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Intern  Joined: 19 Jan 2014
Posts: 24
Re: When is |x - 4| = 4 - x?  [#permalink]

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Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

This might sound silly but i just started preparing for GMAT, and I have a question. Why is it then in some cases we take x<0 or x>0 and in this problem we have x<=0 and x>=0

Thank you.
Math Expert V
Joined: 02 Sep 2009
Posts: 59730
Re: When is |x - 4| = 4 - x?  [#permalink]

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bytatia wrote:
Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

This might sound silly but i just started preparing for GMAT, and I have a question. Why is it then in some cases we take x<0 or x>0 and in this problem we have x<=0 and x>=0

Thank you.

Well, it all depends on the problem at hand. For this problem, we need = sign because x=4 also satisfies |x-4| = 4-x.

Below posts might help to brush up fundamentals on modulus:
Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

_________________
Manager  Joined: 04 Oct 2013
Posts: 150
Location: India
GMAT Date: 05-23-2015
GPA: 3.45
Re: When is |x - 4| = 4 - x?  [#permalink]

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When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

$$|x-4| = 4-x$$

Or, $$\frac{|x-4|}{x-4}=-1$$

=> $$(x-4) < 0$$ or $$x = 0$$

Or, $$x <= 4$$

Director  G
Joined: 23 Jan 2013
Posts: 522
Schools: Cambridge'16
Re: When is |x - 4| = 4 - x?  [#permalink]

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1
Question asks when distance between x and 4 expressed as |x-4| is equal to 4-x

until x is left to 4 or equal to 4 we get the equation right so |x-4|=4-x

Number line is

----------x--------------------4------------->

if x goes righter we get

-----------4-------------x------------------>

|x-4| will continue to be positive, but 4-x will be negative

Algebraically:

x-4=4-x
2x=8,
x=4

-(x-4)=4-x
-x+4=4-x
0=0, so infinitely many solutions when x<4

x<=4

D
Intern  Joined: 13 Dec 2013
Posts: 38
GPA: 2.71
Re: When is |x - 4| = 4 - x?  [#permalink]

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Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

I am still new to modulus so please do bare with me if I sound stupid.

This problem can be solved easily by picking numbers but to understand the concepts I tried to solve it using the books I read.
So according to the book, I need to take into account when the modulus is positive and negative when solving

$$x-4>0, x>4$$

x-4=4-x
x=4

and when $$x+4<0, x<=4$$
-(x+4)=4-x
-x-4=4-x
Just lost here.

My question is why do we chose X<=4 why do we chose one condition over the other.
Math Expert V
Joined: 02 Sep 2009
Posts: 59730
Re: When is |x - 4| = 4 - x?  [#permalink]

### Show Tags

Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

I am still new to modulus so please do bare with me if I sound stupid.

This problem can be solved easily by picking numbers but to understand the concepts I tried to solve it using the books I read.
So according to the book, I need to take into account when the modulus is positive and negative when solving

$$x-4>0, x>4$$

x-4=4-x
x=4

and when $$x+4<0, x<=4$$
-(x+4)=4-x
-x-4=4-x
Just lost here.

My question is why do we chose X<=4 why do we chose one condition over the other.

For the second case, when x - 4 < 0 (x < 4), |x - 4| becomes -(x - 4), so we'd have -(x - 4) = 4 - x, which gives 4 = 4. Since 4 = 4 is true, then it means that for x < -4, |x-4| = 4-x holds true.

Combining x = 4 from the first case and x < 4 from the second, we'll have x <= 4.

Hope it's clear.
_________________
Intern  Joined: 13 Dec 2013
Posts: 38
GPA: 2.71
Re: When is |x - 4| = 4 - x?  [#permalink]

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Bunuel wrote:

I am still new to modulus so please do bare with me if I sound stupid.

This problem can be solved easily by picking numbers but to understand the concepts I tried to solve it using the books I read.
So according to the book, I need to take into account when the modulus is positive and negative when solving

$$x-4>0, x>4$$

x-4=4-x
x=4

and when $$x+4<0, x<=4$$
-(x+4)=4-x
-x-4=4-x
Just lost here.

My question is why do we chose X<=4 why do we chose one condition over the other.

For the second case, when x + 4 < 0 (x < -4), |x - 4| becomes -(x - 4), so we'd have -(x - 4) = 4 - x, which gives 4 = 4. Since 4 = 4 is true, then it means that for x < -4, |x-4| = 4-x holds true.

Combining x = 4 from the first case and x < -4 from the second, we'll have x <= -4.

Hope it's clear.

Makes a lot more sense, Thankyou Intern  Joined: 27 Jan 2015
Posts: 5
Re: When is |x - 4| = 4 - x?  [#permalink]

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The equation holds true for every x value < 0. IMO answer choice E satisfies the equation as well. How can we cross out E?
Math Expert V
Joined: 02 Sep 2009
Posts: 59730
Re: When is |x - 4| = 4 - x?  [#permalink]

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lucky1829 wrote:
The equation holds true for every x value < 0. IMO answer choice E satisfies the equation as well. How can we cross out E?

E is not correct because |x-4| = 4-x also holds for 0 <= x <= 4. Check discussion on previous page. Hope it helps.
_________________
Math Expert V
Joined: 02 Aug 2009
Posts: 8326
Re: When is |x - 4| = 4 - x?  [#permalink]

### Show Tags

Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

Hi Bunuel,
I feel the way Q is asked, even x= 4, x=0 or x<0 may fit in..

the Q asks " When is |x-4| = 4-x?
ofcourse when x=4, ans is yes..
when x= 0... ans is yes..
yes x<=4 gives the entire range, BUT the Q does not ask that..

when all is |x-4| = 4-x?
for which all values is |x-4| = 4-x? OR
What is the range of x for |x-4| = 4-x?

Would in ACTUAL GMAT, the wordings of this kind MEAN what we are inferring here?
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 59730
Re: When is |x - 4| = 4 - x?  [#permalink]

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chetan2u wrote:
Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

Hi Bunuel,
I feel the way Q is asked, even x= 4, x=0 or x<0 may fit in..

the Q asks " When is |x-4| = 4-x?
ofcourse when x=4, ans is yes..
when x= 0... ans is yes..
yes x<=4 gives the entire range, BUT the Q does not ask that..

when all is |x-4| = 4-x?
for which all values is |x-4| = 4-x? OR
What is the range of x for |x-4| = 4-x?

Would in ACTUAL GMAT, the wordings of this kind MEAN what we are inferring here?

You are right the wording of the question is poor.
_________________
Manager  Joined: 10 Apr 2016
Posts: 54
Concentration: Strategy, Entrepreneurship
GMAT 1: 520 Q29 V30 GPA: 3.01
Re: When is |x - 4| = 4 - x?  [#permalink]

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For those struggling to understand how this works with the equal / lesser signs

X=4
|x-4|=4-x --> 4-4=4-4
X=3
|x-4|=4-x --> |-1|=1
X=2
|2-4|=4-2 --> |-2|=2
X=1
X=0
And so on .
_________________
Took the Gmat and got a 520 after studying for 3 weeks with a fulltime job. Now taking it again, but with 6 weeks of prep time and a part time job. Studying every day is key, try to do at least 5 exercises a day.
Director  B
Joined: 04 Jun 2016
Posts: 547
GMAT 1: 750 Q49 V43 Re: When is |x - 4| = 4 - x?  [#permalink]

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nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

|x-4| = 4-x?
Since a Mod always returns +ve value |x-4| can be seen as >0 for all mathematical purposes
so our equation becomes
0<4-x
4-x >0
x<4
Answer is D {more or less; there seems to be a sign problem in either in option or in the original question}
_________________
Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly.
FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.
Intern  Joined: 25 Jul 2017
Posts: 7
Re: When is |x - 4| = 4 - x?  [#permalink]

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Well let us begin with knowing -
|a| = -a when a<= 0
Therefore, |x-4| = 4-x when x-4 <=0
Hence, D
Manager  G
Joined: 22 Nov 2016
Posts: 205
Location: United States (CA)
Schools: Haas EWMBA '22
GMAT 1: 640 Q43 V35 GPA: 3.4
Re: When is |x - 4| = 4 - x?  [#permalink]

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|a - b| = b - a ; when a<b or a=b (includes 0)
Manager  B
Joined: 11 Feb 2017
Posts: 181
Re: When is |x - 4| = 4 - x?  [#permalink]

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nkimidi7y wrote:
When is |x - 4| = 4 - x?

A. x = 4
B. x = 0
C. x > 4
D. x <= 4
E. x < 0

Why not just A? i was confused b/w A and D but I agree with D but why not A?
Manager  B
Joined: 27 Apr 2011
Posts: 52
Location: India
GMAT Date: 06-13-2017
Re: When is |x - 4| = 4 - x?  [#permalink]

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Bunuel wrote:
nkimidi7y wrote:
When is |x-4| = 4-x?

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;

So, $$|x-4|=4-x=-(x-4)$$ to be true should be that $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.

Hi Bunuel,

This is a good quality question and bit tricky. I selected A as an answer.

After looking at answer choice, i can understand solution perfectly.

As per my thought process, whenever the question has equal to sign then we get either 2 values or one value (x=4).

But how can i avoid mistake on this type of question? Which triggering point forces you to think on non-negative value?
Non-Human User Joined: 09 Sep 2013
Posts: 13744
Re: When is |x - 4| = 4 - x?  [#permalink]

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_________________ Re: When is |x - 4| = 4 - x?   [#permalink] 25 Nov 2019, 04:09

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