Is x^2 > 1/x?

Hi Paddy41,

This is a typical inequality DS question in GMAT.

It’s a YES-NO DS question.

Before moving into the statements, start understanding for what values of x, x^2 < 1/x?

Question: x^2 < 1/x?

Let’s first try with positive integers, For x = 2, 3, 4… x^2 > 1/x, here the answer to the question is NO.

For negative values, that is, x < 0, x^2 > 1/x, here the answer to the question is NO.

For “x” a proper(positive) fraction, x = ½, ¼... x^2 < 1/x, here the answer to the question is YES.

Sometimes knowing the YES and NO answer to the question helps to solve the question better.

So, NET-NET, if 0<x<1 then the answer to the question is YES.

Or else Answer to the question is NO.

Let’s move onto statements now,

Statement I is sufficient:

x^2 > x

Except for the values between, 0<x<1, x^2 > x

So, answer to the question is always NO here.

Hence sufficient.

To understand the expression, x^2 > x

We can find the roots and draw the number line.

x^2 > x

x^2-x > 0

x(x-1) > 0

So, roots (critical points) are zero and 1.

See the below number line, to understand how it works.

Just choose values from the given (in the pic below) three ranges and check which satisfy the expression x^2 > x.

Statement II is sufficient:

1 > 1/x

That is “x” has to be negative or x has to greater than 1.

You can do some plugging in here to check it.

For x is negative, 1(positive) > negative.

For x is greater than 1, 1 > 1/x. For x = 2, 3, 4… 1 > 1/x,

So, answer to the question is always NO here.

Hence sufficient.

So, the answer has to be D (Each alone sufficient).

Hope this helps.

by solving the equation we will get
is x^3 -1 /x >0

A) will give x >1 and x<0 in both the cases answer will be YES
B clearly sufficient
D is the answr

A says

X^2 > x

so x>1 and -1<x

for any value of x in the above range , yes would be the answer

hence A is sufficient

B says

1> 1/x

so x>1 and -1<x

for any value of x in the above range , yes would be the answer

hence B is also sufficient

hence the answer would be D
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I feel like this is easiest to solve if we visualize the problem on a number line.

Question stem asks (Y/N) if x belongs to the below Y region.

|-------------(-1)------(0)------(1)-------------|
|-------Y------|---Y---|---N---|--------Y-------|

Both from statement 1 & 2, we'll get the same YYNY on number line.

(D)

Target question: Is x² > 1/x ?

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

Answer: D

RELATED VIDEO

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Why have you not considered value of X in the range 0>x>-1??
Even though the answer does not change, the explanation will be different.

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)

So D. No need to insert all the values and check

Isn't the question x^2 > 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.

This is true only if variable value of X is positive.

Sent from my CPH1609 using GMAT Club Forum mobile app

\(x^2 > \frac{1}{x}\) 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.

9. Inequalities

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  Inequalities - Quadratic Inequalities
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