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# x ≠ 0. Is x^2>x^4?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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x ≠ 0. Is x^2>x^4?  [#permalink]

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10 Aug 2018, 02:33
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[Math Revolution GMAT math practice question]

$$x ≠ 0$$. Is $$x^2>x^4?$$

$$1) x^2<1$$
$$2) x^2<2x$$

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" VP Status: Learning stage Joined: 01 Oct 2017 Posts: 1028 WE: Supply Chain Management (Energy and Utilities) Re: x ≠ 0. Is x^2>x^4? [#permalink] ### Show Tags 10 Aug 2018, 05:28 2 MathRevolution wrote: [Math Revolution GMAT math practice question] $$x ≠ 0$$. Is $$x^2>x^4?$$ $$1) x^2<1$$ $$2) x^2<2x$$ Re-phrasing question stem:- $$x^2>x^4$$ Or, $$x^4-x^2<0$$ Or, $$x^2\left(x+1\right)\left(x-1\right)<0$$ -1<x<0 or 0<x<1 St1:- $$x^2<$$ Or, x^2-1<0 Or, (x+1)(x-1)<0 -1<x<1 Since $$x\neq{0}$$, hence -1<x<0 or 0<x<1 Sufficient. St2:- $$x^2<2x$$ Or, x(x-2)<0 So, 0<x<2 Insufficient. Ans. (A) _________________ Regards, PKN Rise above the storm, you will find the sunshine VP Joined: 09 Mar 2016 Posts: 1273 x ≠ 0. Is x^2>x^4? [#permalink] ### Show Tags Updated on: 13 Aug 2018, 06:54 PKN wrote: MathRevolution wrote: [Math Revolution GMAT math practice question] $$x ≠ 0$$. Is $$x^2>x^4?$$ $$1) x^2<1$$ $$2) x^2<2x$$ Re-phrasing question stem:- $$x^2>x^4$$ Or, $$x^4-x^2<0$$ Or, $$x^2\left(x+1\right)\left(x-1\right)<0$$ -1<x<0 or 0<x<1 St1:- $$x^2<$$ Or, x^2-1<0 Or, (x+1)(x-1)<0 -1<x<1 Since $$x\neq{0}$$, hence -1<x<0 or 0<x<1 Sufficient. St2:- $$x^2<2x$$ Or, x(x-2)<0 So, 0<x<2 Insufficient. Ans. (A) hi PKN hope my questions find you well is it necessary to rephrase this question to solve it ? looking at initial form of both statements, i thought correct answer was E because in both cases we have even roots that yield both negative and positive values why Also, why you changed inequality sign here $$x^4-x^2<0$$ We didn't divide by or multiply with negative value, did we ? And one more question I understand what factoring out is but want to doublecheck it with you hey gmatbusters when you get the chance can you please answer my questions above many thanks in advance! Or, $$x^4-x^2<0$$ Or, $$x^2\left(x+1\right)\left(x-1\right)<0$$ $$x^4-x^2<0$$ from here i factor out $$x^2$$ and get $$x^2(x^2-1)$$ is it correct than I apply this formula $$a^2-b^2= (a+b)(a-b)$$ is my understanding correct ? many thanks for taking time to explain Originally posted by dave13 on 12 Aug 2018, 02:32. Last edited by dave13 on 13 Aug 2018, 06:54, edited 1 time in total. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7597 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: x ≠ 0. Is x^2>x^4? [#permalink] ### Show Tags 13 Aug 2018, 06:24 1 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. $$x^2>x^4$$ $$=> x^4 - x^2 < 0$$ $$=> x^2 ( x^2 – 1 ) < 0$$ $$=> ( x^2 – 1 ) < 0$$ $$=> ( x + 1 )( x – 1 ) < 0$$ $$=> -1 < x < 1$$ This is equivalent to condition 1). Thus, condition 1) is sufficient. Condition 2) $$x^2<2x$$ $$=> x^2 - 2x < 0$$ $$=> x(x – 2) < 0$$ $$=> 0 < x < 2$$ In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient However, the solution set of the question -1 < x < 1 does not include the solution set of the condition, 0 < x < 2. Thus, condition 2) is not sufficient. Therefore, A is the answer. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: x ≠ 0. Is x^2>x^4?  [#permalink]

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13 Aug 2018, 07:04
1
@dace13

Please find my response in GREEN

[/quote]

hi PKN

hope my questions find you well

is it necessary to rephrase this question to solve it ?- REPHRASING THE QUESTION MAKES IT EASIER
looking at initial form of both statements, i thought correct answer was E because in both cases we have even roots that yield both negative and positive values why

Also, why you changed inequality sign here $$x^4-x^2<0$$ - THIS IS BECAUSE THE TERMS ARE TAKEN TO THE OTHER SIDE OF INEQUALITY.
We didn't divide by or multiply with negative value, did we ?- MULTIPLY BY (-)1 IS SAME AS TAKING THE TERMS TO OTHER SIDE OF INEQUALITY. BOTH ARE EQUIVALENT.
And one more question I understand what factoring out is but want to doublecheck it with you

hey gmatbusters when you get the chance can you please answer my questions above many thanks in advance!

Or, $$x^4-x^2<0$$
Or, $$x^2\left(x+1\right)\left(x-1\right)<0$$

$$x^4-x^2<0$$ from here i factor out $$x^2$$ and get $$x^2(x^2-1)$$ is it correct than I apply this formula $$a^2-b^2= (a+b)(a-b)$$ is my understanding correct ?- YOUR UNDERSTANDING IS CORRECT

many thanks for taking time to explain [/quote]
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x ≠ 0. Is x^2>x^4?  [#permalink]

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Updated on: 13 Aug 2018, 07:52
gmatbusters many thanks for your prompt and helpful reply one question regarding flipping signs if you say we flip the sign when we take one term to other side, then in my example below should i change inequility sign ?

$$x+5>8$$

$$x<8-5$$ (here i flipped sign because i took one term to the other side)

$$x<3$$

is it correct ? if not why ?

Originally posted by dave13 on 13 Aug 2018, 07:42.
Last edited by dave13 on 13 Aug 2018, 07:52, edited 1 time in total.
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Joined: 09 Mar 2016
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x ≠ 0. Is x^2>x^4?  [#permalink]

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13 Aug 2018, 07:49
MathRevolution wrote:
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

$$x^2>x^4$$
$$=> x^4 - x^2 < 0$$
$$=> x^2 ( x^2 – 1 ) < 0$$
$$=> ( x^2 – 1 ) < 0$$
$$=> ( x + 1 )( x – 1 ) < 0$$
$$=> -1 < x < 1$$

This is equivalent to condition 1).
Thus, condition 1) is sufficient.

Condition 2)
$$x^2<2x$$
$$=> x^2 - 2x < 0$$
$$=> x(x – 2) < 0$$
$$=> 0 < x < 2$$

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
However, the solution set of the question -1 < x < 1 does not include the solution set of the condition, 0 < x < 2. Thus, condition 2) is not sufficient.

Therefore, A is the answer.

MathRevolution, Greetings Revolutioner

you say " Forget conventional ways of solving math questions" but you are using pure conventonal algebraic method to solve question (step by step), arent you?
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Posts: 1273
Re: x ≠ 0. Is x^2>x^4?  [#permalink]

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13 Aug 2018, 08:15
MathRevolution wrote:
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

$$x^2>x^4$$
$$=> x^4 - x^2 < 0$$
$$=> x^2 ( x^2 – 1 ) < 0$$
$$=> ( x^2 – 1 ) < 0$$
$$=> ( x + 1 )( x – 1 ) < 0$$
$$=> -1 < x < 1$$

This is equivalent to condition 1).
Thus, condition 1) is sufficient.

Condition 2)
$$x^2<2x$$
$$=> x^2 - 2x < 0$$
$$=> x(x – 2) < 0$$
$$=> 0 < x < 2$$

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
However, the solution set of the question -1 < x < 1 does not include the solution set of the condition, 0 < x < 2. Thus, condition 2) is not sufficient.

Therefore, A is the answer.

gmatbusters have a look at highlighted part. the inequility sign was NOT FLIPPED WHEN THE TERM WAS TAKEN TO THE OTHER SIDE. it contradicts your explanation can you provide detailed explanation please
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Re: x ≠ 0. Is x^2>x^4?  [#permalink]

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13 Aug 2018, 08:27
1
You didn't understand it correctly,

You need not flip it while taking the variable on the other side
See, x+5>8
Now if we take x on other side, 5-8>-x, or -3>-x

Now, the pointed part of the inequality is towards x earlier it was open part of the inequality sign.

1) To find x we can either multiply the inequality by (-)1 , we get 3<x ( flip the inequality while multiplying or dividing by negative)

other approach

2) swap the x on other side to make it positive (no flipping of inequality sign is required now)
x>3

dave13 wrote:
gmatbusters many thanks for your prompt and helpful reply one question regarding flipping signs if you say we flip the sign when we take one term to other side, then in my example below should i change inequility sign ?

$$x+5>8$$

$$x<8-5$$ (here i flipped sign because i took one term to the other side)- INEQUALITY IS NOT TO BE FLIPPED

$$x<3$$

is it correct ? if not why ?

_________________
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Joined: 27 Oct 2017
Posts: 1229
Location: India
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Re: x ≠ 0. Is x^2>x^4?  [#permalink]

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13 Aug 2018, 08:34
1
1) Flip the inequality sign if multiply or divide by a negative
2)when you take the variable to other side, the Flipping of inequality sign not required.

See if 2-x > 3 ....(eq1)
taking variable on the side side, -1>x ....(eq2) ( i didn't flip the sign)
BUT , We generally write the variable on left, S
o the final inequality is x<-1... (eq3)

eq 2 & eq3 are same, but it seems that inequality sign has been flipped , actually it is not flipped.

dave13 wrote:
MathRevolution wrote:
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

$$x^2>x^4$$
$$=> x^4 - x^2 < 0$$
$$=> x^2 ( x^2 – 1 ) < 0$$
$$=> ( x^2 – 1 ) < 0$$
$$=> ( x + 1 )( x – 1 ) < 0$$
$$=> -1 < x < 1$$

This is equivalent to condition 1).
Thus, condition 1) is sufficient.

Condition 2)
$$x^2<2x$$
$$=> x^2 - 2x < 0$$
$$=> x(x – 2) < 0$$
$$=> 0 < x < 2$$

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
However, the solution set of the question -1 < x < 1 does not include the solution set of the condition, 0 < x < 2. Thus, condition 2) is not sufficient.

Therefore, A is the answer.

gmatbusters have a look at highlighted part. the inequility sign was NOT FLIPPED WHEN THE TERM WAS TAKEN TO THE OTHER SIDE. it contradicts your explanation

1) Flip the inequality sign if multiply or divide by a negative
2)when you take the variable to other side, the Flipping of inequality sign not required.

can you provide detailed explanation please

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Joined: 16 Aug 2015
Posts: 7597
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GPA: 3.82
Re: x ≠ 0. Is x^2>x^4?  [#permalink]

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15 Aug 2018, 23:41
1
dave13 wrote:
MathRevolution wrote:
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

$$x^2>x^4$$
$$=> x^4 - x^2 < 0$$
$$=> x^2 ( x^2 – 1 ) < 0$$
$$=> ( x^2 – 1 ) < 0$$
$$=> ( x + 1 )( x – 1 ) < 0$$
$$=> -1 < x < 1$$

This is equivalent to condition 1).
Thus, condition 1) is sufficient.

Condition 2)
$$x^2<2x$$
$$=> x^2 - 2x < 0$$
$$=> x(x – 2) < 0$$
$$=> 0 < x < 2$$

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
However, the solution set of the question -1 < x < 1 does not include the solution set of the condition, 0 < x < 2. Thus, condition 2) is not sufficient.

Therefore, A is the answer.

MathRevolution, Greetings Revolutioner

you say " Forget conventional ways of solving math questions" but you are using pure conventonal algebraic method to solve question (step by step), arent you?

The mentioned "conventional way" means checking condition 1), condition 2) and both conditions together and so on. It takes too much time.
The first step of VA method of Math Revolution is modifying a original condition and a question. By that step, 30% of questions can be solved.
When we modify them, we should use algebraic method, of course.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Re: x ≠ 0. Is x^2>x^4?   [#permalink] 15 Aug 2018, 23:41
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