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.
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
