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In fact what you ask is the use of Fourier-Motzkin elimination. It can be obtain directly because Maxima implement it. But I have not been able to obtain all the solution in one program :

for [6,9] & [9, 6] you may use

var('x, y')
assume(x,"integer")
assume(y, "integer")
solve([x + y <= 15, x + y >= 15, x >=1, y>=1, x<=9, y<=9], x, y)

for [7,8] & [8, 7]you may use

var('x, y')
assume(x,"integer")
assume(y, "integer")
solve([x + y <= 15, x + y >= 15, x >=7, y>=7, x<=9, y<=9], x, y)

In fact I was persuaded that Maxima gives all the solution in one shot