Find the value of z such that the area between -z and z is 85%

We want the integral of the standard normal from -z to z to be 0.85.  Let's look at some standard normal tables and pick the right one.

It's easy to find the Erf one, which is the integral of the unit normal from 0 to z.  That will be exactly half of the integral from -z to z.  So we look for the z value corresponding to a probability of 0.425 in that table and find

z = 1.44

1.44

[figure from Wikipedia]