Official ExplanationCorrect Answer: B|x+1| represents the distance of x from -1 on the number line.
So, the question is asking, ‘Is the distance of x from -1 on the number line greater than 6 units or not?’
We see that:
The answer is YES if
i) x < -7 or
ii) x > 5
Else, the answer is NO
Analyzing Statement 1
(x-2)(|x| - 5) > 0The product of two terms is positive.
This means, either both the terms on Left Hand Side are positive or both are negative.
Case 1: x – 2 is positive AND |x| - 5 is positiveThis means, x > 2 AND |x| > 5
Now, |x| is greater than 5 for x < -5 and for x > 5. But x < -5 violates the condition of this case that x – 2 should be positive.
Therefore, the range that satisfies Case 1: x > 5
In this case, the answer to the asked question is YES.
Case 2: x – 2 is negative AND |x| - 5 is negativeThat is, x < 2 AND -5 < x < 5
Upon combining the 2 pieces of information, the resultant inequality is:
-5 < x < 2
In this case, the answer to the asked question is NO.
Thus, we have not been able to get a unique answer to the asked question from St. 1. Therefore, Statement 1 is insufficient.
Analyzing Statement 2
\(x^2 < 4\)That is, \(x^2 - 2^2 < 0\)
\((x+2)(x-2) < 0\)
By using the Wavy Line method, we can easily find that this inequality holds for -2 < x < 2
In this range, the answer is NO.
Since St. 2 does provide us with a unique answer, it is sufficient.
_________________