You can solve this quickly by using algebra.

The target statement Is |x| < 1 ? can be rephrased as Is -1 < x < 1 ?

Statement 2) 1/(1-|x|) > 0 , This implies that (1-|x|) has same sign as 1, to have a overall positive value. Since numerator is positive so (1-|x|) > 0. Solving we get |x| < 1. Sufficient!!!

Statement 1) x^4 - 1 > 0, means (x-1)(x+1)(x^2 + 1) > 0. As (x^2 + 1) is always positive, so we have to consider only (x-1)(x+1) > 0.

Above expression implies x < -1 or x > 1, hence x can never be between -1 and 1 (both non inclusive), so our target statement -1 < x < 1 will not hold true. Sufficient!!!

Answer D.

Can you please rephrase the second question clearly?

Thanks
Puneet

puneetkhurana2000 wrote:Can you please rephrase the second question clearly?

Thanks
Puneet

Nice approach , Thanks Puneet !

2) Is x>3 ?

a)x>0
b) sqrt(x^3 -9x+4) >2 i.e sqare root of x^3-9x+4) > 2

Hope its clear this time .

For Second Question:-

Is x>3 ?

a)x > 0
b)square root of (x^3- 9*x + 4) > 2

Statement 1) x > 0, so x can be 1 or 4. Not Sufficient!!!

Statement 2) (x^3- 9*x + 4) > 2, Taking square both sides we get x(x-3)(x+3)>0, For inequality to hold true either x > 3 or -3 < x < 0. Not Sufficient!!!

Together we have, (x > 0) and (x > 3 or -3 < x < 0), so we get x > 3. Sufficient!!!

Answer C.

puneetkhurana2000 wrote:For Second Question:-

Is x>3 ?

a)x > 0
b)square root of (x^3- 9*x + 4) > 2

Statement 1) x > 0, so x can be 1 or 4. Not Sufficient!!!

Statement 2) (x^3- 9*x + 4) > 2, Taking square both sides we get x(x-3)(x+3)>0, For inequality to hold true either x > 3 or -3 < x < 0. Not Sufficient!!!

Together we have, (x > 0) and (x > 3 or -3 < x < 0), so we get x > 3. Sufficient!!!

Answer C.

Puneet , Thank you so much !

I could not follow the part where you get -3 < x < 0

Since , x(x-3)(x+3)>0 is the equation we get after squaring and simplifying -

i.e x>0 or x>-3 or x>3 then how can we write "-3 < x < 0" ?

Also, Together we have, (x > 0) and (x > 3 or -3 < x < 0) the bolded part doesn't satisfy the condition is x>3 as "Yes ".

Appreciate if you could explain a little .

thanks again!

Is |x| < 1 ?

1)x^4-1 > 0
2)1/1-|x| >0

1) Statement is equivalent to x^4>1. You can use number to visualize what numbers when raised to the fourth power would be greater than 1, this would be all positive numbers > 1 and negative numbers less than -1, which is equivalent to x>1 and x<-1 and for all these values the |x|>1, so the answer to the question is a definite No. Sufficient.

2) The inequality is equivalent to 1-|x|>0, the denominator has to be positive, and this can be further rearranged to |x|>1, again answer is No. Sufficient.

Answer is D.

Is x>3 ?

1)x>0
2) sqrt(x^3 -9x+4) >2 i.e sqare root of x^3-9x+4) > 2

Statement 1: Given x>0, x could be 2, which would answe the question as No, or x could be 5, which would answer the question as Yes. Don't have a definite outcome to the question, Insufficient.

Statement 2: Rewrite it as Sqrt[ x(x^2 - 9) + 4] >2, here we just have to pay attention to the piece x(x^2-9) because if x is 3, then sqrt[x(x^2-9) + 4] is 2. This means to satisfy statement 2, the piece x(x^2-9) must be positive, otherwise we won't be able to satisfy statement 2. You can also square both sides to obtain, x^3 - 9x + 4 > 4, or x^3 - 9x>0.

Therefore, statement 2 is equivalent to x(x^2-9)>0, this would be true if x>3 or -3<x<0. Here x could be > 3 or less than 3.

One could also come up with numbers quickly, x=4 would satisfy statement 2 and answer the question as Yes, and x = -1 would also satisfy statement 2 and answer the question as No.

Statements 1 and 2: Combining 1 and 2 narrows the common values of x to x>3. And the answer to the question Is x>3, is a definite Yes. Sufficient.

Answer is C.

Cheers,
Dabral

Puneet , Thank you so much !

I could not follow the part where you get -3 < x < 0

Since , x(x-3)(x+3)>0 is the equation we get after squaring and simplifying -

i.e x>0 or x>-3 or x>3 then how can we write "-3 < x < 0" ?

Also, Together we have, (x > 0) and (x > 3 or -3 < x < 0) the bolded part doesn't satisfy the condition is x>3 as "Yes ".

Appreciate if you could explain a little .

thanks again!

Since , x(x-3)(x+3)>0 is the equation we get after squaring and simplifying -

i.e x>0 or x>-3 or x>3, you are not right here. Please have a look at the attachment for the values of x where the inequality is > 0 and where it is < 0.

Hope it will make some sense!!!