Starting claim (at the left): |2x - 5| < 1
Thus, 2x - 5 must lie between -1 and +1, that is:
-1 < 2x - 5 and 2x - 5 < +1
-1 + 5 < 2x - 5 + 5 and 2x - 5 + 5 < +1 + 5
4 < 2x and 2x < 6
Finally, divide all sides by (+2):
2 < x and x < +3

In other words: 2 < x < 3

Solution Set in interval notation: ( 2, 3 )
Solution set pictured on a number line