mode inequality

[vipulgoyal]

In the number line, is r between s and t?
1) | r - s | < | r - t |
2) | r - s | > | s - t |

[Atekihcan]

Statement 1: The following two cases are possible,
<----------r----s-----------t--------->
<----------s----r-----------t--------->

So, r may or may not lie between s and t.
So, statement 1 is not sufficient.

Statement 2: This means distance between r and s is greater than the distance between s and t.
So, it is not possible that r is between s and t.
So, statement 2 is sufficient.

Answer : B

[vipulgoyal]

thanks very well explained just wondering is there any algebric way to check if we havent misseed any possible positions of r,s,t on number line , I mean by opening mode with real values or abslute values

[Atekihcan]

thanks very well explained just wondering is there any algebric way to check if we havent misseed any possible positions of r,s,t on number line , I mean by opening mode with real values or abslute values

Off course there is.
But absolute value problems are better solved by interpreting them as the distances on the number line. That way it is much easier and less time consuming than algebraic method.

And for this particular problem, algebraic approach, i.e. opening absolute value brackets and considering all possible scenarios separately will be a mess. (Just thinking of the scenario sends shivers down my spine )

[vipulgoyal]

hi atekihcan following same way as shown by you, just wanted to make sure the ans of the following inequality is C
Whether S is in between R and T?
(1) | R - T | > | R - S |
(2) | R - T | > | T - S |

[Atekihcan]

hi atekihcan following same way as shown by you, just wanted to make sure the ans of the following inequality is C
Whether S is in between R and T?
(1) | R - T | > | R - S |
(2) | R - T | > | T - S |

Yes, indeed.

For statement 1, following scenarios are possible...
<--------r----s---------t--------->
<----s---r--------------t--------->


For statement 2, following scenarios are possible...
<--------r----------s---t--------->
<--------r--------------t---s----->

Both statements together, distance between r and t is greater than both the distances between r and s and s and t. So, s must be between r and t.

So, both statements together is sufficient.

Answer : C