You may want to review solving inequalities and operations on intervals first.

There are two kinds of paired inequalities. One kind looks like “1 ≤ (2 ⁄ 3)(−x − 2) < 2”. This means that 1 ≤ (2 ⁄ 3)(−x − 2) and (2 ⁄ 3)(−x − 2) < 2. So you need to solve both inequalities. The solution to the first is (−∞, −7 ⁄ 2]. The solution to the second is (−5, ∞). Then we take their intersect, which gives us the answer: (−5, −7 ⁄ 2].

The other kind looks like “2 < (2 ⁄ 3)(−x − 2) or (2 ⁄ 3)(−x − 2) ≤ 1”. This means that at least one of the two inequalities is true. Again, we need to solve both inequalities. The solution to the first is (−∞, −5), and the solution to the second is [−7 ⁄ 2, ∞). Now we need to take their union. But be careful — their union isn't an interval. intervals come all in one piece, but the union contains −6 (since it's in the first interval) and −3 (since it's in the second interval) but not −4. So just write the two intervals separately, with a ‘∪’ between them. (On this website, I've provided two spots for your answer, so you can write two intervals. If you only need one, leave the second blank.)