The only negative integers in that interval is -2 and -1, and their sum is -3.
Edit:
Easier but less general solution:
Since we know that l 2x+1 l grows indefinitely for smaller and smaller numbers of x (greater negative) we know that there is a point after which there are no more solutions, so we can just test all negative integers up to that point and sum all the solutions:
x=-1:
l -2+1 l ≤ 3
l -1 l ≤ 3
1 ≤ 3
x=-2:
l -4+1 l ≤ 3
l -3 l ≤ 3
3 ≤ 3
x=-3:
l -6+1 l ≤ 3
l -5 l ≤ 3
5 ≤ 3
So -2 is the point mentioned above. The only solutions are -1 and -2 and their sum is -3.
Answers & Comments
Verified answer
We solve the inequality:
l 2x+1 l ≤ 3
-3 ≤ 2x+1 ≤ 3
Subtract 1 in both inequalities
-4 ≤ 2x ≤ 2
Divide by 2
-2 ≤ x ≤ 1
The only negative integers in that interval is -2 and -1, and their sum is -3.
Edit:
Easier but less general solution:
Since we know that l 2x+1 l grows indefinitely for smaller and smaller numbers of x (greater negative) we know that there is a point after which there are no more solutions, so we can just test all negative integers up to that point and sum all the solutions:
x=-1:
l -2+1 l ≤ 3
l -1 l ≤ 3
1 ≤ 3
x=-2:
l -4+1 l ≤ 3
l -3 l ≤ 3
3 ≤ 3
x=-3:
l -6+1 l ≤ 3
l -5 l ≤ 3
5 ≤ 3
So -2 is the point mentioned above. The only solutions are -1 and -2 and their sum is -3.