You can solve numerically.Try:
With[{a = 1}, NSolve[Abs[x]*BesselK[1, Abs[x]/a] == a/2 && -10 < x < 10, x, Reals]]
f[n_, a_] := FindRoot[Abs[x]*BesselK[1, Abs[x]/a] == a/2, {x, n}]; {f[-1, 1], f[1, 1]}
And
With[{a = 1/2, Eta = 3/10},
NSolve[Eta/(a*Pi)*1/(1 + (x/a)^2) + (1 - Eta)/a*
Sqrt[Log[2]/Pi] E^(-Log[2]*(x/a)^2) ==
Eta/(2*a*Pi) + (1 - Eta)/(2*a)*Sqrt[Log[2]/Pi] && -10 < x < 10, x,
Reals]]
g[n_, a_, Eta_] :=
FindRoot[Eta/(a*Pi)*1/(1 + (x/a)^2) + (1 - Eta)/a*
Sqrt[Log[2]/Pi] E^(-Log[2]*(x/a)^2) ==
Eta/(2*a*Pi) + (1 - Eta)/(2*a)*Sqrt[Log[2]/Pi], {x, n}]; {g[-1,
1/2, 3/10], g[1, 1/2, 3/10]}