Question 1208929
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If |x + 1| <= 3, then a <= 1/(x + 5) <= b.
Find a and b.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The formulation of the problem in the post is incorrect.


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


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Or even simpler: "Find the range of the expression {{{1/(x+5)}}}".


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below is the solution for this modified formulation.



<pre>
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/(x+5)}}} <= 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 

     {{{1/7}}} <= {{{1/(x+5)}}}.    (6)



From  (4)  and  (6)  we get the final  <U>ANSWER</U>

    {{{1/7}}} <= {{{1/(x+5)}}} <= 1.    


So,  a = {{{1/7}}};  b = 1.
</pre>

Solved &nbsp;(in the right modified formulation).


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