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Sage (9.2.rc0) does exactly what you want :

sage: %cpaste
Pasting code; enter '--' alone on the line to stop or use Ctrl-D.
:x_1, x_2=var('x_1','x_2')
:eq2=solve(5*x_1+6*x_2<-8,x_1)
:eq4=solve(2*x_1-3*x_2<-8,x_1)
:l1=solve(eq4[0][0].rhs()>eq2[0][0].rhs(),x_1)
:l2=solve(eq4[0][0].rhs()>=eq2[0][0].rhs(),x_2)
:print(l1) 
:print(l2)
:--
[[x_2 > (8/9)]]
[[x_2 >= (8/9)]]

The first line is the (single-line) solution to your first inequatio, the second line is the (single-line) solution to your second inequation.

Would you have fries with that ?