To solve the equation |-|x + 1| + |x| = 1, we'll need to consider the different cases based on the signs of the absolute values.
Case 1: x ≥ -1
In this case, both x + 1 and x are non-negative, so the absolute value signs can be removed. The equation becomes:
-(x + 1) + x = 1
Simplifying:
-x - 1 + x = 1
-1 = 1
Since -1 ≠ 1, there are no solutions in this case.
Case 2: x < -1
In this case, x + 1 is negative, while x is non-negative. We need to keep the absolute value signs and change the signs of the expressions inside them. The equation becomes:
-(-(x + 1)) + x = 1
Simplifying:
x + 1 + x = 1
2x + 1 = 1
2x = 0
x = 0
Since x = 0 is a solution that satisfies x < -1, it is a valid solution in this case.
Therefore, the solution to the equation |-|x + 1| + |x| = 1 is x = 0.
