Is a

Question

Is  a<-1 ?

(1)  |a|>a+1

(2)  a^2>1

Self made : tricky

akshata19 · Answer

My try on this.. not sure approach is optimal

(1) |a|>a+1

Checking with examples for validity

Positive:
a=1

1>1+1
1>2 -----> No

Negative
a=-1
1>-1+1
1>0 ------> Yes

So above eqtn is satisfied when
a<=-1

Hence A cannot be the answer

(2) a^2>1

a cannot take value 1,-1,0

So we can conclude a can take value
1<a<-1

Hence B cannot be the answer

Combining both I and II
C gives a<-1

Posted from my mobile device

niks18 · Answer

Statement 1 : implies that a>a+1 =>0>1 . Not Possible or a<-(a+1) => a<-0.5 but a can be less than -1 or greater than -1 . Hence insufficient Another way to analyse this statement - |a|>a+1 , square both sides to get a^2>a^2+2a+1 => a<-0.5 Statement 2 : a^2>1 => a>1 or a<-1 . Hence Insufficient Combining 1 & 2: a<-0.5 & a<-1 , the overlapping region is a<-1 . Sufficient Option C

hellosanthosh2k2 · Answer

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

niks18 · Answer

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.

amanvermagmat · Answer

(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

Is a < -1?

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