SOLUTION: If |x + 1| <= 3, then a <= 1/(x + 5) <= b. Find a and b.

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Question 1208929: If |x + 1| <= 3, then a <= 1/(x + 5) <= b.
Find a and b.

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
If |x + 1| <= 3, then a <= 1/(x + 5) <= b.
Find a and b.
~~~~~~~~~~~~~~~~~~~~~~~~~


        The formulation of the problem in the post is incorrect.

        The correct formulation is  " find the maximum a and the minimum b such that a <= 1/(x+5) <= b. "

        Or even simpler: "Find the range of the expression ".

        Below is the solution for this modified formulation. 


If  |x+1| <= 3,  it means that

    -3 <= x+1 <= 3.    (1)


Add 4 to each of the 3 sides of this compound inequality.
You will get then

    1 <= x+5 <= 7.     (2)



Consider left part of the compound inequality (2)

    1 <= x+5.          (3)


Right side of (3) is positive, so we can divide both sides of (3) by (x+5).
You will get then

     <= 1.      (4)



Next consider right part of the compound inequality (2)

    x+5 <= 7.          (5)


Right side of (5) is positive, so we can divide both sides of (5) by 7*(x+5).
You will get then 

      <= .    (6)



From  (4)  and  (6)  we get the final  ANSWER

     <=  <= 1.    


So,  a = ;  b = 1.

Solved  (in the right modified formulation).

When solving and explaining, my task was to perform only necessary
calculations and explanations, without making unnecessary work.



Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


The prescribed values for x are defined by





The interval of x values is thus [-4,2].

For all the values of x in that interval, x+5 is positive, so 1/(x+5) is positive and monotonically decreasing. So the maximum value of 1/(x+5) on [-4,2] is at the left end of the interval and the minimum value is at the right end of the interval.

ANSWERS:

a = minimum value = 1/(2+5) = 1/7
b = maximum value = 1/(-4+5) = 1/1 = 1
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