Maybe you can start by determining N from the graph instead of algebraically. Post your answers here.
You start with the quadratic inequality, which skips a lot of steps. I don't agree with the inequality, but I can't point out the step with an error because these steps are skipped. I wrote a couple of intermediate results in post #6. I also recommended multiplying both sides by 2 (or 4) to get rid of fractions like 1.5.
Here is a sequence of steps I recommend. Start with .
1. Multiply both sides by (x + 1). (The direction of the inequality does not change because we are looking for x > -1, where x + 1 > 0.)
2. Multiply both sides by 2 to get rid of 1.5.
3. Take square of both sides. (This could lead to appearance of spurious solutions, but we ignore this for now.)
4. Move everything to the left-hand side and add like terms.
5. Solve the quadratic equation obtained when > is replaced with =. Let's denote the solutions by and where and .
6. Since the leading coefficient of the quadratic polynomial f(x) is positive and the inequality has the form f(x) > 0, the solutions to the inequality are and . We are interested in .
Edit: Sorry, I missed that there are two attached images and not one. I'm looking at the second one...
OK, I see that you started from , which you converted into . In fact, if we denote with z, is equivalent to . Adding 2 to all sides, we get . So you are right that is a part of the original inequality. However, if you saw the graph from post #2, you must have seen that as x tends to infinity, the graph approaches 2 from below. This means that for positive x, so the inequality is automatically true for positive x. What we are interested in is the other part: . It becomes true only starting from some N > 0. I assumed you saw all this after post #2 and therefore I recommended solving
(*)
in post #4. Later, in post #6, I wrote a couple of inequalities you obtain while solving (*). You ignored all this and started solving a different inequality.