Kapalci

2022-06-26

Does $\frac{x-2}{3x-6}$ really equal $\frac{1}{3}$?

In my maths lesson today we were simplifying fractions by factorising. One question was something like this: $\frac{x-2}{3x-6}$, which I simplified as $\frac{x-2}{3x-6}=\frac{x-2}{3(x-2)}=\frac{1}{3}$. It got me wondering however, whether these expressions are really equal, specifically in the case $x=2$, where the former expression is undefined but the latter takes the value $\frac{1}{3}$

Since the expressions only differ at a single point are they for all intents and purposes equal, or are they theoretically different? If I wanted to be entirely correct would I have to write $\frac{x-2}{3x-6}=\frac{1}{3}$ where $x\ne 2$?

My maths teacher explained that at $x=2$ the expression evaluates to $\frac{0}{3\times 0}$ and the zeros effectively cancel out. I wasn't altogether satisfied with this explanation because as far as I know $\frac{0}{0}$ is undefined.

Thanks in advance!

In my maths lesson today we were simplifying fractions by factorising. One question was something like this: $\frac{x-2}{3x-6}$, which I simplified as $\frac{x-2}{3x-6}=\frac{x-2}{3(x-2)}=\frac{1}{3}$. It got me wondering however, whether these expressions are really equal, specifically in the case $x=2$, where the former expression is undefined but the latter takes the value $\frac{1}{3}$

Since the expressions only differ at a single point are they for all intents and purposes equal, or are they theoretically different? If I wanted to be entirely correct would I have to write $\frac{x-2}{3x-6}=\frac{1}{3}$ where $x\ne 2$?

My maths teacher explained that at $x=2$ the expression evaluates to $\frac{0}{3\times 0}$ and the zeros effectively cancel out. I wasn't altogether satisfied with this explanation because as far as I know $\frac{0}{0}$ is undefined.

Thanks in advance!

drumette824ed

Beginner2022-06-27Added 19 answers

I believe putting it this way could be helpful:

One one hand, $\frac{x-2}{3x-6}$is a function that is defined on $\mathbb{R}$except at $x=2$, and at every point it is defined it is equal to $\frac{1}{3}$

On the other hand, $\frac{1}{3}$can be viewed as a function that needs to accept ANY input and provide you with $\frac{1}{3}$back. In particular, the domain restriction at $x=2$ prevents $\frac{x-2}{3x-6}$to be the same thing as $\frac{1}{3}$

Hence, simplifying wouldn't be appropriate $\frac{x-2}{3x-6}$to $\frac{1}{3}$unless we are working in a domain where $x\ne 2$

gvaldytist

Beginner2022-06-28Added 12 answers

Yes... and no.

It is impractical to properly lay out all of the subtleties of what one intends with math notation, which is ambiguous. We've had ages to learn how to create notations where ambiguities frequently don't matter but do in the fine print.

There are several distinct interpretations of what $\frac{x-2}{3x-6}$; The status of "evaluation" is where the alternatives diverge most noticeably at " $x=2$".

The most basic interpretation is that $\frac{x-2}{3x-6}$is a recipe for performing a sequence of arithmetic operations upon a given input; in this interpretation, evaluation at $x=2$ is, in fact, undefined.

There are also more ways to interpret "take the continuous extension": essentially, you take the function's graph and fill in all the gaps; here you would add in $(2,1/3)$. Also, if you were using the extended real numbers you would add in $(+\mathrm{\infty},1/3)$ and $(-\mathrm{\infty},1/3)$, so that evaluation at $\pm \mathrm{\infty}$ is defined. (similarly if you were using the projective real numbers)

Your teachers description is nonsense when taken literally; however, the likely intention is that he is using "$0$" as a stand-in for some sort of 'witness' of vanishing; e.g. we factor out $(x-2)$ from both the numerator and denominator to get

$\frac{x-2}{3x-6}=\frac{1}{3}\cdot \frac{x-2}{x-2}=\frac{1}{3}\cdot 1$

The witnesses do "cancel" to depart if we adopt one of these "continuous extension" interpretations $1/3$

Which expression has both 8 and n as factors???

One number is 2 more than 3 times another. Their sum is 22. Find the numbers

8, 14

5, 17

2, 20

4, 18

10, 12Perform the indicated operation and simplify the result. Leave your answer in factored form

$\left[\frac{(4x-8)}{(-3x)}\right].\left[\frac{12}{(12-6x)}\right]$ An ordered pair set is referred to as a ___?

Please, can u convert 3.16 (6 repeating) to fraction.

Write an algebraic expression for the statement '6 less than the quotient of x divided by 3 equals 2'.

A) $6-\frac{x}{3}=2$

B) $\frac{x}{3}-6=2$

C) 3x−6=2

D) $\frac{3}{x}-6=2$Find: $2.48\xf74$.

Multiplication $999\times 999$ equals.

Solve: (128÷32)÷(−4)=

A) -1

B) 2

C) -4

D) -3What is $0.78888.....$ converted into a fraction? $\left(0.7\overline{8}\right)$

The mixed fraction representation of 7/3 is...

How to write the algebraic expression given: the quotient of 5 plus d and 12 minus w?

Express 200+30+5+4100+71000 as a decimal number and find its hundredths digit.

A)235.47,7

B)235.047,4

C)235.47,4

D)234.057,7Find four equivalent fractions of the given fraction:$\frac{6}{12}$

How to find the greatest common factor of $80{x}^{3},30y{x}^{2}$?