Is it possible to take the absolute value of both sides of an equation? 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? 

What you've written is an inequality, not an equation.  If you have an equation, say $a=b$, you can conclude that $|a|=|b|$.
But notice that $3>-5$, although $|3|\not>|-5|$.
If $3x^2+bx+7>0$ for every value of $x$, then the quadratic equation $3x^2+bx+7=0$ has no solutions that are real numbers.  THat implies that the discriminant $b^2-4ac=b^2-4\cdot3\cdot7$ is negative.  If $b^2-84<0$ then $b^2<84$, so $|b|<\sqrt{84}$.
Now observe that $\sqrt{84}=\sqrt{4}\sqrt{21}=2\sqrt{21}$.
