The absolute value function |x| is defined by
|x| = {(x if x>=0), (-x if x<0):}
As such, we will consider three cases:
Case 1: x < -1
=> x+1 < 0 and x-1 < 0
=>|x+1| = -(x+1) and |x-1| = -(x-1)
=> -(x+1) - (x-1) <= 2
=> -x-1-x+1 <= 2
=> -2x <= 2
=> 2x => 2
=> x>= 1
As this contradicts the premise that x < -1, there are no solutions on (-oo, -1)
Case 2: -1 <= x <= 1
=> x+1 >= 0 and x-1<= 0
=> |x+1| = x+1 and |x-1| = -(x-1)
=> x+1-(x-1) <= 2
=> x+1-x+1 <=2
=> 2 <= 2
As this is true in all cases, every value in [-1, 1] is a solution.
Case 3: x > 1
=> x+1 > 0 and x-1 > 0
=> |x+1| = x+1 and |x-1| = x-1
=> x+1+x-1 <= 2
=> 2x <= 2
=> x <= 1
This contradicts the premise that x>1, meaning there are no solutions on (1, oo).
Taken together, we have the solution set as [-1, 1]. The inequality will hold for any x in that interval, and will not for any x outside of it.
