Guide3:Inequalities:When shud we REALLY ADD two inequalities

[mbolisetty]

I always get confused when we should really perform valid math ops on TWO inequalities.

Eg: if I have an in-equality a < b; b < c;
Now, a < b
b < c
by adding them up, a < 2b < c (combing inequalities & add, which is ABSOLUTELY wrong)
by lining them up, a < b < c (combing inequalities & line, is correct)

My question:
1]When do we have to [Combine & Line] & When do we have to [Combine & Add]
2]Under what circumstances [Combine & Line] works and under what circumstances [Combine & Add] works

[rajkapoor]

you can align them to combine then
a < b
b < c
=> a < b < c

but you cannot add them the way you did after aligning them.

the correct way is -
a < b
b< c
a+b < c +b

and NOT a < 2b < c
( just take a = 1 , b = 2 , c = 3 and you will see the mistake you made )

The answer to your question of when to combine / add or line them up , is simple -
do whatever it takes to get the answer :)

you can think of it as these inequalities as Lego blocks that can be combined in different way to get different output.

couple of rules though -

i) Rule 1 - make sure that the inequalities being added are in the direction.
ii) Rule 2 - if the inequalities are not in the same direction , go back to comply rule 1.
iii) Rule 3 , to make sure Rule 2 follows rule 1 , either change the inequalities or multiply by -1 across and flip the signs.

e.g.
ineq 1 : a < b
ineq 2: c > d => d < c

and add them up
a + d < c +b


or multiply (-1) across the ineq 2 => -c < -d
and then add them up
a - c < b - d

[RonPurewal]

the above poster pretty much nailed this one, but i'll give a more succinct summary.

there are basically two ways in which you can combine inequalities:

1) ADD THEM IF THEY FACE THE SAME WAY
if you have two "less than"s or two "greater than"s, then you can just add them together in the same way in which you would add equations: i.e., add the left sides together, and add the right sides together.

if they don't face the same way, don't worry about subtraction (the rules are unnecessarily complicated for that); just multiply one of them by -1 so that they face the same way, and then follow the rule above.

2) PUT THEM TOGETHER TO MAKE A "CHAIN"
this is what is known as the "transitive property" (you can look this up online if you wish to find further examples).
for instance, if al is taller than bob, and bob is taller than charlie, then al must be taller than charlie.
you can do the same thing with inequalities, as long as they face the same way:
if a > b and b > c, then a > c.
note that it's the quantities on the ENDS of the inequalities that will produce the final inequality.