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#26 2012-08-02 17:25:52

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

I've been offline for 3 days with a messed up internet provider so I just got to look at this problem.
The solution looks fine to me.  However it might be a bit easier to get at it noting that |x-8| = |8-x|
so that the inequality can be reduced from 3|8-x|+2<7-2|x-8| to 5|x-8|<5 to |x-8|<1 and then
consider the cases x-8<0,  x-8=0 and x-8>0.

Reading "-x"  as "negative x" is the beginning of woes dealing with absolute value for many folks.
It gives the impression that -x is negative when it need not be.  It would probably be better to read
"-x" as "minus x" which is just reading symbols.  The "symbol" for negative is "<0".  So to say that
-x is negative is to say -x<0 which is a sentence ascribing a property to -x.  But of course -x could
be zero or positive depending on what x itself is. 

The field axioms establish "opposites" but there is no mention of positive or negative.  These ideas
are introduced later in the order axioms.  So the "-" should be associated with opposites not the
concept of negative.  The way to say something is negative is to say that it is less than zero.
x<0, x=0 and x>0 say that x is negative, x is zero and x is positive, respectively.

When we see "-2" we just happen to know that this is negative, but the symbolism "-2" does
not say it.  We must write something like -2<0 so say that -2 is negative.  The word "Negative" is
an adjective and as such must be used in a sentence to ascribe this property to a number or expression.

Too many years teaching.  Tends to make me long winded.  Oh Well!  smile

I had to edit this since for some reason |x-8| = |8-x| gave a smily face between the two absolute
values when I left no spaces between the absolute value symbols and the equality symbol.
Like this: |x-8|=|8-x|  Apparently an "=" immediately followed by "|" generates neutral.

Last edited by noelevans (2012-08-02 17:34:38)


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#27 2012-08-02 17:49:51

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

Hi noelevans

Negtive is negative. The human mind is the one interpreting it incorrectly. Negative was never defined as <0. -x (read as "minus x" or "negative x") is a number such that x+(-x)=0, where zero is a natural/real number such that x+0=x. This is kinda confusing, now when I think of it, but luckily math has its way dealing with stuff like this.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#28 2012-08-02 23:49:23

295Ja
Member
Registered: 2012-07-28
Posts: 39

Re: Inequality having Absolute Value and Constant on its 2 side

Thank you noelevans smile

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#29 2012-08-03 00:04:28

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

Hi 295Ja

How were you classes?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#30 2012-08-03 00:15:58

295Ja
Member
Registered: 2012-07-28
Posts: 39

Re: Inequality having Absolute Value and Constant on its 2 side

Hi anonimnystefy!

We had a long exam in math. smile
How about you? Do you still have classes later?

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#31 2012-08-03 00:34:12

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

Hi 295Ja

We are on a summer break now, so I am not going to school at all. smile

How was the exam?

Last edited by anonimnystefy (2012-08-03 00:34:24)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#32 2012-08-03 00:42:49

295Ja
Member
Registered: 2012-07-28
Posts: 39

Re: Inequality having Absolute Value and Constant on its 2 side

Oh, I see. I'll know the result on Monday, maybe. But I think the exam that I took earlier is a little bit simplier than I expected, fortunately. smile

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#33 2012-08-03 00:50:12

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

Good then. I hope the results are good. smile


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#34 2012-08-03 01:05:31

295Ja
Member
Registered: 2012-07-28
Posts: 39

Re: Inequality having Absolute Value and Constant on its 2 side

Thanks! I'll let you know.

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#35 2012-08-03 01:16:19

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

Great. Good luck with your classes. smile


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#36 2012-08-03 09:07:58

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

Hi 295Ja!  You are most welcome.

Hi to you too stefy!
   I agree it is the human mind that has the problem with these ooncepts of opposite and negative.
"-x" can be read as "the additive inverse of x" or "the opposite of x" both of which are mouthfuls.
"minus x" does not necessarily convey the idea of opposite.  It is more just reading the symbols.
But it is certainly less syllables.  I often read "-x" as "op x" short for "opposite of x".  But it is just
one syllable prefixed to the x and so is easier to say.
   "Negative x" is three syllables prefixed to the x and the word negative brings up the idea of less
than zero and so messes with our minds a bit.  Also it is hard to break the habit of reading "-x"
as "negative x" because it is so heavily ingrained in our language, thought processes and minds.

Many aspiring math students when asked to write the absolute value of "-x" will say it is x; that
is, they will write |-x|=x which is not necessarily true.  They are confusing the "-" for negative
instead of thinking it as saying opposite.  They are so used to "stripping it off" of specific negative
numbers like -2, -3/4, -12, etc. that they just "naturally" want to strip it off of the -x as well.

Sorry for beating a dead horse, but reading "-x" as "negative x" has caused problems for many of
my students over the years.  As such this has been one of my pet peeves.

Have a fantastic day both of you!  smile  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#37 2012-08-03 09:11:11

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

I understand your point and I agree with you.

Have a good day, too.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#38 2012-08-06 05:49:30

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

The language of math was created over many hundreds of years by people who were not able for the most part to communicate effectively with each other (like we can today!).  As such we have sort of a hodgepodge of language that is not nearly as perfect as most believe.  Sometimes symbols and words are misleading, sometimes we don't have words or symbols to communicate ideas, sometimes we are given algorithms that are not as good as others available, sometimes we are given notation that is cumbersome.  My main interest is to find examples of such and try to help overcome these problems.

Here's a simple question.  So far over the years I have asked this question of  people of all levels
of mathematics up to and including PhD's.  None have had an answer yet. 

Here goes!  Nearly everyone knows that going from 4/8 to 1/2 is called reducing.  What is it called
when one goes from 1/2 to 4/8?  I used to call it antireducing for lack of a better term, but I don't
like that, so I have begun to call it enlarging.  There seems to be no common name for the
process.  And of course when we are taught to add two fractions like 3/4 and 2/3 by the LCD
process we do this operation on both fractions in getting fractions with a least common denominator.

Also reducing and enlarging are both operations we are doing on fractions, but there seems to be
no symbol for these operations. 

Have you seen a name for the "antireducing" or symbols for these operations?  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#39 2012-08-06 06:22:56

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,532

Re: Inequality having Absolute Value and Constant on its 2 side

hi noelevans

I call reducing simplfying.  So I have occasionally said 'complicating' for antireducing.

Trouble with finding a word for it is this:

Reducing a fraction to its lowest terms gives a unique answer.  But there's no one result for antireducing.

But, in maths, you are allowed to make up new terms, as long as you define them clearly.  So here's your chance to put your stamp on the process, and become famous for devising the term.

Maybe you could make it a compound word, incorporating the amount by which you are enlarging the top and bottom of the fraction.

eg.  evans-three-ing the fraction 1/2 would give the result 3/6

You define your term and I'll (for one) use it.  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#40 2012-08-06 06:39:35

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

Hi bob bundy!

Yes indeed!  One would have to specify by what factor to "enlarge" the fraction.  So given a word
line enlarging or complicating we would probably naturally say for 1/2 to 6/12 "enlarge 1/2 by a
factor of six."  But we could also reduce 6/12 by a different factor, so we could say something like
"reduce 6/12 by a factor of three."

But still there is no standard mathematical notation for either of these.  To make the typing fairly
easy I do this:
1 x      3
-   3 = -   (Spacing gets squirrely unless a monospaced font is available.)
2 x      6

  6  /      2
---   3 = -  (But I use the regular division symbol with the two dots about a "-" instead of "/")
12  /      4

This way if we are reducing or enlarging by a more complicated expression, we would only have
to write it once.

Most of the typing I do is on Word Perfect with Courier New font.  That way I can get most of the
normal arithmetic and algebra to come out looking fairly nice without a strain.

Last edited by noelevans (2012-08-06 14:48:31)


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#41 2012-08-06 19:51:38

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,532

Re: Inequality having Absolute Value and Constant on its 2 side

You've got to be able to make clear this difference:

For reducing I would show:

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#42 2012-08-07 10:17:17

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

Hi bob!
Thanks for the input.
True.  One might have to do some 'splainin' to make sure the difference between regular multiplication
of fractions and enlarging are understood.  It could save a bit of writing when the argument is large:

2x+3 *      2
------    (3x  - 4x + 7)  instead of
3x+4 *         
               2
(2x+3)(3x - 4x + 7)
---------------------- 
               2
(3x+4)(3x - 4x + 7)

The LaTex for the complex fraction 3/3 over 6/3 has 31 characters if I have counted right.  The
output of the Latex is only 7 characters.  So the two dimensional output is much easier to read
and understand than the one dimensional Latex input.

Do you know of a good "front end" for Latex that could be used easily to get the LaTex code for this
forum?  I know there are several programs that allow inputting the math in a mathematical format
but I don't know of one that then allows you to view the equivalent LaTex.


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#43 2012-08-07 15:28:35

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,915

Re: Inequality having Absolute Value and Constant on its 2 side

Hi noelevans,

I use the Online LaTeX Equation Editor by Codehogs, found here.

bobbym gave me this link quite some time ago, and it's the only LaTeX editor I've ever used or tried. I don't know what others use, nor its strengths and weaknesses, but it's done everything I've asked of it.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#44 2012-08-08 04:13:03

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

Hi phrontister!

Thanks a bunch for showing me the link and you too bobbym for showing phrrontister.  Seems to be
just what I was looking for!  smile  Yeah!  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#45 2012-08-08 04:41:29

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

Hi again phrontister smile

I found it easy to get the expression below.  I couldn't get the "*'s" to float between the fraction and
the quadratic so I just left them in the fraction.

Also I tried copy/paste for the latex command, but alas it wouldn't paste.  So I tried separate panes
for the forum and for the code site.  Then I selected and dragged the code over to the forum and it
worked great!  Thanks again.

I finally figured out how to get the "*" to "float."

Last edited by noelevans (2012-08-08 14:18:53)


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#46 2012-08-08 12:05:59

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 3,915

Re: Inequality having Absolute Value and Constant on its 2 side

Hi noelevans,

Also I tried copy/paste for the latex command, but alas it wouldn't paste.

Were you trying to copy the LaTeX output, maybe? Just do this:
- left click in your equation entry box;
- do Ctrl+a (keyboard) to select all your entered text;
- do Ctrl+c (keyboard) to copy that selection to your clipboard;
- do Ctrl+v (keyboard) in your MIF Message panel to paste the clipboard contents there;
- select the pasted text and click on the 'Math' button under the Message pane.

That should work.

To use the 'times' symbol for multiplication instead of the asterisk (as shown below), that is available from the drop down menu above the brackets' menu, and is the symbol directly above the asterisk.


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#47 2012-08-08 14:36:51

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: Inequality having Absolute Value and Constant on its 2 side

Thanks phrontister!
That copy and paste worked OK.  I'm not sure what I was doing that didn't work.  I think I was using
the quick post and then copying from it and trying to go to the post reply.  At any rate I was jumping
around amonst pages and lost something in the process. 

Yeah, I like the x symbol for the multiplication better than the *.  Thanks for the suggestion.  smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#48 2012-08-14 23:15:50

295Ja
Member
Registered: 2012-07-28
Posts: 39

Re: Inequality having Absolute Value and Constant on its 2 side

Hello anonimnystefy! It's been a long time! I just want to keep my promise to you... big_smile
I just got the result of my exam earlier this day.
I didn't got the perfect score due to carelessness but anyway, I still passed. Better luck next time, I guess.

Have a great day!

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#49 2012-08-15 00:03:29

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,658

Re: Inequality having Absolute Value and Constant on its 2 side

Hi 295Ja

First of all, getting a perfect score should never be a necessity.

Second, congrats on passing your exam. I hope you do good on all your exams.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#50 2012-08-15 00:17:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,658

Re: Inequality having Absolute Value and Constant on its 2 side

Hi 295Ja;

That is a very good result, well done!


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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