How many integers less than 2015 are multiples of 2 or 3 (or both)? Here is what I did. 

To find all the multiples of 2 that is less than 2015 all we need to do is
divide by 2. The same can be done for multiples of 3 that is less than 2015:  

2015 / 2 = 1007 (Integers multiples of 2)

2015 / 3 = 671 (Integers multiples of 2)

2015 / 6 = 335

( 1007 + 671 ) - 355 = 1344 Answer
Your method (using the in-and-out formula) is correct, although your work (at least the version you posted) is kind of sloppy. Here is another way:

Out of every six consecutive integers, four are divisible by $2$ or $3$; namely, $6n+2,6n+3,6n+4,6n+6$ are divisible by $2$ or $3$ (or both), while $6n+1,6n+5$ are not. Simce $2014=335\times6+4,$ the number of integers in $[1,2015)$ which are divisible by $2$ or $3$ is $335\times4+3=1343.$
an alternate method would be using AP

for example divisors of 2

2,4,6........2014 

using 

nth term = first term +(number of terms - 1)common difference

2014=2+(n-1)2 

n=1007

similarly you can easily solve for 

3,6,9........,2013
and thus for 
6,12,18........2010