Independence Day - 78INDIA

# How to actually solve the Inequalities with Examples

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.

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:
Rule 1 – write down the signs ignoring ‘=’ and according to the direction of the equation.
Rule 2 – 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 3 – when there are opposite signs – like [< >, > <, ≥ ≤, ≤ ≥, ≥ <, ≤ >]
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 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!)
Rules 4 and 5 are fairly easy to comprehend.

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

Question 1.     A > B, B ≥ C, C ≥ D.
Options: Which is correct?
(a) A>D          (b) B>D

Solution:

Step 1 – write down the sign ignoring ‘=’
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.)

Step 2 – are there opposite signs? 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…

Step 3 – which sign is maximum?
[>, ≥, ≥] 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.
Step 1 = the signs – [ ≥, ≥ ]

Step 2 = Opposites? No. Proceed.

Step 3 = Maximum sign?
[ ≥, ≥ ] = [ > =, > = ]

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:

Step 1 – for option (a) we are going from A to D
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.)

Step 2 – opposite signs? No. Proceed.

Step 3 – maximum sign?
[>, ≥, ≥] = [>, > =, > = ]

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.
Step 1 = the signs – [ ≥, ≥ ] after changing the sign of the relation between D and C.

Step 2 = Opposites? No. Proceed.

Step 3 = Maximum sign?
[ ≥, ≥ ] = [ > =, > = ]

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 3.
P < Q, T > Q, T < S, S ≤ U
Options: Which is correct?
(a) P > U          (b) U > T

Solution:
Solving option (a) P > U
Step 1 – 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!

so our signs are – [ <, <, <, ≤ ] – (Rule 1.)

Step 2 – opposite signs? No. Proceed.

Step 3 – maximum sign?
[<, <, <, ≤] = [<, <, <, < =]

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
Step 1 = the signs – [ ≥, > ]
Step 2 = Opposites? No. Proceed.
Step 3 = Maximum sign?
[ ≥, > ] = [ > =, >]

Thus, U is greater than T option (b) is correct.

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Question 4.
A > B, B = C, C > D, E = D, E > F.
Options: Which is correct?
(a) A = F          (b) A > E

Solution:
Solving option (a) A = F
Step 1 – so our signs are – [ >, >, > ] – ignoring the ‘=’ - (Rule 1.)

Step 2 – opposite signs? No. Proceed.

Step 3 – maximum sign? [ >, >, > ] = ‘>’  (Rule 2)

Therefore for option (a) A > F is incorrect, as correct relation is A > F.

Next, Option (b), A > E
Step 1 = the signs – [ >, > ] – ignoring the ‘=’ - (Rule 1.)
Step 2 = Opposites? No. Proceed.
Step 3 = Maximum sign? [ >, > ] = ‘ > ’

Thus, option (b) is correct, as A > E.

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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
Step 1 – so our signs are – [ >, >, > ] – ignoring the ‘=’ and changing signs- (Rule 1.)

Step 2 – opposite signs? No. Proceed.

Step 3 – maximum sign? [ >, >, > ] = ‘>’  (Rule 2)

Therefore for option (a) H > B is correct.

Next, Option (b), A > D
Step 1 = the signs – [≥, <, < ] - (Rule 1.)

Step 2
= Opposites? YES!  [≥, <]  are opposite signs.

Step 3 = Maximum? Since, opposite signs are present – the maximum rule cannot be applied.

Step 4 = when you find ‘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.

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Question 5.           I = K > F < J, J ≥ Q.
Options: Which is correct?
(a) F > Q          (b) F = Q

Solution:
Solving option (a) F > Q
Step 1 – so our signs are – [ <, ≥ ] going from F to Q - (Rule 1.)

Step 2 – opposite signs? Yes.

Step 3 – maximum sign? Not applicable

Step 4 – check for Rule 3’s conditions.
Condition 1 = same elements in both options (a) and (b)?
YES, both have Fs and Qs!

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

Condition 2 = all three signs, [ =, <, > ] present?
NO. Only > and = are present

Since both conditions have to be met and condition 2 does not follow - Rule 3- of either/or will not apply.

Therefore, for option (a) and (b) which are both direct relations are WRONG.

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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
Step 1 – so our signs are – [ <, ≥ ] going from F to Q - (Rule 1.)

Step 2 – opposite signs? Yes.

Step 3 – maximum sign? Not applicable

Step 4 – check for Rule 3’s conditions.
Condition 1 = same elements in both options (a) and (b)?
YES, both have Fs and Qs!

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

Condition 2 = all three signs, [ =, <, > ] present?
YES !!! As in (a) we have, ‘>’ and ‘=’ and in (b) we have ‘<’.

Since both conditions are met - Rule 3- of either/or will apply.

Therefore, for correct answer will be ‘Either (a) or (b) follows’. And any direct relation will be wrong.

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Thus, for Rule 3 – if conditions met – then either/or will follow.
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!