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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}.

Answer: D.

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? Thanks in advance

Re: When is |x-4| = 4-x? [#permalink]
29 Oct 2012, 01:56

1

This post received KUDOS

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}.

Answer: D.

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? Thanks in advance

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

Re: When is |x-4| = 4-x? [#permalink]
14 Mar 2013, 12:19

2

This post received KUDOS

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.

Hence, d is the answer.

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

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.

Hence, d is the answer.

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

More:"All I wish someone had told me about GMAT beforehand" There are many things you want to know before doing the GMAT exam (how is exam day, what to expect, how to think, to do's...), and you have them in this blog, in a simple way

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? _________________

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.