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Absolute Value Problem

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by iwillsurvive101 » Tue Apr 10, 2012 12:47 pm
For what value of x does |x-4| = 4-x ?

I solved it as follows:

Case1: +(x-4)= 4-x

=> x-4 = 4-x
=> x=4

Substituting 4 in |x-4| = 4 -x
0=0
True

So x=4 is valid

Case2: -(x-4) = 4-x
-x = -x
Lost here...Is this right?

The right answer is, for all x<=4. But, how? I only got x=4.

Some absolute value fundamentals would help clarify this problem for me.

Thanks in advance.
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by neelgandham » Tue Apr 10, 2012 1:59 pm
iwillsurvive101 wrote:For what value of x does |x-4| = 4-x ?
Case2: -(x-4) = 4-x
-x = -x, Lost here...Is this right?
The right answer is, for all x<=4. But, how? I only got x=4.
Definition of |x|
For any real number a the absolute value or modulus of a is denoted by |"‰a"‰| (a vertical bar on each side of the quantity) and is defined as
|"‰a"‰| = a, if a>0 and
|"‰a"‰| = -a, if a<0

So here,
|x-4| = 4-x;
|x-4| = -(x-4;
if y = x-4,
|"‰y"‰|=-y,

From the definition of |y| if |"‰y"‰|=-y, y < 0,
=> x-4<0
=> x<4
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by iwillsurvive101 » Tue Apr 10, 2012 2:33 pm
Thanks Mr. Quant, appreciate the detailed response. I was close, but just wanted to conceptually take-it-in :-)
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by neelgandham » Tue Apr 10, 2012 2:46 pm
I am glad I could help. Thanks for the kind words. Let me know if you need any further help! and YES you will survive :)
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by GMATGuruNY » Tue Apr 10, 2012 7:03 pm
For what values of x does |x-4| = 4-x?
4-x is the DIFFERENCE between 4 and x.
A DIFFERENCE can be negative, 0, or positive.

|x-4| is the DISTANCE between x and 4.
A DISTANCE must be greater than or equal to 0.

For the DIFFERENCE between two values to be equal to the DISTANCE between the two values, the DIFFERENCE -- like the DISTANCE -- must be greater than or equal to 0:
4-x ≥0
4 ≥ x.

Thus, |x-4| = 4-x when x≤4.
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