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

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Intern
Joined: 19 Jan 2014
Posts: 26
Re: When is |x - 4| = 4 - x?  [#permalink]

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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 $$x-4\leq{0}$$ --> $$x\leq{4}$$.

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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Joined: 04 Oct 2013
Posts: 155
Location: India
GMAT Date: 05-23-2015
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Re: When is |x - 4| = 4 - x?  [#permalink]

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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, $$x <= 4$$

Director
Joined: 23 Jan 2013
Posts: 598
Schools: Cambridge'16
Re: When is |x - 4| = 4 - x?  [#permalink]

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29 Nov 2014, 00:23
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: 48
GPA: 2.71
Re: When is |x - 4| = 4 - x?  [#permalink]

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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 $$x-4\leq{0}$$ --> $$x\leq{4}$$.

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.
Math Expert
Joined: 02 Sep 2009
Posts: 47898
Re: When is |x - 4| = 4 - x?  [#permalink]

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10 Dec 2014, 04:34
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
(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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Intern
Joined: 13 Dec 2013
Posts: 48
GPA: 2.71
Re: When is |x - 4| = 4 - x?  [#permalink]

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10 Dec 2014, 05:28
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
(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
Intern
Joined: 27 Jan 2015
Posts: 6
Re: When is |x - 4| = 4 - x?  [#permalink]

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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?
Math Expert
Joined: 02 Sep 2009
Posts: 47898
Re: When is |x - 4| = 4 - x?  [#permalink]

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24 Mar 2015, 04:18
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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Math Expert
Joined: 02 Aug 2009
Posts: 6509
Re: When is |x - 4| = 4 - x?  [#permalink]

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15 Apr 2016, 20:59
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..

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?
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Math Expert
Joined: 02 Sep 2009
Posts: 47898
Re: When is |x - 4| = 4 - x?  [#permalink]

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16 Apr 2016, 10:14
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..

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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Joined: 10 Apr 2016
Posts: 55
Concentration: Strategy, Entrepreneurship
GMAT 1: 520 Q29 V30
GPA: 3.01
Re: When is |x - 4| = 4 - x?  [#permalink]

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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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Director
Joined: 04 Jun 2016
Posts: 603
GMAT 1: 750 Q49 V43
Re: When is |x - 4| = 4 - x?  [#permalink]

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25 Jul 2016, 03:52
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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Intern
Joined: 25 Jul 2017
Posts: 7
Re: When is |x - 4| = 4 - x?  [#permalink]

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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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Posts: 235
Location: United States
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Re: When is |x - 4| = 4 - x?  [#permalink]

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13 Nov 2017, 14:00
|a - b| = b - a ; when a<b or a=b (includes 0)
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Joined: 11 Feb 2017
Posts: 199
Re: When is |x - 4| = 4 - x?  [#permalink]

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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?
Re: When is |x - 4| = 4 - x? &nbs [#permalink] 26 Nov 2017, 04:56

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