GMAT Prep inequality

Expert replies

adilka: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Thu Oct 23, 2008 9:28 am; Location: Canada; Thanked: 1 times; GMAT Score:700

by adilka » Sat Jan 17, 2009 7:44 pm

If 0<x<1 which of the following inequalities must be true?

I. x^5<X^3
II. X^4 + x^5 < x^3 + x^2
III. X^4 - x^5 < x^2 - x^3

A. None
B. I only
C. II only
D. I and II only
E. I, II and III

OA: E

first 2 are easy, but i struggled with the 3rd condition. Where did I go wrong with my logic?

To check III let's assume it is correct. Since X is a positive number, I divided all sides by X^2, here's the resulting inequality:
X^2 - x^3 < 1 - x
rearrange as:
X^2 - x^3 + x < 1

We are given that if 0<x<1 hence X<1

X^2 - x^3 + X < 1 (a)
X < 1 (b)

Subtract (b) from (a), we get

X^2 - x^3 <0. That is clearly not so if X is between 0 and 1!!!

On the other hand, if I use X>0, multiply it by -1, to get to -x<0 (call is (c)) and add to (a) the I get a perfectly correct X^2-X^3 <1.

I get conflicting results...

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Source: Beat The GMAT — Problem Solving | Sat Jan 17, 2009 7:44 pm

sonu_thekool: Senior | Next Rank: 100 Posts; Posts: 60; Joined: Mon Jun 04, 2007 7:37 am; Thanked: 6 times

by sonu_thekool » Sat Jan 17, 2009 8:15 pm

Hi there,

Since you said Condition II is easy, it probably helps if you rephrase the third condition this way :

III) x^4 + x^3 < x^2 + x^5

Now, Stmt III is similar to statement II.

Hope this clears the confusion.

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adilka: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Thu Oct 23, 2008 9:28 am; Location: Canada; Thanked: 1 times; GMAT Score:700

by adilka » Sun Jan 18, 2009 12:08 am

sonu_thekool wrote:Hi there,

Since you said Condition II is easy, it probably helps if you rephrase the third condition this way :

III) x^4 + x^3 < x^2 + x^5

Now, Stmt III is similar to statement II.

Hope this clears the confusion.

Hi. No it doesn't. Im not scared by the signs my man, so switching from - to + doesn't clear it.
if you arrange 4 numbers on a number line in an increasing order, they look as follows:
x^5...............x^4...............x^3...................x^2
So by adding the middle ones and two outer ones, it's not perfectly clear what's bigger (which you assume to believe it is from your quote).

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DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

by DanaJ » Sun Jan 18, 2009 12:13 am

Maybe this can help you:
x^4 - x^5 = x^4 (1 - x)
x^2 - x^3 = x^2 (1 - x)
You have:

x^4(1 - x) < x^2(1 - x) or x^4 < x^2 or x^2 <1 or -1 < 0 <1, which is consistent with the initial constraint.

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sonu_thekool: Senior | Next Rank: 100 Posts; Posts: 60; Joined: Mon Jun 04, 2007 7:37 am; Thanked: 6 times

by sonu_thekool » Sun Jan 18, 2009 5:25 am

adilka wrote: Hi. No it doesn't. Im not scared by the signs my man, so switching from - to + doesn't clear it.
if you arrange 4 numbers on a number line in an increasing order, they look as follows:
x^5...............x^4...............x^3...................x^2
So by adding the middle ones and two outer ones, it's not perfectly clear what's bigger (which you assume to believe it is from your quote).

I see your point. I should have thought it through a little further.

DanaJ's solution is great. I would stop at the step x^4 < x^2 since x is known to be positive and this condition will always be true.

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adilka: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Thu Oct 23, 2008 9:28 am; Location: Canada; Thanked: 1 times; GMAT Score:700

by adilka » Sun Jan 18, 2009 10:44 am

sonu_thekool wrote: I see your point. I should have thought it through a little further.
DanaJ's solution is great. I would stop at the step x^4 < x^2 since x is known to be positive and this condition will always be true.

Agreed, Sonu.

Dana, you're correct, that's the same solution I found on MGMAT forum. You're good with your logic! (i failed to realize that we can factor this thing).

However, still looking for someone to tell me how did I go wrong with division.
Titans?

Instructors?

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DanaJ: Site Admin; Posts: 2567; Joined: Thu Jan 01, 2009 10:05 amFollowed by: 550 members; Thanked: 712 times; GMAT Score:770

Re: GMAT Prep inequality

by DanaJ » Sun Jan 18, 2009 10:52 am

[quote="adilka"]

We are given that if 0<x<1 hence X<1

X^2 - x^3 + X < 1 [b](a)[/b]
X < 1 [b](b)[/b]

Subtract (b) from (a), we get

X^2 - x^3 <0. That is clearly not so if X is between 0 and 1!!!
[/quote]

Here is where you are wrong. Subtracting (b) from (a) does not give you the equation you've written. Because this is an INEQUALITY! Subtracting will get you nowhere. Let me give you an example:
0.6 < 1
0.3 < 1
BUT that doesn't mean that 0.6 - 0.3 < 0. In fact, that is quite incorrect, since 0.3 is greater than 0.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

Re: GMAT Prep inequality

by Ian Stewart » Sun Jan 18, 2009 11:24 am

adilka wrote:
first 2 are easy, but i struggled with the 3rd condition. Where did I go wrong with my logic?

[...]

X^2 - x^3 + X < 1 (a)
X < 1 (b)

Subtract (b) from (a), we get

X^2 - x^3 <0. That is clearly not so if X is between 0 and 1!!!

You can't subtract inequalities, at least not in the way you've done above. To see why this will lead to incorrect conclusions, take the following inequalities, which are clearly true:

10 > 9
10 > 8

If you subtract, you find 0 > 1, which is nonsense.

If you are tempted to subtract inequalities, instead multiply one of the inequalities by -1 (reversing the inequality, of course), and then add.

Edit - I guess Dana beat me to it! Nice one.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

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adilka: Senior | Next Rank: 100 Posts; Posts: 79; Joined: Thu Oct 23, 2008 9:28 am; Location: Canada; Thanked: 1 times; GMAT Score:700

by adilka » Sun Jan 18, 2009 7:38 pm

Thank you very much Dana and Ian!
Silly mistake on my part. The funny thing is I actually recommended to someone on this forum to multiple by -1 and then add instead of subtracting!
Good catch, just in time for my actual test

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singalong: Senior | Next Rank: 100 Posts; Posts: 95; Joined: Wed Feb 09, 2011 6:42 amFollowed by: 1 member; Thanked: 1 times

by singalong » Thu Jan 26, 2012 6:16 am

I still haven't understood how statement 3 holds.

III. X^4 - x^5 < x^2 - x^3

I reduced it as x^(-1)< x^(-1) and got stuck. I probably did something very silly.what did i do wrong?

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Jan 26, 2012 8:06 am

singalong wrote:I still haven't understood how statement 3 holds.

III. X^4 - x^5 < x^2 - x^3

I reduced it as x^(-1)< x^(-1) and got stuck. I probably did something very silly.what did i do wrong?

We can't reduce as you did.
x^4 - x^5 â‰ x^(-1).
Consider x=1:
1^4 - 1^5 = 0.
1^(-1) = 1.
Not equal.

The following is true:
x^4/x^5 = x^(-1).

Statement III: x^4 - x^5 < x^2 - x^3
Since it is given that x is positive, we can divide by xÂ², which also must be positive:
(x^4 - x^5)/xÂ² < (x^2 - x^3)/xÂ²
xÂ² - xÂ³ < 1-x
xÂ²(1-x) < 1-x

Since x is a positive fraction, we can divide by 1-x, which also must be a positive fraction:
xÂ²(1-x)/(1-x) < (1-x)/(1-x)
xÂ² < 1.

Since the square of a positive fraction must be between 0 and 1, statement III must be true.

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royrijit1: Junior | Next Rank: 30 Posts; Posts: 18; Joined: Mon Mar 23, 2015 7:56 am; Location: India; Thanked: 1 times

by royrijit1 » Wed Aug 26, 2015 10:19 am

Dear Experts,

I came across the wavy curve concept to solve inequalities and found it very helpful (because I can omit choosing numbers to solve such problems). I am trying to solve this inequality using wavy-curve and I am stuck.

I. x3 < x -> solved using wavy curve
II. x2 < |x| -> solved using wavy curve
III. x4 - x5 > x3 - x2 stuck!!

I'm stuck at this point:

X^2.(x^2+1).(x-1)<0

Question: Can I disregard (x^2+1) because square of a number will never be negative? i.e. x^2 = -1 (not possible)?

Please advise.

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