Final answer:
To find the largest number x that satisfies |x - 1| < 1 and hence |2x - 2| < 2, we deduce the interval from the first inequality (0 < x < 2) and find that the largest possible value for x that fits both conditions is 2.
Step-by-step explanation:
The question essentially asks us to determine the largest possible value for x given the absolute inequality constraints |x - 1| < 1 leading to |2x - 2| < 2. First, we address the given inequality |x - 1| < 1. This implies that x is 1 unit of distance away from 1 on the number line, so the bounds of x are 0 and 2. Next, |2x - 2| simplifies to |2(x - 1)|, meaning that it will be less than 2 when x is within the interval (0, 2).
To find the largest x that still satisfies the condition, we set the upper bound of the first interval equal to the upper bound of the second interval as follows: 2(x - 1) < 2. Solving for x, we get x < 2. Thus, the largest possible value for x that satisfies both conditions is the upper boundary of the interval, which is 2. To confirm, when x = 2, the original absolute inequality holds true because |2 - 1| = 1 < 1 and |2(2) - 2| = 2 < 2.