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Saw my textbook cancel a 2x with an x so you have just a 2 left, but 2x in my mind is x and x, so how I one x cancelling x + x

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Hi u/scillywoba,

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2x/x

= (x + x)/x

= x/x + x/x

= 1 + 1

= 2

(As long as x is not 0.)
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OP, there are several explanations here, and I don't need to add another. I just wanted to say that you're doing great. The fact you're asking this question, that you noticed something you thought was confusing, that you explained it using terms you understand, and that you're looking to make sense of it is all praiseworthy.

The types of people who become mathematicians think like you do. In this moment of confusion, you did **not** just repeat to yourself some shortcut or rule, you did **not** just take the textbook's/teacher's word for it -- you *started thinking*.

People who see math as steps to do, shortcuts to memorize, rules to obey, and procedures to follow will do *fine* in high school (because of the sad way high school math is usually taught), but they will not go far past that.

Never stop thinking.

(The only thing I'd add to your thought process is this: try to answer the question, "what does *cancel* mean in this situation?" I avoid using the word "cancel" when I am teaching because it means different things in different scenarios. What's really happening when you cross things out in this expression?)
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x is not cancelling (x + x), because that would lose the 2. Think of x as "number of apples" and 2x is then "twice the number of apples". This way no matter how many apples you have, x / 2x is ALWAYS 1/2.  4 out of 8 apples, 46 out of 92 apples, 104/208 apples, doesn't matter. For any x (other than 0), x/2x = 1/2.
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take a number, multiply it by 2. now divide the original number. the number cancels out leaving 2. basically 2x/x = 2(x/x) = 2(1) = 2

If doesn’t help I think the other commenter does a good job doing the algebra out
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Take out a factor of x from (x+x). That should leave you with x/x(1+1). The x’s cancel to leave 1/2
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How would you simplify 3/6?   Hopefully you've learned that to simplify a fraction, you first factor the numerator and denominator and then you can cancel any factors in common.

So for 3/6 you can factor the denominator to get 3/(2*3) and then you can cancel the 3's to get 2.  In other words you can only cancel factors.

Now you can also write 3/6 = 3/(3+3) but then it's not in a form that lets you cancel.  You're correct that you can't cancel parts of a sum.  Only factors can be cancelled.

x/(2x) is exactly the same.  Sure that's also equal to x/(x+x) but x/(x+x) is just like 3/(3+3).  If you wanted to simplify x/(x+x) you'd first rewrite that as x/(2x) and then you cancel the x's because x is now a factor of the numerator and denominator.

More generally there are many equivalent ways to write any expression.  And different ways are useful in different situations.  If you want to simplify a fraction then the most useful thing to do is to write the numerator and denominator as a bunch of factors multiplied together and not as a sum.
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When you canceled the x+x you forgot the 2, x+x is the same as x(1+1). Your x may cancel and (1+1)=2
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So, if 2x = x + x we can rewrite the equation to be (x + x)/x

And since dividing is just the same as multiplying by the reciprocal we can rewrite this as (x + x) * (1/x)

Which when multiplied is (x/x + x/x)

Simplified that's (1 + 1), which is just 2
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2x = x + x. => 2x/x = (x+x)/x = x/x + x/x = 1 + 1 = 2.

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