I cannot accept 6 ÷ 2x as 3x, though. Blindly using order of operations, I guess you could write 6 ÷ 2 * 3, but that is not what it asking. "2x" is a single term, as defined by many algebra books, therefore, there are 2 terms here. 6 & 2x. hence, 6 ÷ 2x = 3/x. People are translating 2x into 2 * x. 2 * x however, are 2 terms separated by an operator, and 2x is one term. Evaluated the same? Yes. But they mean 2 different things from every book I have used in my studies. And, if 2x is one term, then 6 ÷ 2x = 6 "allover 2x" or 6/(2x). Let x = 1+2, and the answer is one. One would have to disprove 2x is a single term.
How about a problem with area of a rectange:
The area is 2(2+1) sq ft
. When asked how many times it can fit another area, the units should cancel out.
6 sq ft ÷ 2(2+1) sq ft = 1
If you tried it the other way:
6 sq ft ÷ 2 ft * (2+1) ft = 9 ft. It can fit into 6 sq ft, 9 ft
I am trying to drive home the point that 2(2+1) is one value, along with the example of factoring/distributing.
In that picture, I am also trying to reinforce that a fractional coefficient absolutely requires parentheses. This notation is undoubtedly used in all texts that use the / as a fraction bar.