DS Inequalities – Modulus

Is |a| > a?
1. a² < a
2. (a/2) > (2/a)

The magnitude of ‘a’ will be greater than a only if ‘a’ is a negative number. For positive numbers the magnitude of ‘a’ will be equal to ‘a’.

So, what we have to determine using the two statements is whether a is negative.

Statement 1: a² < a

This inequality holds good only when 0 < a < 1.  i.e., we can conclude that a is positive. 
Hence, we can answer the question – whether a is negative with a definite NO.

Therefore, statement 1 is SUFFICIENT.

Statement 2: (a/2) > (2/a)

a is a variable and can therefore, take both positive and negative values.

We can derive two options from the information given in statement 2

Option 1:  a² > 4 only if a > 0
i.e.,  a² – 4 > 0 and a > 0
or (a + 2) (a – 2) > 0 and a > 0
or a > 2 or a < -2 and a > 0

Given that option 1 holds good only if a > 0, a > 2 and a cannot be less than -2. 
Option 1 therefore, points to the result that a is positive.

Option 2:  a² < 4 if a < 0
i.e.,  a² – 4 < 0 and a < 0
or (a + 2) (a – 2) < 0 and a < 0
or -2 < a < 2 and a < 0

Given that option 2 holds good only if a < 0, the range of values that a can take narrows down to -2 < a < 0.
Option 2 therefore, points to the result that a is negative.

Statement 2 leaves us with both the possibilities – ‘a’ could be positive or ‘a’ could be negative. 

Therefore, statement 2 is NOT SUFFICIENT.

Choice A is the correct answer.