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
