Finding Maximum value of consecutive positive integers with constraints

Let, $x,y,z$ be consecutive positive integers such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} > \frac{1}{10}$I need to find the maximum value of $x+y+z$ ?

My attempt:

I tried guessing first, To maximise $x+y+z$ it is necessary to have high value for $x$, But not too high which will contradict the constrain of the problem. As we can roughly split the $0.1$ into three parts so I started with numbers closer to $30$ with trial an error I got that for $x=29$ we get the optimum and final result would be 90 (=max. value of $x+y+z$).

Is there any way to arrive at this conclusion with using inequalities?Any hints will be appreciated...