Write down the given terms and the mean.

The three terms are $x-1$, $2x+1$, and $3x+1$.
The mean of these terms is 18.

The mean of a set of numbers is the sum of the numbers divided by the count of the numbers.

In this case, there are 3 terms.

First, sum the terms in the numerator:

Combine like terms ($x$ terms and constant terms):

Multiply both sides by 3:

Subtract 1 from both sides:

Divide by 6:

The value of $x$ is $\boxed{\frac{53}{6}}$.