If the equation | X-1 | - KX = 0 has and only has one positive real root, then the value range of real number k is______ ．

If the equation | X-1 | - KX = 0 has and only has one positive real root, then the value range of real number k is______ ．

The equation | X-1 | - KX = 0 can be transformed into | X-1 | = KX; let Y1 = | X-1 |, y2 = KX, and draw the function image as shown in the figure. To make the equation have and have only one positive real root, then the images of Y1 and Y2 only need to have a unique intersection on the right side of Y axis; when k = 0, y2 = 0, two images have an intersection (1, 0) on the right side of Y axis, satisfying the condition; when k > 0, if K < 1, then two images have a unique intersection on the right side of Y axis If K ≥ 1, there is an intersection on the right side of y-axis, which satisfies the condition; when k < 0, there is no intersection on the right side of y-axis, which does not satisfy the condition; therefore, the value range of K is k = 0, or K ≥ 1, so the answer is: {K | k = 0 or K ≥ 1}