[Math] Is it possible to take the absolute value of both sides of an equation

Question

I have a problem that says:
Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$.

Since the quadratic is greater than 0, I assume that there are no real solutions since
$y = 3x^2+bx+7$, and $3x^2+bx+7 > 0$, $y > 0$
since $y>0$ there are no x-intercepts. I would use the discriminant $b^2-4ac<0$.

I now have $b^2-4(3)(7)<0$
$b^2-84<0$
$b^2<84$
$b<\pm\sqrt{84}$

Now how do I change $b$ to $|b|$? Can I take the absolute value of both sides of the equation or is there a proper way to do this?

Answer 1

Use the result that $a^2>b^2$ if and only if $|a|>|b|$.