When the point m (x, y) moves in the triangle ABC (including boundary) as shown in the figure, the optimal solution of the maximum value of the objective function z = KX + y is (1,2), then the value range of the real number k is______ ．

According to the feasible region, the slope of the line AC = 2 − 11 − 0 = 1, and the slope of the line BC = 2 − 11 − 2 = − 1. When the slope of the line z = KX + y is between AC and BC, C (1,2) is the optimal solution of the objective function z = KX + y, so - K ∈ [- 1,1], {K ∈ [- 1,1], then the value range of the real number k is [- 1,1]. So the answer is: [- 1,1]

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