Expert: Paul Klarreich - 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
Questioner:   John
Category:  Advanced Math
 
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
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Hi, John,

Sorry, I don't think you are correct.  Things look like this, I believe:

************* USE COURIER FONT TO VIEW THIS *************

Let A={x:2<=x<=7}, i.e, the closed interval  [2,7]
and B={x:|x-4|<=h}, i.e the closed interval centered at  x = 4 and  +-h on each side.
         0   1   2   3   4   5   6   7   8             
A:    +---+---+---<===+===+===+===+===>---+-
B:    +---+---+---+---<===C===>---+---+---+- h=1  OK
B:    +---+---+---<===+===C===+===>---+---+- h=2  OK
B:    +---+---<===+===+===C===+===+===>---+- h=3  Not OK
B:    +---<===+===+===+===C===+===+===+===>- h=4  Not OK

I placed  C at  x = 4, the center of  | x - 4 | <= h

How wide can B be (BB? bee-bee?  Sorry.) and still be inside A?  Obviously you can go 3 units to the right before you hit the right bound of A, but only 2 units to the left before you hit the left bound of A.

Therefore, h cannot exceed 2.

Another approach:

| x - 4 | <= h   can be written (and this is important to know for other problems)

- h <= x - 4 <= h,  which you can manipulate:
+ 4      + 4    + 4
------------------------
4 - h <= x <= 4 + h

Obviously, 4 - h is the left boundary and should be at LEAST 2, to match A:

4 - h >= 2
- h >=  -2  
 h <= 2    *** Note required switch of inequality.

and  4 + h is the right boundary and should be at MOST 7, to match A:

4 + h <= 7
   h <= 3   ** No switch here.

Since you want everything inside A, you must take the smaller h.