Expert: Ahmed Salami - 4/27/2007

Question
hi i have this problem i'm trying to solve.

Let A={x:2<=x<=7} and B={x:|x-4|<=h},h is an element of R.Find the LARGEST value of h for which B subset A.
this is what i tried: when x=2; |2-4|=|-2|=2<=h
when x=3; |3-4|=|-1|=1<=h    when x=4; |4-4|=|0|=0<=h     when x=5; 1<=h   when x=6; 2<=h   when x=7; 3<=h therefore LARGEST value oh h for which B is a subset A is h=3. Is my reasoning correct? thank You in advance

Answer
Hi,
For |x-4| <= h
Then either x-4 <= h
or -(x-4) <= h
The latter resulting in x-4 >= -h
Combining the two expressions gives
-h <= x-4 <= h
i.e 4-h <= x <= 4+h
So for the set B to be a subset of A where
A = {x: 2 <= x <= 7} and
B = {x: 4-h <= x <= 4+h}
4 - h >= 2
AND
4 + h <= 7
The first expression gives h <= 2
The second gives h <= 3
We can therefore see that h = 2 is the highest value satisfying both expressions.
This is the way to go about obtaining a sound solution.
Your reasoning is incorrect.
I hope i have helped you.
You can always get back to me.
Regards.