The energy for absortion would be given by the energy difference of 2 fine energy levels.
So delta_E = g_j * B_0 * mu_B * 1 = 9.26 E-5 eV
Using the formula on slide 18.
There are now 2 possible energy differences
For the large one we can e.g. use m_j -3/2; m_i 1/2 to m_j -1/2; m_i 1/2
We get an energy value of 1.89 E-7 eV for the first case
and an energy value of 8.404 E-8 eV for the second case
The first value will increase the overall energy value (it is higher than the previous splitting), while the second one goes between the previous 2 energy levels and therefore will lower the overal energy
Or in other words, take the 9.26 E-5 eV, add the first value and substract the second one and we get: 9.27 E-5 eV
For the lower energy value we need to (looking at the figure) do the exact opposite, therefore the new energy splitting is 9.2495 E-5 eV