I am a 3-digit whole number. If you double me, I remain a 3-digit number. If you add 2 to me after I am doubled, I become a 4-digit number. What number am I?

Let's look at the question carefully.  If you double the 3 digit number, it stays a 3 digit number.
The possible numbers that would work include 100 - 499.
The second part of the question states, if you add 2 after I am doubled, I become a 4 digit number.  
The smallest possible 4 digit number is 1000.  2 less than 1000 is 998.  Half of 998 is 499.
Our final answer to this question is 499.
We know this is true when looking at any other possible answers.  If we try 2 less than 1001, we get to 999.  Unfortunately, if we divide 999 by 2, we do not get an integer.  Thusly, that would not work.  Also, if we try 500 and double it, we get 1000 which is a 4 digit number.  Anything less than 499, for example 498, does not work either (498 x 2 = 996 + 2 = 998)

The number doubled is a 3-digit number so 2x<1000 or x<500.
2 more than the number doubled is a 4-digit number so 2x+2>999 or x>498
The solution to 498(Alternatively 2x+2>=1000 ==> x>=499

The answer to your question is 499.
First, we are told that we are dealing with a three digit number. At this point, this can be any number from 100 to 999. We are told that, if this number is doubled, the number will remain a three digit number. From this, we can deduce that the doubled number cannot be more than 999.
We are also told that, if we add 2 to the doubled number, the number immediately becomes a four digit number. To experiment, we can add 2 to 999 (999+2= 1001).
However, we soon realize that 999 does not divide by 2 without a remainder. So, now we know that we can't use 999 as the doubled number. At this point, the best thing to do is to try 998, which divides by 2 easily, giving us 499. Also, when we add 998 with 2, we get 1000 (a four digit number). So the original three digit number is 499.

You can do this a couple of ways.  Let's try this. . .Let x = the number.  Then, we have:
x < 1000
since x is specified to be a 3 digit number.  Then, if we double it, x is still a 3 digit number.  So:2x < 1000
Then, if you add 2 to it, x is a 4 digit number.  So, we would have:
2x+2 > 1000
We have "greater than or equal to" since 1000 is a 4 digit number.Solving this for x, we get:x > 499But, then, if you consider any other numbers greater than 499 and double them, they would be 4 digit numbers.  So, the number has to be 499.