Do the equations
$|x|<a$ and $x<|a|$ have the same solutions , i.e
$-a<x<a$
In general , do the two equations mean the same thing , that the absolute value of $x$ is less that the distance of $a$ from the origin , on either side of the origin ?
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3$x<|a|$ includes all $x<0$. – lulu Feb 04 '18 at 12:13
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@lulu Oh so they do not mean the same thing ? – Aditi Feb 04 '18 at 12:14
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1@Aditi no, they do not. the set of solutions to $|x|<a$ is included inside the set of solutions to $x<|a|$ however. (assuming $a$ is positive) – John Doe Feb 04 '18 at 12:15
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@John Oh alright I get it . It means $|x|<a$ is a smaller set than $x<|a|$ – Aditi Feb 04 '18 at 12:16
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1The set of $x$ is bounded in the first case but it is not in the second one. – Piquito Feb 04 '18 at 12:20
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@Aditi yes, that is correct – John Doe Feb 04 '18 at 12:22
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@Piquito okay I understood ! In the second case $x$ can take all the negative values from negative infinity $|a|$ – Aditi Feb 04 '18 at 12:23
2 Answers
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If $a=-1$, then$$x<|a|\iff-1<x<1,$$whereas there is no $x$ such that $|x|<a$.
José Carlos Santos
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Because $| t|=\max\{t,-t\}$, the inequality $|x|<a$ is equivalent to $$x<a\quad\land\quad -x<a $$ whereas $x<|a|$ is equivalent to $$ x<a\quad\lor\quad x<-a.$$ I suppose tha tthe lack of equivalence becomes obvious from this.
Hagen von Eitzen
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