ K, K @ N, N © R, R # W

Conclusions: 1) W @ K  2) M $ R  3) K @ W  4) M @ N

Step 1: M $ K @ N © R # W 

(You don’t need to draw table again and again. I am pasting it for your convenience.)

Analyzing Conclusions

1) W @ K – Reverse line. Symbols are #, ©, and @. In table # and @ are in different Rows. False.

2) M $ R – Forward Line. Symbols are $, @, ©. They are in different rows, False.

3) K @ W – Forward Line. Symbols are @, ©, #. They are in different rows, False.

4) M @ N – Forward line. Symbols are $ and @. Highest priority is for @. So M @ N is True.

Only Conclusion 4 is True.

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Example 4

Statement: H @ T, T # F, F ∂ E, E © V.

Conclusions: 1) V $ F.   2) E @ T.   3) H @ V.   4) T # V

Modified Statement – H @ T # F ∂ E © V

Conclusions

1) V $ F – Reverse Line – Symbols are © and ∂. Both are in Row 2. Since it forms a reverse line, and highest priority is for ©, we should mark the symbol which is exactly opposite to ©. So the correct symbol is $. Conclusion 1 is True.

2) E @ T – Reverse Line. Symbols are ∂ and #. Both are in Row 2. High priority is #. Symbol opposite to # is @. So E @ T is True.

3) H @ V – Forward line. Symbols are in different rows. So False.

4) T # V – Forward line. Symbols are #, ∂, and ©. All are in Row 2. High Priority is for #. So T # V is True.

Answer – Conclusion 1, 2, and 4 are True.