Intro 
In all cases, there is a common idea: Two clues have overlapping areas, which makes 3 sections:



Suppose we put only one Filled square in the Aonly section (yellow). Then, to complete the 2, we would need to put another one Filled square into the Common section (green) Then, to complete the 5, we would need to put four Filled squares in the Bonly section (blue). 

5 is three greater than 2 (remember that) In the end there will be a certain amount of Filled squares in each section (possibly zero). We'll use simple algebra and call the amounts B, G and Y (for blue, green and yellow, respectively).
the Blue section has three more Filled squares than the Yellow section (see the connection?) 
Part 1  assuming the puzzle has no squares marked 

We know that 7 is three greater than 4 so the Blue section will have three more Filled squares than the Yellow section therefore ALL of the Blue section will be marked with Filled squares and ALL of the Yellow section will be marked with Empty squares. 

In this second example (no color this time), the top 3 squares MUST be marked with Filled squares and the bottom 3 MUST be marked with Empty squares. (Did you do the math?) 

If both numbers are on an edge of the puzzle and have values 2 apart, you must mark 2 Filled squares and 2 Empty squares. (you know which ones) 

The bold 3 is on the edge of the puzzle, and will have three Filled squares in the green area. Those three squares will also satisfy the nonbold 3, so the three yellow squares MUST all be Empty What would happen if we had a 6 instead of the nonbold 3? (this is a test) 

(2 squares, touching diagonally,  A  B  = 5 ) Here, the yellow area will all be marked Empty, the green area will eventually get three Full squares, and the blue area will all be marked with Full squares. 


(1st & 3rd squares in a line of 3,  A  B  = 6 ) Which area will all be marked with Empty squares? Which area will all be marked with Full squares? How many Full squares go in the other section? 


(2 squares, chessknight's move apart,  A  B  = 7 ) I hope you know what to do by now 

Since we only have two squares in the blue section, we are looking for clues whose values are two apart. For example, if A is 5 and B is 3 then the blue area would all be marked Full and the yellow area will all be marked with Empty squares. If A is 3 and B is 5 then it doesn't help. Why not? 
Part 2  but the puzzle does have squares marked 
X  2  1  
X  X 

In this first view I removed some of the information to show the general case. We know that 6 is four greater than 2, so the Blue section will have four more Filled squares than the Yellow section. This doesn't give us enough to be able to do anything 

However, by the time we get to this, five squares have been marked as Empty (grey), and two have been marked as Filled (orange), and it looks like this: 


again: The Blue section (including the upper orange one) will have four more Filled squares than the Yellow section (which includes those four grey ones). So . . . you do the math. It doesn't even matter that we know the two full ones already, the one empty one (with the 3) gives us all the info we need. 