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

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bytatia
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Re: When is |x - 4| = 4 - x? [#permalink] New post   22 Jan 2014, 00:30
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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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.
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Re: When is |x - 4| = 4 - x? [#permalink] New post   23 Jan 2014, 03:50
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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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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
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Re: When is |x - 4| = 4 - x? [#permalink] New post   24 Jan 2014, 01:03
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, \(x <= 4\)

Answer: (D)
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Re: When is |x - 4| = 4 - x? [#permalink] New post   29 Nov 2014, 00:23
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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
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Re: When is |x - 4| = 4 - x? [#permalink] New post   10 Dec 2014, 00:29
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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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
(not sure if this value has to be rejected or not. Please help)

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.
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Re: When is |x - 4| = 4 - x? [#permalink] New post   10 Dec 2014, 04:34
Expert Reply
saadis87 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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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
(not sure if this value has to be rejected or not. Please help)

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.
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Re: When is |x - 4| = 4 - x? [#permalink] New post   10 Dec 2014, 05:28
Bunuel wrote:
saadis87 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
(not sure if this value has to be rejected or not. Please help)

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 :)
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Re: When is |x - 4| = 4 - x? [#permalink] New post   24 Mar 2015, 04:12
The equation holds true for every x value < 0. IMO answer choice E satisfies the equation as well. How can we cross out E?
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Re: When is |x - 4| = 4 - x? [#permalink] New post   24 Mar 2015, 04:18
Expert Reply
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.
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Re: When is |x - 4| = 4 - x? [#permalink] New post   15 Apr 2016, 20:59
Expert Reply
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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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..

Had the Q been.
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?
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Re: When is |x - 4| = 4 - x? [#permalink] New post   16 Apr 2016, 10:14
Expert Reply
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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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..

Had the Q been.
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.
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Re: When is |x - 4| = 4 - x? [#permalink] New post   17 Apr 2016, 04:54
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 :).
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Re: When is |x - 4| = 4 - x? [#permalink] New post   25 Jul 2016, 03:52
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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}
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Re: When is |x - 4| = 4 - x? [#permalink] New post   06 Aug 2017, 02:55
Well let us begin with knowing -
|a| = -a when a<= 0
Therefore, |x-4| = 4-x when x-4 <=0
Hence, D
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Re: When is |x - 4| = 4 - x? [#permalink] New post   13 Nov 2017, 14:00
|a - b| = b - a ; when a<b or a=b (includes 0)
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Re: When is |x - 4| = 4 - x? [#permalink] New post   26 Nov 2017, 04:56
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?
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Re: When is |x - 4| = 4 - x? [#permalink] New post   13 Sep 2018, 03:52
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 \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

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?
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Re: When is |x - 4| = 4 - x? [#permalink] New post   10 Mar 2021, 23:41
nkimidi7y wrote:
When is |x - 4| = 4 - x?

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


I think the answer options combined with the wording of the question are a bit misleading.
For example, take Option B: x = 0

\(|0 - 4| = 4 - 0\)?
\(|-4| = 4\)
\(4 = 4\)

That is true: in this case |x - 4| = 4 - x

It seems like there are several options that are true.
I doubt that you will see something like this on the real GMAT
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Re: When is |x - 4| = 4 - x? [#permalink] New post   21 Jun 2023, 05:38
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