[tex]\text{if }x>7\text{, then }|x|>7[/tex] is a valid argument
[tex]|y|>7\text{, so }y=7[/tex] is not a valid argument
For the first argument: [tex]\text{if }x>7\text{, then }|x|>7[/tex]
From the definition of absolute value function
[tex]|x|=x[/tex] if [tex]x\ge0[/tex]
That is every positive number is its own absolute value. Since
[tex]x>7\implies x\ge0[/tex],
we can argue that
[tex]x>7\implies |x|>7[/tex]
so the first argument is valid
For the second argument: [tex]|y|>7\text{, so }y=7[/tex]
From the definition of absolute value function
[tex]|y|:=\left \{ {y\text{ if }y\ge0}\atop{-y\text{ if }y<0} }\right[/tex]
This means that
[tex]|y|>7:=\left \{ {y>7\text{ if }y\ge0}\atop{-y>7\text{ if }y<0} }\right[/tex]
or
[tex]|y|>7:=\left \{ {y>7\text{ if }y\ge0}\atop{y<-7\text{ if }y<0} }\right[/tex]
no part of the definition allow for the option [tex]y=7[/tex]. So the second argument is not valid.
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