GTNN= |x-1|+|x-2|+|x-3|+|x-4|=|x-5|

To solve this equation, we can break it down into different cases based on the value of x.

Case 1: x < 1
In this case, all the absolute values on the left side of the equation will be negative. Therefore, we can rewrite the equation as:
-(x-1) - (x-2) - (x-3) - (x-4) = -(x-5)
Simplifying this equation gives:
-4x + 10 = -x + 5
Combining like terms gives:
-3x = -5
Dividing both sides by -3 gives:
x = 5/3

Case 2: 1 ≤ x < 2
In this case, the absolute value of (x-1) will be positive, while the absolute values of (x-2), (x-3), and (x-4) will be negative. Therefore, we can rewrite the equation as:
(x-1) - (x-2) - (x-3) - (x-4) = -(x-5)
Simplifying this equation gives:
-3x + 6 = -x + 5
Combining like terms gives:
-2x = -1
Dividing both sides by -2 gives:
x = 1/2

Case 3: 2 ≤ x < 3
In this case, the absolute values of (x-1) and (x-2) will be positive, while the absolute values of (x-3) and (x-4) will be negative. Therefore, we can rewrite the equation as:
(x-1) + (x-2) - (x-3) - (x-4) = -(x-5)
Simplifying this equation gives:
2x - 4 = -x + 5
Combining like terms gives:
3x = 9
Dividing both sides by 3 gives:
x = 3

Case 4: 3 ≤ x < 4
In this case, the absolute values of (x-1), (x-2), and (x-3) will be positive, while the absolute value of (x-4) will be negative. Therefore, we can rewrite the equation as:
(x-1) + (x-2) + (x-3) - (x-4) = -(x-5)
Simplifying this equation gives:
3x - 6 = -x + 5
Combining like terms gives:
4x = 11
Dividing both sides by 4 gives:
x = 11/4

Case 5: x ≥ 4
In this case, all the absolute values on the left side of the equation will be positive. Therefore, we can rewrite the equation as:
(x-1) + (x-2) + (x-3) + (x-4) = -(x-5)
Simplifying this equation gives:
4x - 10 = -x + 5
Combining like terms gives:
5x = 15
Dividing both sides by 5 gives:
x = 3

Therefore, the solutions to the equation GTNN are x = 5/3, 1/2, 3, and 11/4.