Originally Posted by mathman26
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.
There is a lot of fact in there, so I'll address everything both correct and incorrect.
2x is indeed one term, otherwise know as a monomial. However, 6/(2x) and (6/x)2 are both also considered to be monomials. What is the splitter of terms is addition/subtraction. Just because there is an operator between them, doesn't mean that it breaks it into 2 terms. A perfect example is exponentiation. x^2 is considered to be one term, but x^2 = x*x. The same applies to 2x; 2x = 2*x = x + x. Simplifying x+x is referred to as combining like terms. However simplifying x*x is not because multiplication is not a splitter of terms. It's weird and seems arbitrary, but it is what it is. Blame ancient Babylonians and ancient Greeks.
I'm am not disputing at all that 6/(2x) where x = 2+1 isn't 1 at all; it is absolutely equal to 1. The discrepancy occurs because the way that the equation is written, the (2+1) is actually on the side of 6/2 fraction and thus making multiplication of the numerator and not the denominator. Also, there is no such thing as "blindly" using order of operations. They're absolute unless second hand stipulations have been stated in the problem. It's not something that comes with conditions on where it works and where it doesn't. Division and exponentiation do have conditions where they fail (division by zero, negative base of a rational exponent with an even denominator), but the order of operations do not. If logic dictates that the order of operations are wrong, then you add parenthesis. That is why they are given highest priority, and their purpose over all: to alter the order of operations.
As for your example, they are actually two different questions that are trying to be posed as the same. In fact, there is an inconsistency that causes a contradiction. In the left picture, the width is 2ft, but in the right picture it is 3. The left picture calculates the area beforehand and poses the question and asks how many times the area can be divided by itself. The right picture asks to calculate the area of a new rectangle. The logic of the first question is correct, but the notation is wrong. Even though the problem does require the multiplication to occur before the division, the notation signals that the division must occur before the multiplication. The correct equation should be written as 6 ÷ [2(2+1)].
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