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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

nkimidi7y wrote:

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, \(|x-4|=4-x=-(x-4)\) đồ sộ be true should be that \(x-4\leq{0}\) --> \(x\leq{4}\).

Answer: D.

Hope it's clear.
_________________

nkimidi7y wrote:

|x-4| = 4-x

1) |x-4| is ALWAYS none negative.
so , 4-x must be also more or equal đồ sộ 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
_________________

4-x is always >=0.
So x is always <=4.

Is this what you have meant?

Bunuel wrote:

Hi Bunuel
I am trying đồ sộ understand theese two properties, but how is it possible đồ sộ have |X|=-X, in order that absolute value has đồ sộ be always positive?
Could you please provide u an explaination in more details?
Thanks in advance

mario1987 wrote:

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

For more kiểm tra Absolute Value chapter of Math book:

math-absolute-value-modulus-86462.html

Hope it helps.
_________________

|x-4| = 4-x

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

Answer: D

There is no need đồ sộ test values.

Hi brunel,

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

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

Thanks!

Xem thêm: trường đại học văn hoá tp hcm

LiquidGrave wrote:

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.
_________________

Why not E?

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

Victorlp89 wrote:

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 \(|x-1|=-(x-1)=1-x\)
_________________

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 đồ sộ -(x-4)=4-x.
Right?

We know that x-4 would have đồ sộ be negative for the equation in question đồ sộ be true. This would imply that x would have đồ sộ be a small positive number smaller kêu ca 4 or a negative number. You can take examples đồ sộ 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 đồ sộ your question, if a question đơn hàng with equality it also indirectly đơn hàng with inequality. If you say the equation is satisfied when x=0,x=4,x=-5 and sánh on, it also implies that the equation is true for all values of x less kêu ca or equal đồ sộ 4.

An equation exists only at certain points. We have đồ sộ 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 u 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

thanks Kris. that does help. i guess my follow up question would be, is there a way đồ sộ solve it algebraically rather kêu ca plugging in numbers?

Kris01 wrote:

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

Critical Values method:

The Critical Value here is x=4 (we make the absolute value term equal đồ sộ zero), sánh we have this conditions đồ sộ 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 đồ sộ 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 đồ sộ solve it algebraically rather kêu ca plugging in numbers?

GMAT 1: 530 Q50 V14

GMAT 2: 630 Q48 V29

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 đồ sộ solve this problem with above method.

A. x=4
B. x=0
C. x>4
D. x<=4
E. x< 0

Bunuel wrote:

Hi Bunuel,

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

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

Is this property correct or what are its limitations?

Thanks
J

jlgdr wrote:

\(|4-x| = 4-x\) --> \(4-x\geq{0}\) --> \(x\leq{4}\).
_________________

Silly u, thats correct
Thanks
Cheers
J:)

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