Is x^4 - x^3 + x^2 - x > 0? (1) x > 1 (2) |x^3| > |x|

\(x^4-x^3+x^2-x > 0\)

\(x^3(x-1)+x(x-1) > 0\)

\((x-1)(x^3+x) > 0\)

\(x(x-1)(x^2+1) > 0\)

\(x^2 + 1\) is always positive, hence for the expression to hold true, \((x)(x-1) > 0\)

Case 1: \(x > 0\)

If \(x > 0, x - 1 > 0.\)

Therefore \(x > 1\)

Case 2: \(x < 0\)

If \(x < 0, x - 1 < 0.\)

Therefore \(x < 1\).

Combining the information above, either \(x > 1\) or \(x < 0\)

Statement 1

x > 1

We know from the analysis above that if x > 1, the expression holds true. Hence, we can conclude that the information provided in Statement 1 is sufficient to answer the question.

We can eliminate B, C, and E.

Statement 2

\(|x^3|>|x|\)

This statement tells us that on a number-line \(x\) lies in the highlighted region.

------------- -1 ------------- 0 ------------- 1 -------------

In both regions, x is either less than -1 (satisfies the condition \(x < 0\)) or x is greater than 1 (satisfies the condition \(x > 1\)). Hence, this statement is also sufficient to answer the question.

Option D