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Statement 1: x² > x First off, this inequality tells us that x ≠ 0 Second, we can conclude that x² is POSITIVE. So, we can safely divide both sides of the inequality by x² to get: 1 > 1/x If 1 > 1/x, then there are two possible cases: Case a: x > 1. If x is a positive number greater than 1, then 1/x will definitely be less than 1. Case b: x is negative. If x is negative, then 1/x will definitely be less than 1.

IMPORTANT: So how do these two cases affect the answer to the target question? Let's find out. Case a: If x > 1, then x² is greater than 1, AND 1/x is less than 1. This means x² > 1/x Case b: If x is negative, then x² is positive, AND 1/x is negative. This means x² > 1/x Perfect - in both cases, we get the SAME answer to the target question Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 1 > 1/x Notice that this inequality is the SAME as the inequality derived from statement 1 (we got 1 > 1/x) Since we already saw that statement 1 is sufficient, it must be the case that statement 2 is also SUFFICIENT

Its a direct question. First we will rephrase the question. Since x^2 is greater than one we can cross divide it without changing sign now question become 1>1/x^3

option1-->(x^2)>x cross divide with x^2 it becomes 1>(1/X) ..Since it is true then 1>(1/x^3) option 2 is same as option 1 1>(1/X)

Hi could someone let me know if my solution is correct.

The question asks is x^2 > 1/x I reduced the question to is x^3 > 1 by multiplying both sides by x

Option 1: x^2 > x , Divide both sides by x it reduces to x>1 , If x>1 then it implies x^3 > 1 Hence Sufficient

Option 2: 1>1/x ,

Multiplying both sides by x it reduces to x>1, If x>1 then it implies x^3 > 1 Hence sufficient

Bunuel or anyone guide me by letting me know if my approach is correct.

\(x^2 > \frac{1}{x}\) is not the same as \(x^3>1\). You cannot multiply the inequality by x because you don't know its sign. If x is positive, then yes, \(x^2 > \frac{1}{x}\) is equivalent to x^3 > 1 but if x is negative, then when you multiply by negative value you should flip the sign, so in this case you'll get: x^3 < 1.

The same for (1) and (2): never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know its sign.