Hi,
When dealing with inequalities and absolute values, it helps to know these standard forms.
1. If
x^2 < a^2 then the range for x will always be
-a < x < a.
2. If
x^2 > a^2 then the range for x will always be
x > a and x < -a.
3. If
|x| < a then the range for x will always be
-a < x < a4. If
|x| > a then the range for x will always be
x > a and x < -aNote : The above standard forms hold good even if we have <= or > =. Also the ranges for x^2 < a^2 and x^2 > a^2 are the same for |x| < a and |x| > a. The reason for this is because |x| = \(\sqrt{x^2}\)
So the question here asks us 'Is |x-1| < 1'?
We can break down |x-1|< 1 to -1 < x - 1 < 1. Adding 1 throughout we can rephrase the question to
'Is 0 < x < 2'Statement 1 : (x - 1)^2<=1(x - 1)^2 <=1 -----> -1 <= x - 1 <= 1. Adding 1 throughout we get
0 <= x <= 2. This clearly is not sufficient. As x can be equal to 0 and 2.Statement 2 : x^2 - 1 > 0x^2 > 1 -----> x > 1 and x < -1. This again is clearly insufficient.
Combining 1 and 2 : From Statement 1 we have 0 <= x <= 2 and from Statement 2 we have x > 1 and x < -1. So to satisfy both the statements the only possible range for x will be 1 < x <= 2, which again is insufficient since the target question is
'Is 0 < x < 2'. x can be 1,5 which gives us a YES and x can also be 2 which gives us a NO.
Answer: E
Hope this helps!
CrackVerbal Academics Team
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