I see many of you have doubts with regards to yesterday’s article on solving inequalities. It is actually pretty easy – the easiest way to solve – and the rules aren’t actually confusing – it is just a little difficult to explain and learn without the chalk and board!

Hence, I decided to host another piece solely on solving some particular examples – for every rule for you better understanding.

Hence, I decided to host another piece solely on solving some particular examples – for every rule for you better understanding.

But I have a sincere question did you read it once more and tried solving it any how? My sincere advise would be that you should read the rules once more – I’m sure it’ll make a little bit more sense than the last time – and it will help you to follow the sums solved in this article!

Quick recap:

**– write down the signs ignoring ‘=’ and according to the direction of the equation.**

*Rule 1***– in the jotted down signs – if all signs are in the same direction, i.e., all are greater/greater than, or all are lesser/lesser than – then the sign appearing maximum is the correct ‘relation’ between the characters.**

*Rule 2***– when there are opposite signs – like [< >, > <, ≥ ≤, ≤ ≥, ≥ <, ≤ >]**

*Rule 3*then, we check for these conditions:

(1) if the characters in both the options are same

(2) all the three signs, =, <, > are to be present in any combination

if these 2 cumulatively follow – then answer is

**.**

__either/or__If the two conditions do not meet or one of them does not meet then the answer is none follow or wrong, depending on how the question is given. (we’ll do this one in the examples for sure!)

**are fairly easy to comprehend.**

*Rules 4 and 5*Now, Rule 1 will be applicable in every question – so All the ‘Step 1’ is Rule 1 – i.e., jotting down the signs.

Please solve simultaneously in your copies line by line to help with your speedy understanding!

**Question 1**. A > B, B ≥ C, C ≥ D.

Options: Which is correct?

(a) A>D (b) B>D

Solution:

**– write down the sign ignoring ‘=’**

__Step 1__Solving first –

**option (a)**we need to go from A to D (as we are checking if A is greater than D),

thus the signs are – [ >, ≥, ≥ ] – (

**.)**

*Rule 1***– are there opposite signs?**

__Step 2__**No**. All the signs are greater/greater than equal to! They all point towards the right! (this point clear?)

Since there are all same direction signs, we can safely proceed…

**– which sign is maximum?**

__Step 3__[>, ≥, ≥] can be re-written as [>, > =, > = ]

thus the sign which comes

**maximum is [ > ]**. (

*Rule 2)*Therefore for

__option (a) A>D is correct answer!__Next,

**Option (b)**, B > D. we’ll need to go from B to D.

**= the signs – [ ≥, ≥ ]**

__Step 1__**= Opposites? No. Proceed.**

__Step 2__**= Maximum sign?**

__Step 3__[ ≥, ≥ ] = [ > =, > = ]

no clear winner here – there are 2 ‘>’ and 2 ‘=’

So the relation we can understand by solving is that, B≥D.

Hence, direct relation that B>D is not correct because we do not know for sure!

Thus

__option (b) is incorrect or false.__

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**Question 2.**A > B, B ≥ C, D ≤ C

Options: Which is correct?

(a) A>D (b) B>D

Solution:

**– for option (a) we are going from A to D**

__Step 1__A > B is ‘>’

B ≥ C is ‘≥’. Okay we got these two.

Our destination is to arrive at D. Now we have arrived till C…( A to B to C)

next relation given is D≤C.

we are at C, and we need to go to D, and our current direction is left to right A – B – C!

So to go to D from C, we simply change the signs of D≤C. so I will become C≥D!

now it become A – B – C – D! [phew! I hope you got this!]

so our signs are – [ >, ≥, ≥ ] – (

**.)**

*Rule 1***– opposite signs?**

__Step 2__**. Proceed.**

*No***– maximum sign?**

__Step 3__[>, ≥, ≥] = [>, > =, > = ]

thus the sign which comes

**maximum is [ > ]**. (

*Rule 2)*Therefore for

__option (a) A>D is correct answer!__Next,

**Option (b)**, B > D. we’ll need to go from B to D.

**= the signs – [ ≥, ≥ ] after changing the sign of the relation between D and C.**

__Step 1__**= Opposites? No. Proceed.**

__Step 2__**= Maximum sign?**

__Step 3__[ ≥, ≥ ] = [ > =, > = ]

no clear winner here – there are 2 ‘>’ and 2 ‘=’

So the relation we can understand by solving is that, B≥D.

Hence, direct relation that B>D is not correct because we do not know for sure!

Thus

__option (b) is incorrect or false.__

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Options: Which is correct?

(a) P > U (b) U > T

Solution:

**P < Q, T > Q, T < S, S ≤ U**__Question 3.__

Options: Which is correct?

(a) P > U (b) U > T

Solution:

Solving option (a) P > U

**– we are going from P to U … c’mon write down the signs! C’mon! it is all DIY – DO IT YOURSELF from now on!**

__Step 1__so our signs are – [ <, <, <, ≤ ] – (

**.)**

*Rule 1***– opposite signs?**

__Step 2__**. Proceed.**

*No***– maximum sign?**

__Step 3__[<, <, <, ≤] = [<, <, <, < =]

thus the sign which comes

**maximum is [ < ]**. (

*Rule 2)*Therefore for

__option (a) P > U is incorrect.__Next,

**Option (b)**, U > T … going from U to T … right to left … sign change

**= the signs – [ ≥, > ]**

__Step 1__**= Opposites? No. Proceed.**

__Step 2__**= Maximum sign?**

__Step 3__[ ≥, > ] = [ > =, >]

Thus, U is greater than T

**------------------------------------------------------------------------------------------------------------**

__option (b) is correct.__

**A > B, B = C, C > D, E = D, E > F.**

__Question 4.__

Options: Which is correct?

(a) A = F (b) A > E

Solution:

Solving option (a) A = F

**– so our signs are – [ >, >, > ] – ignoring the ‘=’ - (**

__Step 1__**.)**

*Rule 1***– opposite signs?**

__Step 2__**. Proceed.**

*No***– maximum sign? [ >, >, > ] = ‘>’ (**

__Step 3__

*Rule 2)*Therefore for

**as correct relation is A > F.**

__option (a) A > F is incorrect,__Next,

**Option (b)**, A > E

**= the signs – [ >, > ] – ignoring the ‘=’ - (**

__Step 1__**.)**

*Rule 1***= Opposites? No. Proceed.**

__Step 2__**= Maximum sign? [ >, > ] = ‘ > ’**

__Step 3__Thus,

**as A > E.**

__option (b) is correct,__

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**Question 5.**A ≥ B < C < D < F = H.

Options: Which is correct?

(a) H > B (b) A > D

Solution:

Solving option (a) H > B

**– so our signs are – [ >, >, > ] – ignoring the ‘=’ and changing signs- (**

__Step 1__**.)**

*Rule 1***– opposite signs?**

__Step 2__**. Proceed.**

*No***– maximum sign? [ >, >, > ] = ‘>’ (**

__Step 3__

*Rule 2)*Therefore for

__option (a) H > B is correct.__Next,

**Option (b)**, A > D

**= the signs – [≥, <, < ] - (**

__Step 1__**.)**

*Rule 1***= Opposites?**

Step 2Step 2

**[≥, <] are opposite signs.**

*YES!***= Maximum? Since, opposite signs are present – the maximum rule**

__Step 3__**.**

*cannot be applied***= when you find ‘**

__Step 4__*opposite signs*’ make a quick

*check*– of the two conditions.

Condition 1 – are the elements in options (a) and (b) same?

**No.**

As in (a) we had ‘H’ and ‘B’ and in (b) we have ‘A’ and ‘D’.

Since, condition 1 is ‘no’ – we do not proceed to check condition 2 (if all 3 signs are present).

Thus

*Rule 3*will not apply.

Therefore, any

__direct relation__given in the options will be

__WRONG.__

Thus,

**------------------------------------------------------------------------------------------------------------**

__option (b) is incorrect.__

**Question 5.**I = K > F < J, J ≥ Q.

Options: Which is correct?

(a) F > Q (b) F = Q

Solution:

Solving option (a) F > Q

**– so our signs are – [ <, ≥ ] going from F to Q - (**

__Step 1__**.)**

*Rule 1***– opposite signs?**

__Step 2__**.**

*Yes***– maximum sign? Not applicable**

__Step 3__**– check for Rule 3’s conditions.**

__Step 4__Condition 1 = same elements in both options (a) and (b)?

**both have Fs and Qs!**

*YES,*(

*Please note: Where Condition 1 follows, as you can see both options (a) and (b) get solved simultaneously.)*

Condition 2 = all three signs, [ =, <, > ] present?

**. Only > and = are present**

*NO*Since both conditions have to be met and condition 2 does not follow -

*Rule 3-*of either/or will not apply.

Therefore, for

Options: We have

(a) F ≥ Q (b) F < Q

then,

Solution:

**------------------------------------------------------------------------------------------------------------**__option (a) and (b) which are both direct relations are WRONG.__

**Question 6.**If in the same question -**I = K > F < J, J ≥ Q.**Options: We have

(a) F ≥ Q (b) F < Q

then,

Solution:

Solving option (a) F ≥ Q

**– so our signs are – [ <, ≥ ] going from F to Q - (**

__Step 1__**.)**

*Rule 1***– opposite signs?**

__Step 2__**.**

*Yes***– maximum sign? Not applicable**

__Step 3__**– check for Rule 3’s conditions.**

__Step 4__Condition 1 = same elements in both options (a) and (b)?

**both have Fs and Qs!**

*YES,*(

*Please note: Where Condition 1 follows, as you can see both options (a) and (b) get solved simultaneously.)*

Condition 2 = all three signs, [ =, <, > ] present?

**!!! As in (a) we have, ‘>’ and ‘=’ and in (b) we have ‘<’.**

__YES__Since both conditions are met -

__Rule 3- of either/or will apply.__
Therefore, for correct answer will be

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Thus, for

If condition(s) are not met – then option with direct relation will be wrong.

Rule 3 is applicable ONLY when signs are opposite.

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I really, really, really…and really hope you’ve got the hang of this finally!

You’ve got to try these out yourself with a pen and paper and some more questions.

You’ll definitely see results and I couldn’t have made it more clearer through this medium of communication!

So there you go, fare well this time!

**And any direct relation will be wrong.**__‘Either (a) or (b) follows’.__------------------------------------------------------------------------------------------------------------

Thus, for

**– if conditions met – then either/or will follow.**__Rule 3__If condition(s) are not met – then option with direct relation will be wrong.

Rule 3 is applicable ONLY when signs are opposite.

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I really, really, really…and really hope you’ve got the hang of this finally!

You’ve got to try these out yourself with a pen and paper and some more questions.

You’ll definitely see results and I couldn’t have made it more clearer through this medium of communication!

So there you go, fare well this time!