The absolute value of the opposite of negative 6?

Question

CONFUSING!!!!!!!

*the l l are absolute value signs* * the - are negative signs*
a. -(-l 6 l)
b. -(-6)
c. l -(-6) l
d -l -6 l

Answer 1

Answer: 6

Start with the opposite of negative 6.
The opposite of a number means you change its sign, so the opposite of negative 6 is positive 6.

The absolute value of positive 6 is just 6.

Answer 2

It doesn't matter about the opposite anyway, because absolute value means it's 6.

The opposite of -6 is just 6. Absolute value ignores the positive/negative portion, and just take the size of the number.

Absolute value means to just take a quantity, and ignore anything like direction. Think of negatives as "left" and positives as "right." Going 6 to the left is not the same as going 6 to the right. However, absolute value means that you don't care about the direction, just how far you went.

Absolute Value is easy:
|6| = 6
|-6| = 6

Answer 3

Ho ho, "the opposite of negative 6", eh? Sounds like your book is doing its utmost to make the question as confusing as possible. As others have already suggested, ignore the red herrings. Just focus on the 6 and your knowledge that an absolute value can't be negative.

Oh, we have choices now, I see. OK, c.

Answer 4

Don't confuse yourself. The absolute value of anything is always positive.

6

Answer 5

The opposite of negative 6 is positive 6.
The modulus of positive number is the positive number.
The answer is therefore 6.

Answer 6

There are at least TWO equally valid ARITHMETICAL or NUMERICAL answers, namely the positive quantities:

6 or 1/6.

How could there POSSIBLY be these TWO valid answers to this particular question? ---

There can be at least TWO valid answers because the operation "opposite" is simply not uniquely defined, mathematically. Therefore it is up to us to give it a definition that does not make your question trivial. (Simply saying "it means the negative of something" is a trivial statement with nothing worth contemplating about it.) At best the term "opposite" can be interpreted as "the arithmetic inverse under a group operation with respect to a chosen identity." (The "identity" in such systems is the quantity that remains UNCHANGED when its OWN "inverse" is constructed.)

But there are TWO such well-known sets of group operations having "opposites" in arithmetic:

(i) Addition and subtraction, with the identity zero (0); and

(ii) Multiplication and division, with the identity one (1).

In system (i), the "opposite" of - 6 is indeed + 6, but that change of sign is itself IRRELEVANT in this case since the final operator "absolute value of " trumps any sign assignment. In this case, you need not have performed the "opposite" operation before applying "absolute value." Whether you applied "opposite" or not, in this case the arithmetical or numerical result would still be

6.

In system (ii), the "opposite" of - 6 is - 1/6 (since their product must be 1), and its absolute value is of course then

+ 1/6.

In this more interesting case, the "opposite" operator produces both a NEGATIVE quantity AND one that is SMALLER than 1, in absolute terms. Thus the final "absolute" operator CHANGES this particular smaller "opposite" quantity back to a positive one, an intrinsically more interesting result.

Live long and prosper.

P.S. I now see that 10 minutes after you posed your original question, you gave multiple choice answers from which the supposedly correct answer could be chosen.

As is often the case, these proffered multiple choice "answers" show that writers of elementary texts or tests are people of limited imagination. In this case the original question poser wasn't even INTERESTED in the final VALUE to assign to the end result of these operations, only the TRANSCRIPTION onto the written page of what he/she had stipulated in words and a figure. How limited that aim is in its scope!

This is a SIMPLY BEAUTIFUL EXAMPLE of an EXCEEDINGLY DULL QUESTION, in which only written transcription rather than evaluation is wanted.

[Note that, as of this writing, ALL of the OTHER 6 responders to this question ASSUMED that only the resulting VALUE was at issue, NOT how to write the original expression on the page! ]

In that case, PROVIDED that your textbook writer somewhere in his/her book DEFINES "the opposite of " to mean "the same size of number but with the 'opposite sign'," then the merely transcriptive answer is indeed

c.) | - ( - 6 ) |,

that is, reading from the outside in and from left to right,

"the absolute value of the 'opposite' (' - ') of 'negative 6'. "

But if your textbook writer DID NOT define what he/she wanted you to understand by "the opposite of," then he/she has fallen into a logical trap of his/her own making.

So please, would you satisfy our natural curiosity?:

Did he/she ever take the trouble to define, for the purposes of his/her readers reading his/her book, what was to be understood and meant by the term "the opposite of " ?!

I think we'd all be grateful to know the answer to that question.

Note that if EVALUATION were the issue at hand, ALL of

a. - ( - l 6 l),

b. - ( - 6), and

c. | - ( - 6 ) |

would be equally valid answers, since their VALUES are all
( + ) 6.

Only

d. - l - 6 l

would NOT be a correct answer for the value, since the " - " sign outside the symbol for "absolute value of " makes it intrinsically negative.

One can only conclude that this was an exceedingly badly posed question. It would have been far better to pose it as "Write down (but DO NOT EVALUATE) a mathematical expression corresponding to the following text: The absolute value of the opposite of negative 6."

But it would still be necessary for the reader to have been informed, somewhere, that "the opposite of " was only to be interpreted as "the negative of."

As I have written or implied several times, this was a very DULL question, only made interesting by its revelation of the limited imagination(s) lying behind it.

Answer 7

Doesn't have to be. Absolute value is always, always, always positive. It doesn't matter what other operations are applied prior to the absolute value operation. As soon as you say absolute value, the number becomes positive.