The sum of five different positive integers is 320. The sum of greatest three integers in this set is 283. The sum of five different positive integers is $320$. The sum of the greatest three integers in this set is $283$. The sum of the greatest and least integers is $119$. If $x$ is the greatest integer in the set, what is the positive difference between the greatest possible value and the least possible value of $x$?
I obtained the equations
$$\begin{align}x+b+c+d+e &= 320\\
x+b+c &=283\\
x+e &= 119\\
d+e &= 37\\
b+c+d &= 201\end{align}$$
How to proceed after this?

The answer is 17. The maximum possible value of x is 118 and the least possible value is 101.
Explanation:
taking reference from the equations you use, I know:
$d + e = 37, d > e, d\neq e$
$e$ can range from 1 to 18. Now, using this information and the equation
$x + b + c = 283, x > b > c, x \neq b \neq c$
$x + e = 119$
Case 1: $e = 1$
$x = 118$, it satisfies the condition $x > b > c$ and there exists a solution where $c > d(=36)$
Case 2: $e = 18$
$x = 101$, it satisfies the condition $x > b > c$ and there exists a solution where $c > d(=19)$\
$\therefore$ the difference between the greatest value and the least value is 17 = (118 - 101)
