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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 \(x-4\leq{0}\) --> \(x\leq{4}\).

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

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).

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

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

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?

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.