How Do You Calculate Probability in a Two-Dimensional Sample Space?

[mdnazmulh]

Let’s select two numbers x and y such that 1<=x<=4 and 2<=y<=6. What is the probability of x +y>=5?
I solved the problem and attached it. Please check my solution and tell me whether my approach of solution is correct or wrong. The most puzzling matter is that I'm getting two different answers. Any suggestion would be of great help to me regarding my attempted solution. Thanks.

 

[mathman]

The thumbnails are almost unreadable.

 

[diggy]

If I'm reading what you are doing correctly, it looks like the sample space you drew is an "or" rather than an "and". That is your first picture looks like 1<x<4 "or" 2<y<6. If you do an "and" instead then you should get a box, not a tetris piece.

 

[Bacle]

I think if you graph all three equations, you will see a geometric way
of getting the answer, and use integration to do it.