The book first introduces a property: if a > b > 0, then a^2 > b^2
By assuming that if a > b > 0, then a^2 > b^2 is true. This is because both a,b>0 making aa>bb true as no negative is introduced (inequality sign does not flip).
The book then introduces another property: if a < b < 0 then a^2>b^2
I tried to approach this with a < b < 0 by first finding a,b<0 meaning that if I squared both sides, I would get a*-|a|< b*-|b|, the sign would have flipped twice as both sides were multiplied by a negative number, making it return to normal so I thought a^2 < b^2.
Although I know if examples were given/by common sense that the property is true, I just wondered why the sign flipped only once rather than twice when 2 negative were multiplied in the equation. Could you guys help me prove the unique concept of squaring both sides of inequalities? Do I lack basic understanding towards the concept of inequalities?