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I am a bit confused about this, I am reading a math textbook and it says something that I dont agree with. Maybe I am missing something,If $-14$ or (ii)$y<6$ . In my understanding, the given inequality implies that $44$. for example, 8>4 but 8 does not lie between 4 and 6. I have the same argument about y<6.

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    $4 < y < 6$ means $y > 4$ ***and*** $y < 6$. – 2017-02-28
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    Yes, I agree with you, but that does not answer my question. if _14? – 2017-02-28
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    I think it did answer your question, but that's a moot point as you have accepted an answer that ways the same. – 2017-02-28

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It means both conditions have to be satisfied. You can't choose y=8 because it violates the condition that y has to be less than 6

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For two sets $A,B$ the statement

"$x\in A$ necessarily means $x\in B$"

is equivalent to the statement "$A\subset B$".

This does not mean: "the region defined by the condition $ -1 < y-5 < 1$ and the region defined by the condition $y> 4$ are the same.".