With your first solution, the problem is that in your logic, you have introduced spurious solutions. Just try plugging in a few numbers here and there in your final domain, and you'll see that there are numbers in your final answer that do not satisfy the initial inequality. What your logic has done is essentially take each individual absolute value sign and get a domain for each of those. Then you take the intersection of those two domains. What you have shown is that the actual domain must be a subset of (-8,9). But you can easily see that -8 doesn't work, and should therefore not be part of the solution. One suggestion might be to think of this problem as follows: x's distance from 3 added to its distance from -2 must be less than 11.

In problem 2, your logic is rather mixed up. Regions 4 and 5, if taken together as both true, can imply anything, because they are mutually exclusive! You can't be in both of those regimes simultaneously. Finally, I'm not sure I agree with your overall approach. All of these cases are "or"-ed together. That is, you're in case 1 or case 2 or... But you're trying to get conditions that are true regardless of which case you're in. In logic, to accomplish that, you would have to show that, no matter which case you're in (and assuming you've done this for all cases), you get the same result every time. But you're coming up with different results for each case. It is logically impermissible, then, to claim that all of the conditions are always true, when they need not necessarily hold in every single case. If I were you, I'd use some different sort of strategy. Honestly, I'd just try poking numbers into the inequality, start seeing a pattern, and then see if you can prove that pattern.