Is lxl=y-z?

[anand]

Hi,

Could someone please help me with this question.



Thanks.

[san]

provide the two statements please

[RonPurewal]

hi -

per the forum rules, TYPE THE QUESTION INTO THE FORUM. do not use image files for problems that don't require image files.

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ironically, because this problem is SO short (and therefore so ridiculous to post as an image file), i'll copy it into the forum myself.

the problem is:
is |x| = y - z ?
(1) x + y = z
(2) x < 0

[RonPurewal]

hi -

per the forum rules, TYPE THE QUESTION INTO THE FORUM. do not use image files for problems that don't require image files.

--

ironically, because this problem is SO short (and therefore so ridiculous to post as an image file), i'll copy it into the forum myself.

the problem is:
is |x| = y - z ?
(1) x + y = z
(2) x < 0

statement 1:
we can rephrase this to x = z - y.
there are thus 3 possibilities for the absolute value |x| :
(a) if z - y is positive, then |x| = z - y, and will NOT equal y - z (which is a negative quantity).
(b) if z - y is negative, then |x| = y - z (the opposite of z - y).
(c) if z - y = 0, then |x| equals both y - z and z - y, since each is equal to 0.
TAKEAWAY: when you consider absolute value equations, you'll often do well by considering the different CASES that result from different combinations of signs.
notice that (a) and (b), or (a) and (c), taken together prove that statement 1 is insufficient.

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statement 2:
we don't know anything about y or z, so this statement is insufficient.**
if you must, find cases: say y = 2 and z = 1. if x = -1, then the answer is YES; if x is any negative number other than -1, then the answer is NO.

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together:
if x < 0, then this is case (b) listed above under statement 1.
therefore, the answer to the prompt question is YES.
sufficient.

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**note that, if i were particularly evil, i could craft a statement that doesn't mention all three of x, y, z and yet IS STILL SUFFICIENT.
here's one way i could do that:
(2) y < z
in this case, y - z is negative and therefore CAN'T equal |x| -- no matter what x is -- since |x| must be nonnegative.
so, this statement is a definitive NO, and is thus sufficient even though it doesn't mention x at all.
this is evil, but i see no reason why it wouldn't be on the test.

[JPG]

Hi Ron,

in your explanation, I don't quite understand the following case:

(c) if z - y = 0, then |x| equals both y - z and z - y, since each is equal to 0.

Can you explain why we need to consider this?

[RonPurewal]

Hi Ron,

in your explanation, I don't quite understand the following case:

(c) if z - y = 0, then |x| equals both y - z and z - y, since each is equal to 0.

Can you explain why we need to consider this?

we need to consider it because it's distinct from the first two cases. in general, when you split a situation up into cases based on signs, there are three different "signs" to consider: positive, negative, and zero.

in this problem, you can get away with not thinking about this case, because cases (a) and (b) already show that statement 1 is insufficient.
but there are plenty of other situations in which a statement is true for both positive and negative numbers, but false for zero (or vice versa). ignoring zero is not a good habit to which to become accustomed.