= Solution
For a nucleus of <spin> $I>0$ and electronic $J=1/2$, the magnetic-dipole <hyperfine interaction> is $A\mathbf I\cdot\mathbf J$, using dimensionless angular-momentum operators. The electric-quadrupole hyperfine contribution vanishes for $J=1/2$. In zero field, $F=I\pm1/2$ labels the levels:
$$
\boxed{E_F=\frac A2[F(F+1)-I(I+1)-3/4],\quad
E_{I+1/2}=AI/2,\quad E_{I-1/2}=-A(I+1)/2.}
$$
Their signed separation is $\Delta=A(I+1/2)$. For $A>0$, the larger-$F$ multiplet lies higher; negative $A$ reverses this order. If $I=0$ there is no such hyperfine splitting.
Let the field define $z$ and write
$$
H=A\mathbf I\cdot\mathbf J+\mu_BB(g_JJ_z+g_I I_z).
$$
Here $g_I$ is the signed nuclear Zeeman coefficient expressed in Bohr-magneton units, including its nuclear-to-Bohr-magneton factor; it is not an unconverted nuclear $g$ factor. Total projection $m=m_I+m_J$ is conserved, while $F$ need not be. In the basis $|m-1/2,+1/2\rangle$, $|m+1/2,-1/2\rangle$, the two-state block is
$$
H_m=\left(-\frac A4+g_I\mu_BBm\right)\mathbf1+
\frac12\begin{pmatrix}
Am+(g_J-g_I)\mu_BB&A\sqrt{(I+1/2)^2-m^2}\\
A\sqrt{(I+1/2)^2-m^2}&-Am-(g_J-g_I)\mu_BB
\end{pmatrix}.
$$
For $A>0$, define $x=(g_J-g_I)\mu_BB/\Delta$. <Diagonalization> gives the <Breit-Rabi formula>,
$$
\boxed{E_\pm(m)=-\frac A4+g_I\mu_BBm
\pm\frac\Delta2\sqrt{1+\frac{4m}{2I+1}x+x^2}.}
$$
For either sign of $A$, the block itself remains valid; its ordered <eigenvalues> are $-A/4+g_I\mu_BBm\pm\tfrac12\sqrt{\Delta^2+2Am(g_J-g_I)\mu_BB+(g_J-g_I)^2\mu_B^2B^2}$.
For a <stretched hyperfine Zeeman state> at projection $m=\pm(I+1/2)$ there is only one state each. Their energies are exactly
$$
\boxed{E_{\rm stretched}=AI/2\pm\mu_BB(g_J/2+g_I I).}
$$
Assigning two branches to each stretched projection would double-count states.
In weak fields the shift is $g_F\mu_BmB$, with $g_F=g_I\pm(g_J-g_I)/(2I+1)$ for $F=I\pm1/2>0$; an $F=0$ state has no first-order Zeeman shift. At strong hyperfine-decoupling fields the <eigenstates> approach product states with
$$
E\simeq A m_Im_J+\mu_BB(g_Jm_J+g_Im_I).
$$
This is the hyperfine regime of the <Paschen-Back effect>, still assuming the field is weak enough not to invalidate the electronic $J=1/2$ manifold. States of the same $m$ mix and avoid crossing when their off-diagonal coupling is nonzero; different-$m$ levels can cross.
\Image[/past-exam-of-the-atomic-and-optical-physics-course-of-the-university-of-cambridge/2013/paper-1-aop-hyperfine-levels.png]
{title=Illustrative J=1/2 hyperfine-to-Zeeman crossover with all stretched states counted once}
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