Is r between s and t? (1) |r -s| < |r - t| (2) |r -s| > |s - t|

The solution already mentioned is a very novel one, it uses the basic definition of Modulus
|a| = distance of point a from Origin that is (0) or (0,0) or (0,0,0) where a can be anything rational, irrational, real or even imaginary.
and some times we spend lots of time in solving the questions algebraically, simplifying the expressions when they can be very easily done if we spend a lil more time understanding the question.
IN Most of the problems in algebra we can narrow down to the range of answer if we define the range and domain of all the functions involved in that expression.

In this example if you are not able to remember what the modulus of a number means you can always use the basic rule that
|a| < |b| implies ....... -|b| < a<|b|

apply it to both the options.

small query
I am used to the general perception that if there is Mod on both sides of the equation then we have 2 cases

1) Both the sides of the equation have the same sign or 2) Both the sides have opposite signs

using the same logic here for statement 2 --> |r -s | > |s - t|.
I thought we could write this as r-s >s-t -->r+t>2s case 1

or

r-s>-s+t --> case 2 ( both the sides opposite signs ) which gives r>t
so if r>t then statement 2 is also satisfied , but here we have nothing about s
so I thought if r>t then statement 2 is also insufficient , as there is nothing about s. what is the flaw here?

why cannot we have r-s>-s+t and hence r>t for statement 2 ?
Thank you

can any body help with this query of mine, thank you

Not a good way to solve this problem.

What is the question you are trying to answer there?
When you say that r-s and s-t have the same/opposite signs, what cases you'd have?

1 ) In the number line,is R between S and T ?
a.) |r-s|<|r-t|
b.)|r-s|>|s-t|

Merging similar topics. Please refer to the discussion above.

Also, please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 3 and 7. Thank you.

Great Question on the core definition of absolute values.
s---r---t or t----r----s ?

1) |r-s| < |r-t|
Distance b/w r & s is less than the distance b/w r & t.
r--s-------t -> No
s-r------t -> Yes
Insufficient.

2) |r-s| > |s-t|
Distance between r & s is greater than the distance b/w s & t
=> r cannot be between s & t.
Sufficient.

B is the answer.

There can be no easier explanation than this. Thanks!

Similar questions:
https://gmatclub.com/forum/is-s-between ... 92599.html
https://gmatclub.com/forum/is-s-between ... 92600.html

#1
we get |r -s| < |r - t|
t>s
s+t>2r
relation btwn r & t not know

so r....s......t or s...t....r

insufficient
#2
|r -s| > |s - t|
2s>t+r
r>t
so we can say that
t....r....s ; sufficient
IMO B

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