How to solve 2 absolute value inequalities abs(x+1)+abs(x-1)<=2?

1 Answer
Alan P.
Apr 7, 2015

With respect to the two absolute value expressions
abs(x+1) and abs(x-1)
there are two values of x for which the absolute value operation becomes significant:
x= -1 and x=+1

This gives us three ranges to consider:
(1) x<=-1
(2)-1 < x < +1 and
(3) x >= +1

Case 1 : x <= -1
Both (x+1) and and (x-1) are zero or negative
so the original inequality can be re-written as
-(x+1)-(x-1) <= 2
-2x <= 2
x >= -1
The only value that satisfies (x <= -1) and (x>=-1)
is (x=-1)

Case 2: -1 < x < +1
Of the two terms involving x only (x-1) will be negative and the inequality can be re-written as
(x+1)-(x-1) <= 2
2 <= 2
This is true for all values of x in the "Case 2" range.

Case 3: x>=+1
Both of the terms involving x are positive and the inequality can be re-written as
(x+1)+(x-1)<=2
2x<=2
x<=1

Solution
Combining all three cases we get
-1 <= x <= +1
as the solution range.

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