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I would like to explain the process that an expression "x*(a-b) < ac-bc" becomes an expression "x < c".

I think one of the following would be correct according to my googling and my dictionary. Which one is correct?

"x < c" is obtained by dividing (a-b) on the both sides of "x*(a-b) < ac-bc".

"x < c" is obtained by dividing the both sides of "x*(a-b) < ac-bc" by (a-b).

Thank you.

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    I think this is taking out a common factor - otherwise referred to as factoring out (a-b). Feb 25, 2021 at 12:44
  • @FumbleFingers Your comment is very good. So, I can say that "After taking out a common facter, which is "c" on the right hand side, "x<c" is obtained by dividing by (a-b) on both sides of "x*(a-b)<ac-bc". Thank you for your comment.
    – Danny_Kim
    Feb 25, 2021 at 12:51
  • In the language of mathematicians (which I'm not really! :), I think your equation needs to be factored and simplified (that's a link to dozens of written instances of the term). Feb 25, 2021 at 13:04

2 Answers 2

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Apart from the incorrect use of "the", the second one is correct:

"x < c" is obtained by dividing the both sides of "x*(a-b) < ac-bc" by (a-b).

We "divide X by Y" to count how many times Y can fit into X. We never "divide Y on X".

However, you can also say:

"x < c" is obtained by dividing by (a-b) on both sides of "x*(a-b) < ac-bc".

Here, "dividing by (a-b)" is an action behaving as a noun, which linguists call a gerund clause. We are doing this action on both sides of the equation.


One might "divide (a-b) on both sides of the equation" if they wrote "(a-b)/(x*(a-b)) < (a-b)/(ac-bc)", but it would be incorrect mathematics. In this case, in my opinion, it would also be more natural to "divide (a-b) by both sides of the equation", or "by each side".

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Both the sentences are incorrect, but only because of the use of the word "the". If you remove that, then both sentences are correct.

  • "x < c" is obtained by dividing (a-b) on both sides of "x*(a-b) < ac-bc".

  • "x < c" is obtained by dividing both sides of "x*(a-b) < ac-bc" by (a-b).

Personally, I think the second sentence sounds better, but both are perfectly understandable.

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  • Firstly, I would like to thank you for the answer. May I ask why "the" should be crossed out? I saw the phrase "the both sides" many times in various papers.
    – Danny_Kim
    Feb 25, 2021 at 11:00
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    I've probably never heard of "dividing on", only "dividing by". You could say "dividing by (a-b) on both sides ..." or "dividing both sides ... by (a-b)"
    – user253751
    Feb 25, 2021 at 11:02
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    @Danny_Kim "the both sides" doesn't sound natural at all, but maybe it made sense in context. You can share an example of that if you want.
    – cigien
    Feb 25, 2021 at 11:03
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    @user253751 "dividing on both sides by" is perfectly fine, as well as ""by dividing on both sides".
    – cigien
    Feb 25, 2021 at 11:06
  • @cigien Thank you very much. Maybe, I think that's because the documents I often come across are scientific journal articles written by non-native people. I believe you much more, so I will remove "the" and using the word "divide" with "by".
    – Danny_Kim
    Feb 25, 2021 at 11:14

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