How do you solve #abs(x+1)+ abs(x-1)<=2#?
With moduli it is useful to split into cases at the point that the sign of the enclosed expression changes.
Case (a) :
Dividing both ends by
Notice that we have to reverse the inequality, because we are dividing by a negative number.
So in Case (a) we have
These conditions cannot be satisfied at the same time, so Case (a) yields no solutions.
Case (b) :
So the target inequality is satisfied for all
Case (c) :
So in Case (c) the inequality is never satisfied.
So the solution is