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

Answer: D.

Hope it's clear.
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|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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4-x is always >=0.
So x is always <=4.

Is this what you have meant?

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).

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

Hope it helps.
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|x-4| = 4-x

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

Answer: D

There is no need to test values.

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

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.

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?

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

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?

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

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

\(|4-x| = 4-x\) --> \(4-x\geq{0}\) --> \(x\leq{4}\).
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Silly me, thats correct
Thanks
Cheers
J:)

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