Answer:
1. We want to write an equation in the form:
Ix - bI ≤ c
Such that the solutions are the range −1 ≤ x ≤ 3
First, the extremes of the range are: - 1 and 3
Half of the difference between these values is:
m = (3 - (-1))/2 = (3 + 1)/2 = 4/2 = 2
Now let's go to the lower extreme, and add this value, we get:
-1 + m = -1 + 2 = 1
(if we go to the upper value and we subtract m, we get the same)
This is the mid-value of the solution interval, and we will define it as M
We now can write the absolute value inequality as:
I x - MI ≤ m
In our case, we get:
Ix - 1I ≤ 2
2.
Now we want to have the solution set given by:
x ≤ -9
x ≥ -5
In this case, we will have an equation of the form:
Ix - MI ≥ m
(before we used the symbol ≤, which means that the solutions are inside the interval, now we use the symbol ≥, which means that the solutions are outside of the interval (-9, -5) in this case).
So let's do the same than in the above case:
m = (-5) - (-9) = (-5 + 9)/2 = 4/2 = 2
M = -9 + 2 = -7
Replacing these in the absolute value inequality we got:
Ix - (-7)I ≥ 2
Ix + 7I ≥ 2
