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

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Manager
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When is |x-4| = 4-x? [#permalink]  03 May 2012, 10:38
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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

I could answer this question by plugging in some numbers.
But how do i prove this using algebra?
[Reveal] Spoiler: OA
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Re: When is |x-4| = 4-x? [#permalink]  03 May 2012, 11:04
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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?

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.
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Re: When is |x-4| = 4-x? [#permalink]  08 May 2012, 09:57
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nkimidi7y wrote:
4-x is always >=0.
So x is always <=4.

Is this what you have meant?

|x-4| = 4-x

1) |x-4| is ALWAYS none negative.
so , 4-x must be also more or equal to zero

4-x>=0
x<=4

2) if x<4 , then 4-x=4-x
if x>=4 x-4=4-x x=4

so x <= 4
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Re: When is |x-4| = 4-x? [#permalink]  03 May 2012, 14:00
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4-x is always >=0.
So x is always <=4.

Is this what you have meant?
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Re: When is |x-4| = 4-x? [#permalink]  14 Mar 2013, 12:19
2
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Absolute value of any number or expression must be positive.
If (x-4) is positive then |x-4| is also positive
What if x-4 is negative? Since the absolute value must be positive, |x-4| would be equal to -(x-4)=4-x.
Right?

We know that x-4 would have to be negative for the equation in question to be true. This would imply that x would have to be a small positive number smaller than 4 or a negative number. You can take examples to test that.
x=-14 (x-4)=-ve
x=1, x-4=-3 -ve
x=4, implies x-4=0 and 4-x=0. Thus, the equation is satisfied.

Coming to your question, if a question deals with equality it also indirectly deals with inequality. If you say the equation is satisfied when x=0,x=4,x=-5 and so on, it also implies that the equation is true for all values of x less than or equal to 4.

An equation exists only at certain points. We have to find those points and if those points range over a large space, the easiest way would be express it as inequality.

Note: An equality question can have answers which might be expressed as inequalities. There is nothing wrong with it.

Hope it helps! Let me know if I can help you any further.

dhlee922 wrote:
how come all of a sudden the answer has inequalities when the question only had equal signs? that's the part i dont understand

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Re: When is |x-4| = 4-x? [#permalink]  29 Oct 2012, 01:56
1
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Expert's post
mario1987 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 $$|some \ expression|\leq{-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|\leq{some \ expression}$$. 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 am trying to understand theese two properties, but how is it possible to have |X|=-X, in order that absolute value has to be always positive?
Could you please provide me an explaination in more details?

When $$x\leq{0}$$, for example when $$x=-5$$, then $$|-5|=5=-(-5)$$ so $$|x|=-x$$ (|negative |=-(negative)=positive).

For more check Absolute Value chapter of Math book: math-absolute-value-modulus-86462.html

Hope it helps.
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Re: When is |x-4| = 4-x? [#permalink]  28 Nov 2014, 23:23
1
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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

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Re: When is |x-4| = 4-x? [#permalink]  28 Oct 2012, 01:56
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 $$|some \ expression|\leq{-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|\leq{some \ expression}$$. 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 am trying to understand theese two properties, but how is it possible to have |X|=-X, in order that absolute value has to be always positive?
Could you please provide me an explaination in more details?
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Re: When is |x-4| = 4-x? [#permalink]  05 Dec 2012, 03:51
|x-4| = 4-x

Absolute values are always positive or greater than 0.
So, 4-x >= 0 ==> x <= 4

There is no need to test values.
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Re: When is |x-4| = 4-x? [#permalink]  06 Jan 2013, 13:26
Hi brunel,

You said some expression when it is > = 0 then l some espression l <= some expression.

In this case how do i know X-4 is >= 0

Thanks!
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Re: When is |x-4| = 4-x? [#permalink]  07 Jan 2013, 02:37
Expert's post
LiquidGrave wrote:
Hi brunel,

You said some expression when it is > = 0 then l some espression l <= some expression.

In this case how do i know X-4 is >= 0

Thanks!

We have that $$|x-4|=4-x=-(x-4)$$. So, $$|some \ expression|\leq{-(some \ expression)}$$, thus $$x-4\leq{0}$$ --> $$x\leq{4}$$.

Hope it's clear.
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Re: When is |x-4| = 4-x? [#permalink]  10 Feb 2013, 05:14
Why not E?

If i am not mistaken when x<0, then |x-1| = x-1?
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Re: When is |x-4| = 4-x? [#permalink]  10 Feb 2013, 05:21
Expert's post
Victorlp89 wrote:
Why not E?

If i am not mistaken when x<0, then |x-1| = x-1?

First of all we have |x-4| = 4-x not |x-1| = x-1.

Next, |x-1| = x-1 for $$x\geq{1}$$.

If $$x\leq{1}$$, then $$|x-1|=-(x-1)=1-x$$
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Re: When is |x-4| = 4-x? [#permalink]  13 Mar 2013, 13:45
how come all of a sudden the answer has inequalities when the question only had equal signs? that's the part i dont understand
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Re: When is |x-4| = 4-x? [#permalink]  14 Mar 2013, 16:34
thanks Kris. that does help. i guess my follow up question would be, is there a way to solve it algebraically rather than plugging in numbers?

Kris01 wrote:
Absolute value of any number or expression must be positive.
If (x-4) is positive then |x-4| is also positive
What if x-4 is negative? Since the absolute value must be positive, |x-4| would be equal to -(x-4)=4-x.
Right?

We know that x-4 would have to be negative for the equation in question to be true. This would imply that x would have to be a small positive number smaller than 4 or a negative number. You can take examples to test that.
x=-14 (x-4)=-ve
x=1, x-4=-3 -ve
x=4, implies x-4=0 and 4-x=0. Thus, the equation is satisfied.

Coming to your question, if a question deals with equality it also indirectly deals with inequality. If you say the equation is satisfied when x=0,x=4,x=-5 and so on, it also implies that the equation is true for all values of x less than or equal to 4.

An equation exists only at certain points. We have to find those points and if those points range over a large space, the easiest way would be express it as inequality.

Note: An equality question can have answers which might be expressed as inequalities. There is nothing wrong with it.

Hope it helps! Let me know if I can help you any further.

dhlee922 wrote:
how come all of a sudden the answer has inequalities when the question only had equal signs? that's the part i dont understand
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Re: When is |x-4| = 4-x? [#permalink]  14 Mar 2013, 17:19
When is |x-4| = 4-x?

Critical Values method:

The Critical Value here is x=4 (we make the absolute value term equal to zero), so we have this conditions to check:

1) x<4: -(x-4)=4-x ---> x-4=x-4 ---> true for all values of x, but only when x<4 the initial condition is satisfied ---> true always that x<4

2) x=4: 0=0 ---> this is true always that x=4

3) x>4: x-4=4-x ---> 2x=8 ---> x=4 ---> initial condition of x>4 is not met

Therefore, there is a solution only when: x<=4

Solution D
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Re: When is |x-4| = 4-x? [#permalink]  15 Mar 2013, 10:26
As Johnwesley said, for |x-4|=4-x, s-4 should be negative or equal to 0.

i.e. x-4<=0
Hence, x<=4

[quote="dhlee922"]thanks Kris. that does help. i guess my follow up question would be, is there a way to solve it algebraically rather than plugging in numbers?
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Re: When is |x-4| = 4-x? [#permalink]  09 Aug 2013, 21:24
I solved it in this way.

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

Choice A: X=4, it is true but X cannot be always 4
Choice B: X=0, it is also true, but X cannot be always 0
Choice C: X>4, it is false, for e.g. X=6, then one side of equation is 2 and the other side is -2
Choice D: X<=4, this choice encapsulate Choice A, Choice B and for all other conditions and is true for above said equation. Hence the answer choice is D.

It took only 1min to solve this problem with above method.

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0
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Re: When is |x-4| = 4-x? [#permalink]  10 Aug 2013, 13:15
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?

.....
picking numbers is best for this sort of problem.
you did well already..... it saves time.....
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Re: When is |x-4| = 4-x? [#permalink]  19 Dec 2013, 04:14
Bunuel wrote:
Victorlp89 wrote:
Why not E?

If i am not mistaken when x<0, then |x-1| = x-1?

First of all we have |x-4| = 4-x not |x-1| = x-1.

Next, |x-1| = x-1 for $$x\geq{1}$$.

If $$x\leq{1}$$, then $$|x-1|=-(x-1)=1-x$$

Hi Bunuel,

Sinetunes you can manipulate say |x-4|=|-(-x+4)|=|4-x| is what you'll end up with.

So then how do you solve when you have |4-x| = 4-x?

Is this property correct or what are its limitations?

Thanks
J
Re: When is |x-4| = 4-x?   [#permalink] 19 Dec 2013, 04:14

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