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

Question

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

bytatia · Answer

This might sound silly but i just started preparing for GMAT, and I have a question. Why is it then in some cases we take x<0 or x>0 and in this problem we have x<=0 and x>=0 Thank you.

Bunuel · Answer

Well, it all depends on the problem at hand. For this problem, we need = sign because x=4 also satisfies |x-4| = 4-x. Below posts might help to brush up fundamentals on modulus: Theory on Abolute Values: math-absolute-value-modulus-86462.html DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37 PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58 Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

arunspanda · Answer

When is |x-4| = 4-x? A. x=4 B. x=0 C. x>4 D. x<=4 E. x< 0 |x-4| = 4-x Or, \frac{|x-4|}{x-4}=-1 => (x-4) < 0 or x = 0 Or, x <= 4 Answer: (D)

Temurkhon · Answer

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

D

saadis87 · Answer

I am still new to modulus so please do bare with me if I sound stupid.

This problem can be solved easily by picking numbers but to understand the concepts I tried to solve it using the books I read.
So according to the book, I need to take into account when the modulus is positive and negative when solving

\(x-4>0, x>4\)

x-4=4-x
x=4
(not sure if this value has to be rejected or not. Please help)

and when \(x+4<0, x<=4\)
-(x+4)=4-x
-x-4=4-x
Just lost here.

My question is why do we chose X<=4 why do we chose one condition over the other.

Bunuel · Answer

For the second case, when x - 4 < 0 (x < 4), |x - 4| becomes -(x - 4), so we'd have -(x - 4) = 4 - x, which gives 4 = 4. Since 4 = 4 is true, then it means that for x < -4, |x-4| = 4-x holds true.

Combining x = 4 from the first case and x < 4 from the second, we'll have x <= 4.

Hope it's clear.

saadis87 · Answer

Makes a lot more sense, Thankyou

lucky1829 · Answer

The equation holds true for every x value < 0. IMO answer choice E satisfies the equation as well. How can we cross out E?

Bunuel · Answer

E is not correct because |x-4| = 4-x also holds for 0 <= x <= 4. Check discussion on previous page. Hope it helps.

chetan2u · Answer

Hi  Bunuel ,
I feel the way Q is asked, even x= 4, x=0 or x<0 may fit in..

the Q asks " When is |x-4| = 4-x?
 ofcourse when x=4, ans is yes..
when x= 0... ans is yes..
yes x<=4 gives the entire range, BUT the Q does not ask that..

Had the Q been.
 when all is |x-4| = 4-x?
for which all values is |x-4| = 4-x? OR
What is the range of x for |x-4| = 4-x?

Would in ACTUAL GMAT, the wordings of this kind MEAN what we are inferring here?

Bunuel · Answer

You are right the wording of the question is poor.

mathiaskeul · Answer

For those struggling to understand how this works with the equal / lesser signs X=4 |x-4|=4-x --> 4-4=4-4 X=3 |x-4|=4-x --> |-1|=1 X=2 |2-4|=4-2 --> |-2|=2 X=1 X=0 And so on

.

LogicGuru1 · Answer

|x-4| = 4-x? Since a Mod always returns +ve value |x-4| can be seen as >0 for all mathematical purposes so our equation becomes 0<4-x 4-x >0 x<4 Answer is D {more or less; there seems to be a sign problem in either in option or in the original question}

himmih · Answer

Well let us begin with knowing -
|a| = -a when a<= 0 
Therefore, |x-4| = 4-x when x-4 <=0
Hence, D

sasyaharry · Answer

|a - b| = b - a ; when a<b or a=b (includes 0)

rocko911 · Answer

Why not just A? i was confused b/w A and D but I agree with D but why not A?

VAKRATUNDA · Answer

Hi Bunuel,

This is a good quality question and bit tricky. I selected A as an answer.

After looking at answer choice, i can understand solution perfectly.

As per my thought process, whenever the question has equal to sign then we get either 2 values or one value (x=4).

But how can i avoid mistake on this type of question? Which triggering point forces you to think on non-negative value?

argerniki · Answer

I think the answer options combined with the wording of the question are a bit misleading.
For example, take Option B: x = 0

|0 - 4| = 4 - 0 ?
 |-4| = 4 
 4 = 4

That is true: in this case |x - 4|  =  4 - x

It seems like there are several options that are true.
I doubt that you will see something like this on the real GMAT

SergejK · Answer

Actually, if we stick to the process of testing the ranges, we also see that x<=4.

|x-4|=4-x: as we would need to multiply the rhs with -1 to have the expression on lhs we know that inside the absolute value brackets the value must be negative. So we know that we have the following range: x-4<=0. Given that, we also know that x<=4. The difference here is we are not testing ranges, we know that |x-4| must lie in the range x-4<=0, hence we are able to come to the conclusion of x<=4.

bumpbot · Answer

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

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