Solve for ##x## in the Given Problem

  • Thread starter Thread starter chwala
  • Start date Start date
AI Thread Summary
The discussion centers around solving the equation -[(3x+6)/x] = 20, with participants analyzing different approaches to isolate x. One participant correctly factors the equation to find x = -6/17, while another attempts a different method that leads to an incorrect solution of x = -6/23. There is confusion regarding the interpretation of mixed fractions in algebraic expressions, with some arguing that juxtaposed terms should be treated as multiplication rather than addition. The conversation highlights the importance of clear notation and communication in mathematical problem-solving. Ultimately, the thread emphasizes the need for consistent interpretation of expressions to avoid misunderstandings.
chwala
Gold Member
Messages
2,825
Reaction score
413
Homework Statement
I normally set and create my own math problems ; This one was a bit interesting ...need your input.

Find ##x## given:



##-2\dfrac{2}{x} - 1\dfrac{4}{x} =20##
Relevant Equations
Equations with unknown variable
In the first approach; which yields the correct solutiuon we have;##-\left[\dfrac{3x+6}{x}\right] =20##

I factored out the negative...solved for ##x## then re-introduced the negative at the end.

##3x+6=20x##

##6=17x##

##x=-\dfrac{6}{17}##

Now to my question: I tried doing this...

##-\left[\dfrac{3x+6}{x}\right] =20## then i multiplied both sides by ##-1## i.e

##\left[\dfrac{3x+6}{x}\right] =-20##

##3x+6=-20x##

i.e
##-\dfrac{3x+6}{x}=\left[-1 ×\dfrac{3x+6}{x}\right]=20##
## \left[-1× -1 ×\dfrac{3x+6}{x}\right]=-20##

##x=-\dfrac{6}{23}## which is clearly not correct! following my checking through substitution...something i am missing here?

I know that;

##-\left[\dfrac{3x+6}{x}\right]=-\dfrac{3x+6}{x} = \dfrac{-3x-6}{x} = \dfrac{3x+6}{-x}## cheers guys!I think i see the mistake the negative only applies to the numerator...was a nice one...time for coffee lol!:cool:
 
Last edited:
Physics news on Phys.org
Hi,

chwala said:
This one was a bit interesting
You can make it a whole lot less interesting if you define ##y={1\over x}## and solve for ##y## !

##\ ##
 
BvU said:
Hi,

You can make it a whole lot less interesting if you define ##y={1\over x}## and solve for ##y## !

##\ ##
I'll check this out...later mate. Cheers...
 
@chwala, neither of your answers are correct!
##-\left[\dfrac{3x+6}{x}\right] =20## is wrong.

And even if the above equation were correct, your next line
##3x+6=20x##
would also be wrong; it should be
##-3x-6=20x## or
##3x+6=-20x##

I would solve your original equation by multiplying both sides by ##x##. Or use @BvU's method.

Also, always check your answer by substituting it back into the original equation.
 
@chwala, neither of your answers are correct!
##-\left[\dfrac{3x+6}{x}\right] =20## is wrong.

And even if the above equation were correct, your next line
##3x+6=20x##
would also be wrong; it should be
##-3x-6=20x## or
##3x+6=-20x##

I would solve your original equation by multiplying both sidesby ##x##. Or use @BvU's method.

Also, always check your answer by substituting it back into the original equation.
 
Steve4Physics said:
@chwala, neither of your answers are correct!
##-\left[\dfrac{3x+6}{x}\right] =20## is wrong.

And even if the above equation were correct, your next line
##3x+6=20x##
would also be wrong; it should be
##-3x-6=20x## or
##3x+6=-20x##

I would solve your original equation by multiplying both sidesby ##x##. Or use @BvU's method.

Also, always check your answer by substituting it back into the original equation.
Interesting you're saying ##x=\dfrac{-6}{17}## is not correct? Note that in my approach i worked with improper fractions throughout.

Let me check my working again...

if i substitute;

##x=\dfrac{-6}{17}## back into original equation i would have;

##2+5+\dfrac{2}{3}+1+11+\dfrac{1}{3}=20##

note that in my working i took;

##-2\dfrac{2}{x}= \dfrac{-2x-2}{x}##

Let me check my working again...back after break.
 
Last edited:
I don't think I've ever seen juxtaposition used to indicate a mixed number with an unknown in the denominator, or indeed anywhere. I would naturally interpret <br /> 2\frac{2}{x} = 2 \times \frac{2}{x} = \frac{4}{x} rather than as 2 + \frac{2}{x}.
 
  • Like
Likes Steve4Physics
chwala said:
note that in my working i took;

##-2\dfrac{2}{x}= \dfrac{-2x-2}{x}##

Let me check my working again...back after break.
Huh ?
##-2-\dfrac{2}{x}= \dfrac{-2x-2}{x}## !!

While you are checking anyway, please check if we are trying to solve
chwala said:
Find ##x## given:

##-2\dfrac{2}{x} - 1\dfrac{4}{x} =20##
or something else entirely !!

##\ ##
 
BvU said:
Huh ?
##-2-\dfrac{2}{x}= \dfrac{-2x-2}{x}## !!

While you are checking anyway, please check if we are trying to solve

or something else entirely !!

##\ ##
My thinking on this is based on addition and subtraction of Mixed fractions...unless you may want to indicate if my way of thought is wrong on this.
 
  • #10
BvU said:
Huh ?
##-2-\dfrac{2}{x}= \dfrac{-2x-2}{x}## !!

While you are checking anyway, please check if we are trying to solve

or something else entirely !!

##\ ##
Yes we are trying to solve that...
 
  • #11
@chwala, I think there is a basic communications problem here! To expand on what @pasmith said in Post #7....

An expression such as
##2\dfrac{2}{x}##
could be interpreted in two ways:

Interpretation-1 (the usual one): simple multiplication:
Take the product of ##2## and ##\dfrac{2}{x}##

Interpretation-2: a mixed fraction:
Take the sum of ##2## and ##\dfrac{2}{x}##

I’m guessing you are using interpretation-2 but I (and maybe others) have used interpretation-1.

Edit: typo'
 
Last edited:
  • #12
chwala said:
Homework Statement:: I normally set and create my own math problems ; This one was a bit interesting ...need your input.

Find ##x## given:

##-2\dfrac{2}{x} - 1\dfrac{4}{x} =20##
Based on comments along the way in this thread, it appears that you intend that the terms on the left side are to be interpreted as mixed fractions akin to 1 7/8; i.e., 1 + 7/8.

I taught algebra (and trig, calculus, linear algebra, and differential equations) for many years, but have never seen any algebra textbook that carried the notion of mixed fractions over to algebraic expressions.

As noted several times here, ##-2\dfrac{2}{x}## would be almost universally interpreted as ##-2 \cdot \frac 2 x##. Making up your own problems is good, but you shouldn't make up your own notation that is in conflict with notation in common usage.
 
  • #13
Mark44 said:
Based on comments along the way in this thread, it appears that you intend that the terms on the left side are to be interpreted as mixed fractions akin to 1 7/8; i.e., 1 + 7/8.

I taught algebra (and trig, calculus, linear algebra, and differential equations) for many years, but have never seen any algebra textbook that carried the notion of mixed fractions over to algebraic expressions.

As noted several times here, ##-2\dfrac{2}{x}## would be almost universally interpreted as ##-2 \cdot \frac 2 x##. Making up your own problems is good, but you shouldn't make up your own notation that is in conflict with notation in common usage.
@Mark44 ...that's true maybe I should have indicated my intention...it depends on the context...when am solely looking at adding and subtraction Fractions, I do incorporate this ...cheers man!
 
  • #14
chwala said:
@Mark44 ...that's true maybe I should have indicated my intention...it depends on the context...when am solely looking at adding and subtraction Fractions, I do incorporate this ...cheers man!
Well you need to stop doing that. Just in case there is any doubt, you cannot simply state that "when I write ## 2 \frac 3 x ## I mean ## 2 + \frac 3 x ##" and expect to communicate with anyone else. It is equally as invalid as saying "when I write ## 6 + 3 = 2 ## I mean ## 6 \div 3 = 2 ##".
 
  • Like
Likes Mark44 and BvU
  • #15
pbuk said:
Well you need to stop doing that. Just in case there is any doubt, you cannot simply state that "when I write ## 2 \frac 3 x ## I mean ## 2 + \frac 3 x ##" and expect to communicate with anyone else. It is equally as invalid as saying "when I write ## 6 + 3 = 2 ## I mean ## 6 \div 3 = 2 ##".
I clearly get your point...
What I meant was when I am looking at fractions...I would occasionally add the mixed fractions as required (no variables involved) then later replace any of the numbers with a letter (variable)...then seek to find the unknown (letter)variable...
 
  • #16
By the way do we have a general rule in Math on how to interprete the problem? it can either be looked at as multiplication ( accepted universally) or as addition of mixed fractions as I had indicated. In both contexts they follow math rules to the latter. Cheers.
 
  • #17
chwala said:
By the way do we have a general rule in Math on how to interprete the problem? it can either be looked at as multiplication ( accepted universally)
This -- based on the problem in post #1.

chwala said:
or as addition of mixed fractions as I had indicated. In both contexts they follow math rules to the latter.
Not this. In my experience with algebra and subsequent textbooks, mixed fractions play no role once we move from ordinary grade-school arithmetic to algebra. Terms that are juxtaposed in algebra expressions are interpreted as being multiplied.
chwala said:
##-2\dfrac{2}{x} - 1\dfrac{4}{x} =20##
This would have been much clearer:
##-2 - \frac 2 x - 1 - \frac 4 x = 20##
 
  • Like
Likes pbuk, chwala and SammyS
  • #18
chwala said:
By the way do we have a general rule in Math on how to interprete the problem? it can either be looked at as multiplication ( accepted universally) or as addition of mixed fractions as I had indicated. In both contexts they follow math rules to the latter. Cheers.
Yes. Mixed fractions are not used in mathematics or physics. They are used in the US in engineering (e.g. a 1⅜" diameter pipe). But you need to understand the difference between a mixed fraction, which is notation for a constant, a particular point on the number line, and an expression like ## \frac 2 x ## which is notation saying "divide 2 by x".

Compare with the "-" sign which means something different in ## a = -1 ## and ## a = 0 - 1 ##.
 
Back
Top