GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Oct 2019, 22:55

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If x is positive, what is the value of |x – 3| – 2|x – 4| +

Author Message
TAGS:

### Hide Tags

Manager
Joined: 09 Feb 2013
Posts: 111
If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

Updated on: 29 Apr 2013, 00:13
8
28
00:00

Difficulty:

95% (hard)

Question Stats:

25% (02:39) correct 75% (02:21) wrong based on 393 sessions

### HideShow timer Statistics

If x is positive, what is the value of |x – 3| – 2|x – 4| + 2|x – 6| – |x – 7| ?

(1) x is an odd integer.
(2) x > 6

_________________
Kudos will encourage many others, like me.
Good Questions also deserve few KUDOS.

Originally posted by emmak on 27 Apr 2013, 22:59.
Last edited by emmak on 29 Apr 2013, 00:13, edited 2 times in total.
VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1019
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

27 Apr 2013, 23:19
3
3
If x is positive, what is the value of $$|x -3| - 2|x - 4| + 2|x - 6| - |x - 7|$$ ?
Studying this equation we found out that
0<x<3 f(x)=0
3<x<4 f(x)=2x-6
4<x<6 f(x)=-2x+10
6<x<7 f(x)=2x-14
x>7 f(x)=0

(1) x is an odd integer.
So x can be 5 and we are in the 4<x<6 range so f(x)=-2*5+10=0
In any other interval, if x is odd, the value is 0.
Take x=3 for example, range 3<x<4 so 2*3-6=0
Take x=7 2*7-14=0
For any other odd integer we are in the 0<x<3 or x>7 range and in those any value of x will make no difference because f(x)=0
Sufficient

(2) x > 6
With this only we have a lot of values, because x can be a non integer. Any value that respect [6<x<7] f(x)=2x-14 is possible for example

A
_________________
It is beyond a doubt that all our knowledge that begins with experience.
Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]
##### General Discussion
Tutor
Joined: 20 Apr 2012
Posts: 97
Location: Ukraine
GMAT 1: 690 Q51 V31
GMAT 2: 730 Q51 V38
WE: Education (Education)
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

28 Apr 2013, 02:42
4
I like this problem. You even don't need to open modulus before. Just go on statements.

(1) Sufficient.
If $$x=1$$ then $$|1- 3|-2|1- 4|+2|1-6|-|1-7|=2-6+10-6=0$$.
If $$x=3$$ then $$|3- 3|-2|3- 4|+2|3-6|-|3-7|=0-2+6-4=0$$.
If $$x=5$$ then $$|5- 3|-2|5- 4|+2|5-6|-|5-7|=2-2+2-2=0$$.
If $$x>=7$$ then $$x-3- 2x+8+2x-12-x+7=0$$.
We have the same answer anyway:0. Statement is sufficient.

(2) Insufficient. You don't know how to open the last modulus $$|x-7|$$.
If $$x=6.5$$ then $$|6.5-3|-2|6.5-4|+2|6.5- 6|-|6.5-7|=3.5-5+1-0.5=-1$$.
If $$x=7$$ then we already know $$|1- 3|-2|1- 4|+2|1-6|-|1-7|=0$$.
So, insufficient.

_________________
I'm happy, if I make math for you slightly clearer
And yes, I like kudos:)
Senior Manager
Joined: 13 May 2013
Posts: 405
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

24 Jun 2013, 14:18
1
I am having difficultly with this problem.

The way I see it, we start by finding the checkpoints in |x -3| - 2|x - 4| + 2|x - 6| - |x - 7|

X = 3,4,6,7

I would think we would set up ranges as follows:

x<3
3<x<4
4<x<6
6<x<7
x>7

The first problem is, my method is unlike yours (I got seemingly incorrect answers) and secondly, it takes a long time just to set up the ranges - much longer than two minutes. There must be a quicker way!

Thanks.

Zarrolou wrote:
If x is positive, what is the value of $$|x -3| - 2|x - 4| + 2|x - 6| - |x - 7|$$ ?
Studying this equation we found out that
0<x<3 f(x)=0
3<x<4 f(x)=2x-6
4<x<6 f(x)=-2x+10
6<x<7 f(x)=2x-14
x>7 f(x)=0

(1) x is an odd integer.
So x can be 5 and we are in the 4<x<6 range so f(x)=-2*5+10=0
In any other interval, if x is odd, the value is 0.
Take x=3 for example, range 3<x<4 so 2*3-6=0
Take x=7 2*7-14=0
For any other odd integer we are in the 0<x<3 or x>7 range and in those any value of x will make no difference because f(x)=0
Sufficient

(2) x > 6
With this only we have a lot of values, because x can be a non integer. Any value that respect [6<x<7] f(x)=2x-14 is possible for example

A
VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1019
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

25 Jun 2013, 00:26
2
WholeLottaLove wrote:
I am having difficultly with this problem.

The way I see it, we start by finding the checkpoints in |x -3| - 2|x - 4| + 2|x - 6| - |x - 7|

X = 3,4,6,7

I would think we would set up ranges as follows:

x<3
3<x<4
4<x<6
6<x<7
x>7

The first problem is, my method is unlike yours (I got seemingly incorrect answers) and secondly, it takes a long time just to set up the ranges - much longer than two minutes. There must be a quicker way!

Thanks.

Your ranges are correct except for the first one that is 0<x<3, because x is positive.

If I remember correctly this was a question from the Challenge archive of MGMAT, much harder and longer to solve than anything you'll find on the real test. A quicker way (but less methodical) is to test some odd values for X, and see what you find.

(1) x is an odd integer.

If you test any odd value, you'll find that that function always equals 0. Generally speaking testing values is a good practice in the test, however you cannot be 100% sure of your answer just by testing values.

Hope it helps
_________________
It is beyond a doubt that all our knowledge that begins with experience.
Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]
Senior Manager
Joined: 13 May 2013
Posts: 405
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

25 Jun 2013, 06:04
1
Then wouldn't the last range be between just 6 and 7, not greater than 7 for the same reason (because x is positive)?

Thanks for the explanation though!

Zarrolou wrote:
WholeLottaLove wrote:
I am having difficultly with this problem.

The way I see it, we start by finding the checkpoints in |x -3| - 2|x - 4| + 2|x - 6| - |x - 7|

X = 3,4,6,7

I would think we would set up ranges as follows:

x<3
3<x<4
4<x<6
6<x<7
x>7

The first problem is, my method is unlike yours (I got seemingly incorrect answers) and secondly, it takes a long time just to set up the ranges - much longer than two minutes. There must be a quicker way!

Thanks.

Your ranges are correct except for the first one that is 0<x<3, because x is positive.

If I remember correctly this was a question from the Challenge archive of MGMAT, much harder and longer to solve than anything you'll find on the real test. A quicker way (but less methodical) is to test some odd values for X, and see what you find.

(1) x is an odd integer.

If you test any odd value, you'll find that that function always equals 0. Generally speaking testing values is a good practice in the test, however you cannot be 100% sure of your answer just by testing values.

Hope it helps
VP
Status: Far, far away!
Joined: 02 Sep 2012
Posts: 1019
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

25 Jun 2013, 10:06
2
WholeLottaLove wrote:
Then wouldn't the last range be between just 6 and 7, not greater than 7 for the same reason (because x is positive)?

Thanks for the explanation though!

Nope. The ranges of the function taken by itself are

x<3
3<x<4
4<x<6
6<x<7
x>7

But the text says that x is positive so is wrong to consider negative values for x; this condition affects only the interval with negative values.

$$x<3$$ becomes $$0<x<3$$, all the others stay the same.
_________________
It is beyond a doubt that all our knowledge that begins with experience.
Kant , Critique of Pure Reason

Tips and tricks: Inequalities , Mixture | Review: MGMAT workshop
Strategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant

Rules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]
Senior Manager
Joined: 13 May 2013
Posts: 405
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

25 Jun 2013, 10:14
1
1
Ahh. I understand now. Thanks!

Zarrolou wrote:
WholeLottaLove wrote:
Then wouldn't the last range be between just 6 and 7, not greater than 7 for the same reason (because x is positive)?

Thanks for the explanation though!

Nope. The ranges of the function taken by itself are

x<3
3<x<4
4<x<6
6<x<7
x>7

But the text says that x is positive so is wrong to consider negative values for x; this condition affects only the interval with negative values.

$$x<3$$ becomes $$0<x<3$$, all the others stay the same.
Manager
Joined: 06 Jun 2013
Posts: 54
Concentration: Entrepreneurship, General Management
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

15 Jul 2013, 08:27
1
just found this question...http://www.manhattangmat.com/blog/index.php/2013/05/22/want-a-750-think-your-way-through-this-challenge-problem/

Just wanted to add that this is supposed to be a >750+ question.. So 600-700 is wrong. Therefore, don't be encouraged if you can't get it (I was...).
_________________
With all the best!
Math Expert
Joined: 02 Sep 2009
Posts: 58347
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

15 Jul 2013, 08:31
TheGerman wrote:
just found this question...http://www.manhattangmat.com/blog/index.php/2013/05/22/want-a-750-think-your-way-through-this-challenge-problem/

Just wanted to add that this is supposed to be a >750+ question.. So 600-700 is wrong. Therefore, don't be encouraged if you can't get it (I was...).

76% of the users answered this question incorrectly, so I changed the difficulty level to 700+.
_________________
Senior Manager
Joined: 13 May 2013
Posts: 405
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

15 Jul 2013, 12:42
1
If x is positive, what is the value of |x – 3| – 2|x – 4| + 2|x – 6| – |x – 7| ?

x≥1
Check points: 3, 4, 6, 7
x<3, 3<x<4, 4<x<6, 6<x<7, x>7

x<3:
|x – 3| – 2|x – 4| + 2|x – 6| – |x – 7|
-(x-3) -2 -(x-4) +2 -(x-6) - -(x-7)
-x+3 -2(-x+4) +2(-x+6) - (-x+7)
-x+3 +2x -8 -2x +12 +x-7
= 0

3<x<4:
|x – 3| – 2|x – 4| + 2|x – 6| – |x – 7|
(x-3) - 2 -(x-4) + 2 -(x-6) - -(x-7)
x-3 -2 -(x-4) +2 -(x-6) - -(x-7)
x-3 +2x -8 -2x +12 +x-7
2x-6

4<x<6
|x – 3| – 2|x – 4| + 2|x – 6| – |x – 7|
(x-3) -2(x-4) + 2 -(x-6) - -(x-7)
x-3 - 2x+8 +2 (-x+6) - (-x+7)
x-3 - 2x+8 + -2x +12 +x-7
-2x+10

6<x<7
|x – 3| – 2|x – 4| + 2|x – 6| – |x – 7|
(x-3) - 2(x-4) +2(x-6) - -(x-7)
(x-3) - 2(x-4) +2(x-6) +x-7
x-3 - 2x + 8 +2x-12 +x-7
2x-14

x>7
|x – 3| – 2|x – 4| + 2|x – 6| – |x – 7|
x-3 -2x+8 +2x-12 -x+7
=0

(1) x is an odd integer.

If x is an odd integer, then x=1, 3, 5, 7, 9, etc. Going through the five ranges:

When x=1 the result = 0
When x=5 (4<x<6) the result is -2(5)+10 = 0
When x>7 (for all odd integers) the result = 0

Therefore, as long as x is a positive, odd integer the result is always zero.
SUFFIFICIENT

(2) x>6
|x – 3| – 2|x – 4| + 2|x – 6| – |x – 7|

If x>6 then |x-6| will be positive but |x-7| may or not be positive (we are not told if x is an integer or not) therefore, the signs for |x-7| may be positive or may be negative and will affect the value of x.
INSUFFICIENT

(A)

P.S. while this problem was fairly straight forward, it took me a long time to solve. Is there any way to solve this type of problem in a quicker fashion? Thanks!
Intern
Joined: 13 Jan 2015
Posts: 3
Location: United States
Concentration: Finance, General Management
WE: Project Management (Commercial Banking)
If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

19 Jul 2015, 03:14
2
If x is positive, what is the value of V = |x – 3| – 2|x – 4| + 2|x – 6| – |x – 7| ?

Clearly:
x>7 or x<3 => V = 0
so we only find whether x>=3 and x<=7 what is V?

(1) if x is an odd integer, if x= 3, 5, 7 => V=0 => sufficient

(2) x > 6 => V = 2(x-1) with 6<x<7, V can be any of many values as x is not an integer.

Hence, A.

Hope my solution saves your time.
Manager
Joined: 01 Sep 2016
Posts: 189
GMAT 1: 690 Q49 V35
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

23 Sep 2017, 13:14
1
OFFICIAL SOLUTION FROM MANHATTAN

1) SUFFICIENT: It’s impractical to take an algebraic approach to this statement; doing so would entail a large number of cases. For instance, |x – 3| is equal to 3 – x if x < 3, but is equal to x – 3 if x > 3; similarly, the other three absolute-value expressions switch at x = 4, 6, and 7, respectively.

Instead, it’s more efficient to consider the first three positive odd integers (1, 3, and 5) individually and then to consider only one algebraic case, the case in which x > 7 (because when x > 7, all values are positive so we can ignore the absolute value symbols).

If x = 1, then the value is (2) – 2(3) + 2(5) – (6) = 0.
If x = 3, then the value is (0) – 2(1) + 2(3) – (4) = 0.
If x = 5, then the value is (2) – 2(1) + 2(1) – (2) = 0.

If x > 7, then drop the absolute value symbols and simplify:
(x – 3) – 2(x – 4) + 2(x – 6) – (x – 7) =
x – 3 – 2x + 8 + 2x – 12 – x + 7 =
(x – 2x + 2x– x) + (-3 + 8 – 12 + 7) =
0

Therefore, the value of the expression is also 0 for all values of x greater than or equal to 7, including the odd integers 7, 9, 11, and so on. The value of the expression is thus 0 for all positive odd integers. The statement is sufficient.

(2) NOT SUFFICIENT:
As determined during the discussion of statement 1, the expression is equal to 0 when x > 7 (whether odd integer, even integer, or non-integer). We still need to test the non-integer values of x between 6 and 7.

If x = 6.5, then the value is (3.5) – 2(2.5) + 2(0.5) – 0.5 = –1, which is not equal to 0. The expression can thus have multiple values, so the statement is insufficient.

_________________
we shall fight on the beaches,
we shall fight on the landing grounds,
we shall fight in the fields and in the streets,
we shall fight in the hills;
we shall never surrender!
Non-Human User
Joined: 09 Sep 2013
Posts: 13162
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +  [#permalink]

### Show Tags

09 Apr 2019, 02:43
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If x is positive, what is the value of |x – 3| – 2|x – 4| +   [#permalink] 09 Apr 2019, 02:43
Display posts from previous: Sort by