STOP!!!!!■■■数学高手留步！！！！追加分！■■■

来源：百度知道 编辑：UC知道 时间：2024/05/29 17:48:45
设[a]表示不大于a的最大证书，例如[√2]=1，[-√2]=-2，那么方程[3x+1]=2x-1/2的所有根的和是（）

请诸位务必写出详细的过程啊！！

x = [x] + {x} , 0 <= {x} < 1 是小数部分。
[3x+1]=2x-1/2 =====〉
3[x] + 1 + [3{x}] = 2[x] + 2{x} - 1/2
[x] + [3{x}] = 2{x} - 3/2 .......(I)

分三种情况讨论：
(1). 0 <= {x} < 1/3 时，(I) 变为
[x] = 2{x} - 3/2
0 - 3/2 <= 2{x} - 3/2 < 2/3 - 3/2 ====>
-3/2 <= 2{x} - 3/2 < -5/6 =====>
-3/2 <= [x] < -5/6 =====>
[x] = -1 ,
-1 = 2{x} - 3/2 ====>
{x} = 1/4
x1 = -1 + 1/4

(2). 1/3 <= {x} < 2/3 时，(I) 变为
[x] = 2{x} - 5/2
2/3 - 5/2 <= 2{x} - 5/2 < 4/3 - 5/2 ====>
-11/6 <= 2{x} - 5/2 < -7/6 =====>
-11/6 <= [x] < -7/6 =====>
[x] 无解。

(3). 2/3 <= {x} < 1 时，(I) 变为
[x] = 2{x} - 7/2
4/3 - 7/2 <= 2{x} - 7/2 < 2 - 7/2 ====>
-13/6 <= 2{x} - 7/2 < -1.5 =====>
-13/6 <= [x] < -1.5 =====>
[x] = -2,
-2 = 2{x} - 7/2
{x} = 3/4
x2 = -2 + 3/4

x1