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**bobey** Problem : prove that $\displaystyle |x|<|y| <=> x^2 < y^2$ by giving a reason for each step :

My teacher give solution

step 1 : $\displaystyle |x|<|y| => |x||x| =< |x||y|$ and $\displaystyle |x||y| < |y||y|$

step 2 : $\displaystyle => |x|^2 < |y|^2$

step 3 : $\displaystyle => x^2<y^2$

can anybody justify the first step? i can't understand which properties values to use in order to get to first step. tq

He's using the basic property of inequalities: $\displaystyle a<b\Longrightarrow ac<bc \,\,\,for\,\,\,c>0$

problem :

suppose $\displaystyle |x| =< 2 $. Use properties of absolute value to show that

$\displaystyle |(2x^2+3x+2)/(x^2+2)| =< 8$