To solve this problem, I need to find all the values of $x$ that make $2x - 23$ a divisor of $67$. I know that $67$ is a prime number, so it has only two positive divisors: $1$ and $67$. Therefore, $2x - 23$ must be either $\pm 1$ or $\pm 67$. I can set up four equations and solve for $x$ in each case: $2x - 23 = 1$, $2x - 23 = -1$, $2x - 23 = 67$, and $2x - 23 = -67$. For the first equation, I get $x = 12$. For the second equation, I get $x = 11$. For the third equation, I get $x = 45$. For the fourth equation, I get $x = -22$. These are the only possible integer values of $x$ that satisfy the given condition. To find the sum of these values, I can add them up: $12 + 11 + 45 - 22 = 46$. 
What is the sum of all integer values of $x$ such that $\frac{67}{2x - 23}$ is an integer?