Quote:
Why are the questions "Is |x – 1| < 1 ?" and "Is x > 0?" the same?
They are the same under the condition that
x – 1 < 0, so that we simplify the absolute value: |
x – 1| = 1 –
x .
In Statement (2) we show that
x < 0 implies
x – 1 < 0, which turns "Is |
x – 1| < 1?" into "Is
x > 0?". The latter contradicts Statement (2),
x < 0.
If to speak about an equivalent inequality for |
x – 1| < 1 with no additional conditions, then it is
-1 <
x – 1 < 1
0<
x < 2
In other words its solution is that
x must be between 0 and 2.