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Analistul
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Post subject: Challenge problems - Set solutions 06/23/03  Posted: Wed Aug 08, 2007 3:11 pm |
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Hi,
Shouldn't x=2, x=3 and x=5 be tested also (either included in the intervals or tested separately?)
Regards and thanks,
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dbernst
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Post subject: Posted: Thu Aug 09, 2007 9:59 am |
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| ManhattanGMAT Staff |
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Posts: 304
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Analistul,
So all students can benefit from the posts, please include the entire problem and its answer choices. Once posted, an instructor will be glad to address your concerns.
Thanks!
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Analistul
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Post subject: Challenge problem Posted: Thu Aug 09, 2007 12:57 pm |
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For the following problem I believe that x=2, x=3 and x=5 should also be considered in the intervals or separately.
[b]"Which of the following sets includes ALL of the solutions of x that will satisfy the equation: ?
|x-2|-|x-3|=|x-5|
Answer
One way to solve equations with absolute values is to solve for x over a series of intervals. In each interval of x, the sign of the expressions within each pair of absolute value indicators does not change.
In the equation , there are 4 intervals of interest:
x < 2: In this interval, the value inside each of the three absolute value expressions is negative.
2 < x < 3: In this interval, the value inside the first absolute value expression is positive, while the value inside the other two absolute value expressions is negative.
3 < x < 5: In this interval, the value inside the first two absolute value expressions is positive, while the value inside the last absolute value expression is negative.
5 < x: In this interval, the value inside each of the three absolute value expressions is positive.
Use each interval for x to rewrite the equation so that it can be evaluated without absolute value signs.
For the first interval, x < 2, we can solve the equation by rewriting each of the expressions inside the absolute value signs as negative (and thereby remove the absolute value signs):
Notice that the solution x = 6 is NOT a valid solution since it lies outside the interval x < 2. (Remember, we are solving the equation for x SUCH THAT x is within the interval of interest).
For the second interval 2 < x < 3, we can solve the equation by rewriting the expression inside the first absolute value sign as positive and by rewriting the expressions inside the other absolute values signs as negative:
Notice, again, that the solution is NOT a valid solution since it lies outside the interval 2 < x < 3.
For the third interval 3 < x < 5, we can solve the equation by rewriting the expressions inside the first two absolute value signs as positive and by rewriting the expression inside the last absolute value sign as negative:
The solution x = 4 is a valid solution since it lies within the interval 3 < x < 5.
Finally, for the fourth interval 5 < x, we can solve the equation by rewriting each of the expressions inside the absolute value signs as positive:
The solution x = 6 is a valid solution since it lies within the interval 5 < x.
We conclude that the only two solutions of the original equation are x = 4 and x = 6. Only answer choice C contains all of the solutions, both 4 and 6, as part of its set. Therefore, C is the correct answer."[/b][/b]
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StaceyKoprince
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Post subject: Posted: Sat Aug 11, 2007 7:25 pm |
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| ManhattanGMAT Staff |
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Location: San Francisco
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If x=2, then |x-2|-|x-3|=|x-5| becomes (2-2) - (2-3) = (2-5) or -1 = -3. So x can't be 2
If x=3, then |x-2|-|x-3|=|x-5| becomes (3-2) - (3-3) = (3-5) or 1 = -2. So x can't be 3
If x=5, then |x-2|-|x-3|=|x-5| becomes (5-2) - (5-3) = (5-5) or 1 = 0. So x can't be 5
_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT
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pranjali.deshpande
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Post subject: Re: Challenge problems - Set solutions 06/23/03 Posted: Mon Jan 03, 2011 12:26 pm |
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what about x=0 ...
If x = 0 then the equation becomes :-
|-2| - |-3| = |-5|
2 - 3 = 5
-1 = 5 => which is clearly wrong....
Is this question even correct????
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jnelson0612
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Post subject: Re: Challenge problems - Set solutions 06/23/03 Posted: Wed Jan 05, 2011 9:17 am |
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Posts: 1857
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pranjali, this is a very old thread--can you please post the question? Thank you.
_________________
Jamie Nelson
ManhattanGMAT Instructor
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