If $$3\lt x\lt 6\lt y\lt 10$$, then what is the greatest possible integer value of $$y-x$$?

Incorrect.
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Incorrect.
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Incorrect.
You fell for the obvious trap! Way to go GMAC.
To avoid this trap, __Plug In__ numbers for $$x$$ and $$y$$. To get the greatest possible difference, use numbers that are as far apart from each other as possible.
The number 5 seems to be the obvious answer: $$x=4$$ and $$y=9$$ are integers that are greater than 3 but smaller than 10, respectively. But who said the numbers you use have to be integers? The question asks for an *integer difference*, not necessarily difference between *integers*. Both $$x$$ and $$y$$ are actually free to be fractions, although we want them to be fractions that create an integer difference between them.
With this in mind, can you come up with values of $$x$$ and $$y$$ that satisfy the requirements of the question but have a greater integer difference than 5?

Incorrect.
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Note that you cannot plug in the extremes for $$x$$ and $$y$$ because they are not included in the range.

Correct.
To get the greatest possible difference, __Plug In__ numbers for $$x$$ and $$y$$ that are as far apart from each other as possible.
But who said the numbers you use have to be
integers? The question asks for an *integer difference*, not necessarily
for *integer numbers*—for instance, $$x=3.5$$ and $$y=9.5$$. That gives a value of $$9.5-3.5 = 6$$.
The only way you could get a difference of 7 is if $$y$$ and $$x$$ equalled their extremes. This is not possible because the inequalities do not allow them to take the values of the extremes exactly. Thus, 6 is the greatest value of $$y-x$$.

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