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

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Joined: 06 Sep 2013
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Re: When is |x-4| = 4-x? [#permalink]  19 Dec 2013, 05:54
Silly me, thats correct
Thanks
Cheers
J:)

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Re: When is |x-4| = 4-x? [#permalink]  21 Jan 2014, 23:30
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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Posts: 30362
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Re: When is |x-4| = 4-x? [#permalink]  23 Jan 2014, 02:50
Expert's post
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

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Re: When is |x-4| = 4-x? [#permalink]  24 Jan 2014, 00:03
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, $$x <= 4$$

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Joined: 23 Jan 2013
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Schools: Cambridge'16
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Re: When is |x-4| = 4-x? [#permalink]  28 Nov 2014, 23:23
1
KUDOS
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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When is |x-4| = 4-x? [#permalink]  09 Dec 2014, 23:29
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

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: 30362
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Kudos [?]: 57219 [0], given: 8808

When is |x-4| = 4-x? [#permalink]  10 Dec 2014, 03:34
Expert's post
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.
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Re: When is |x-4| = 4-x? [#permalink]  10 Dec 2014, 04:28
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

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]  24 Mar 2015, 03:12
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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Posts: 30362
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Re: When is |x-4| = 4-x? [#permalink]  24 Mar 2015, 03:18
Expert's post
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] 24 Mar 2015, 03:18

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