Final answer:
The solution to the equation sqrt((x-1)²) = 1 - x is x = 1.
Step-by-step explanation:
To solve the given equation, we start by simplifying the expression inside the square root. The square root of (x-1)² is the absolute value of (x-1). Therefore, we have |x-1| = 1 - x.
Now, we consider two cases:
1. When (x-1) is non-negative (x-1 ≥ 0):
In this case, |x-1| is equal to (x-1). So, the equation becomes (x-1) = 1 - x. Solving for x, we find x = 1.
2. When (x-1) is negative (x-1 < 0):
In this case, |x-1| is equal to -(x-1). So, the equation becomes -(x-1) = 1 - x. Solving for x, we find x = 1.
Since both cases lead to the same solution, x = 1, we conclude that this is the solution to the given equation. It's important to note the application of absolute value to consider both positive and negative cases.