Since 90% of the distribution area is between -z and +z, that means
that the 90% is in the middle and the remaining 10% is split equally
as 5% below the left green line in the graph below and 5% is above
the right green line.



What we are looking for is the values on the z-axis (the horizontal axis)
where the two red question marks are located.

To do this we can either use a TI graphing calculator or a normal table.
I'll show you both ways:

On the TI-84 calculator, 

press 2ND 
press VARS
press 3

next, depending on what model you have, if you see

               invnorm
           area:
           m:
           s:
           Paste

 Then make it read like this

               invnorm
           area:0.05
           m:0
           s:1
           Paste

Then scroll down to Paste and press ENTER

and read 

invNorm(.05,0,1)

Or if you have an older model TI, you just see invNorm(
so type the rest in make it read like the above, that
is, invNorm(.05,0,1)

Then press ENTER

and read:

-1.644853626

and round to -1.645

That's -z = -1.645, so +z = +1.645.

----------------------------

But if you're using tables, go to this website 

https://www.easycalculation.com/statistics/normal-ztable.php

where there is a table that reads from the y-axis over to
the right green line in the graph above.  Notice that only 
half of the 90% or 45% is between the y-axis and the right 
green line on the graph above.  So we look through the body 
of the table for the closest entry to 0.4500.  We don't find 
that. The closest we can find are the two numbers 0.4495 and
0.4505 which are located side by side in the table. 0.4500 
is exactly half-way between 0.4495 and 0.4505.  So we see 
that the z-value on the far left on the row they are on is 
1.6.  We see that the heading of the vertical column that 
0.4495 is on is 0.04, and the vertical column that 0.4505 is 
on is 0.05.  Since 0.4500 is exactly half-way between those, 
we add 0.045 to 1.6 and get 1.645.  So +z = 1.645, abd so
-z = -1.645.

Edwin