To solve the inequality |x - 1| - x + 2 > x, you can break it down into two cases based on the value of x within the absolute value:
Case 1: If x - 1 is non-negative (x ≥ 1), then the absolute value of x - 1 is simply (x - 1). In this case, the inequality becomes:
(x - 1) - x + 2 > x
Now, simplify it:
x - 1 - x + 2 > x
Combine like terms:
2 - 1 > x
1 > x
So, for x ≥ 1, the solution is x < 1.
Case 2: If x - 1 is negative (x < 1), then the absolute value of x - 1 is -(x - 1), which is -(x - 1) = 1 - x. In this case, the inequality becomes:
1 - x - x + 2 > x
Now, simplify it:
1 - 2x + 2 > x
Combine like terms:
3 - 2x > x
Now, add 2x to both sides:
3 > 3x
Divide by 3 (since it's positive):
1 > x
So, for x < 1, the solution is x < 1.
Now, combine the solutions from both cases:
x < 1 (from Case 1)
x < 1 (from Case 2)
Since both cases yield the same solution, the overall solution to the inequality is x < 1.