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2x = [x + x]

So the statement "2x divided by 2x" is:

[ x + x ] ÷ [ x + x ]

...which, given that x is not zero, has a quotient of 1.

Let's say that in the statement "2x divided by 2x" the value of x is 3

Now the statement is:

[ 3 + 3 ] ÷ [ 3 + 3 ]

Inside each set of brackets, it's "3 + 3" on each side of the division sign, making the statement "6 divided by 6."

A monomial is a single term with ONE VALUE -- which is the PRODUCT of the coefficient multiplied by the variable (factor). With that understanding, a monomial never needs parentheses around it -- it's as if it already has parentheses around it, in much the same way an exponent attached to a base number never needs to be encased inside a set of parentheses. In both cases, you're told how many times to multiply a quantity.

The division sign (obelus), the slash (solidus), and the fraction bar (vinculum) all mean "divided by" -- they are synonymous. Therefore, those division symbols are interchangeable. If you do an internet search for "How to divide one monomial by another monomial," you will find example after example being worked through as a top-and-bottom vertical fraction, even if the division statement was originally written horizontally with an obelus or slash.

The statement "6 ÷ 2(1+2)," the numerator "6" can be factored out as: 2(1+2) ...making the statement: 2(1+2) ÷ 2(1+2) and if we replace what's inside the parentheses with the variable "x," the statement is:
2x ÷ 2x
which can also be written as the horizontal fraction...
2x / 2x
or as the vertical fraction...
...since all of those symbols mean "divided by."

The statement "6 ÷ 2(1+2)" is dividing one monomial by another monomial -- it's "2x divided by 2x." Because a monomial is a SINGLE TERM with a SINGLE VALUE, the coefficient cannot just be "peeled off" & used in some other operation as if it had no bearing on the variable (factor).

2x is equal to: (x + x), Therefore, the "2" in "2x" is not a stand-alone number unto itself -- it's the coefficient in a monomial which tells you how many times to multiply the factor (which is actually adding the factor to itself), in much the same way that an exponent tells you how many times to multiply the base number by itself. Just as the exponent is "attached" to the base number, the coefficient of a monomial is "attached" to the variable (factor). So the monomial division statement "2x divided by 2x" is: [ x + x ] ÷ [ x + x ]

Notice that the coefficient number of 2 "disappears" when the statement is written out in its most basic form (as the indicated additions). That proves, once and for all, that using the coefficient of the monomial (the "2" in "2x") in some other operation is not valid.

Division is fractions. Fractions is division. We all know how to work through a fraction: Do all the operations indicated in the numerator, then do all the operations indicated in the denominator, and finally divide the numerator by the denominator.

PEMDAS is incorrect for "processing" division statements.

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