Is a < -1?

Hi niks18,
I am not sure, if we can square on both sides of this expression without knowing sign of a, |a| > a + 1
if a is negative, after squaring the equality sign stays, but if a is positive, the equality sign is flipped

Please correct me if i am wrong.

Thanks

Hi hellosanthosh2k2

\(a\) can be negative but \(|a|\) is always positive

so we know that \(|a| > a + 1\) implies that some positive number > some number for e.g if \(a=-5\), then \(|a|=|-5|=5\)

\(5>-5+1 =>5>-4\), clearly you can square both sides of the inequality without changing the sign.

In this case \(a\) cannot be positive because in that case the equation will be

\(a>a+1 => 0>1\) which is impossible.

(1) |a| - a > 1
Now if a is positive or 0, then |a| = a, so the above becomes a-a > 1 or 0 > 1, which is NOT possible. So a cannot be positive or 0.
If a is negative, |a| = -a, so the above becomes -a-a > 1 or a < -1/2. So we know that a is < -1/2, but we dont know whether a is < -1 or not (because a also might lie between -1/2 and -1). So Insufficient.

(2) a^2 > 1
This means a > 1 if a is positive, and a < -1 if a is negative. But we dont know whether a is positive or negative. So Insufficient.

Combining the two statements, first statement rules out that a can be positive or zero. So a can only be negative, and so according to statement 2, it must be < -1. Sufficient.

Hence C answer