%used definitions
newcommand{mb}[1]{mathbf{#1}}
newcommand{skp}[2]{langle#1,;#2rangle}
newcommand{cuberoot}[1]{sqrt[leftroot{0}uproot{2}scriptstyle 3]{#1}}

%inequalities
begin{align}
    &&-alphaskp{mb{a}}{mb{n}_mathcal{W}} &;geq;& skp{mb{a}}{mb{n}_mathcal{W}}   left( cuberoot{left(left( 1+ frac{h_mathcal{F}}{a_z}right)^3 -1right)k_t+alpha^3} -alpharight) nonumber \
    &Leftrightarrow& -alpha &;leq;& cuberoot{left(left( 1+ frac{h_mathcal{F}}{a_z}right)^3 -1right)k_t+alpha^3} -alpha nonumber \
    &Leftrightarrow&  0 &;leq;& left(left( 1+ frac{h_mathcal{F}}{a_z}right)^3 -1right)k_t+alpha^3 nonumber \
    &Leftrightarrow&  alpha &;geq;& cuberoot{left( 1- left(1+ frac{h_mathcal{F}}{a_z}right)^3 right)frac{Delta t}{t_mathcal{F}}} .
end{align}