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Using your example:

sage: var('x_1 x_2 x_3')
(x_1, x_2, x_3)
sage: solve([x_1+x_2+x_3==10, x_1 < x_2, x_2 < x_3], x_1, x_2, x_3)
[[x_1 == -x_2 - x_3 + 10, -1/2*x_3 + 5 < x_2, x_2 < x_3, (10/3) < x_3]]