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I was reading Spivak's Calculus and in the introductions to limits, it was stated that we assumed $|x-3|<1$ and hence $2<x<4$. Then this implies $5<x+3<7$ and then he says "this gurantees that $|x+3|<7$", but I don't understand how he went from $5<x+3<7$ to $|x+3|<7$, because from my understanding, $|x+3|<7$ is a larger interval than $5<x+3<7$.

Saran Wrap
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    The implication is in that direction just because the second interval is larger. If $x$ were outside the larger interval, it would also be outside the smaller one. – dfnu Oct 09 '19 at 13:43

2 Answers2

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$5<x+3<7$ means that $x+3$ is somewhere in the interval $(5, 7)$. On the other hand, $|x+3|<7$ means that $x+3$ lives somewhere in the interval $(-7, 7)$. Clearly, if $x+3$ lives in the first interval, then that implies that it lives in the second interval as well. There is no issue here. He goes from a stronger restriction and says it implies a looser restriction. That's completely fine.

Arthur
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Put simply, if $5 < x+3$ then $x+3$ is a positive number and thus $|x+3|=x+3$

David
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