# Explaining the process where "x*(a-b) < ac-bc" becomes "x < c" by dividing both sides by (a-b)

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.

• 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. 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

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".

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.

• 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. Feb 25, 2021 at 11:00
• 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)" Feb 25, 2021 at 11:02
• @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. Feb 25, 2021 at 11:03
• @user253751 "dividing on both sides by" is perfectly fine, as well as ""by dividing on both sides". 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". Feb 25, 2021 at 11:14