
Upper and Lower Bounds
Find an upper and lower bound for the reciprocal of the modulus of $\displaystyle z^45z^2+6$ if $\displaystyle z=2$
$\displaystyle n\leq\frac{1}{z^45z^2+6}\leq m$
Out of all the the ways to break modulus how do I know which structure of the triangle inequalities to use?
$\displaystyle z^45z^2+6=(z^23)(z^22)\geqz^23z^22=z^23*z^22$$\displaystyle =4342=1*2=2$
$\displaystyle z^45z^2+6\leqz^45z^2+6=2^45*2^2+6=1620+6=2$
Why am I getting the same value for my upper and lower bounds?

It is not necessary to do that. A is an upper bound for 1/f(x) if and only if 1/A is a lower bound (not 0) for f(x) and B is a lower bound for 1/f(x) if and only if 1/B is an upper bound for f(x).

I don't understand since the bounds are $\displaystyle \frac{1}{42}\leq\frac{1}{modulus}\leq\frac{1}{2}$