Inequalities and Square Root

Question

My maths book says that $y^2 > a^2$ implies that $y > a$ or $y < -a$

and that $y^2 < a^2$ implies that $-a < y < a$

Please can someone explain this to me as I do not understand this. Thanks.

Answer 1

Because $$0<y^2-a^2=(y-a)(y+a).$$ Now, we can assume that $a\geq0$ and it's obvious enough:

We have or $y+a>0$ and $y-a>0$, which gives $y>a$ or

$y+a<0$ and $y-a<0$, which gives $y<-a$.

The second statement we can get by the similar way.

Answer 2

The square operation disregards the sign and only deals with the magnitude.

We take for granted that $a$ is positive. Then $y^2>a^2$, i.e. $y$ has a larger magnitude if $y$ is positive and $y>a$ or $y$ is negative, i.e. $|y|=-y>a$, which is also $y<-a$.