Solving Nonlinear Inequalities

I have a question regarding nonlinear inequalities. Here is the problem, and what I did:
4x / (2x + 3) > 2
Subtract 2 to each side, in the form 2(2x + 3) / (2x + 3)
(4x - 4x - 6) / (2x + 3) > 0
Simplify.
-6 / (2x + 3) > 0

So I believe that's the simplified form. I need to graph it as well as state the solution in interval notation. To do this, I found the critical value (by solving for x), which is -3/2. Then I graphed it with a sign chart, and thought the solution would be anything to the right of -3/2 (to make the expression greater than 0). This would be the solution set (-3/2, ∞). However, in the book, the correct answer they give is (-∞, -3/2). Where did I go wrong in the problem?

I'm asking this again because I am wondering how to solve it algebraically, without graphing it. I understand that if you graph the function, you can observe the asymptotes to see why the solution is (-∞, -3/2). But how do you find this algebraically, using the inequality -6 / (2x + 3) > 0?

Thanks :)

I see nothing wrong with your method, but your answer was incorrect. You say that you used a sign chart. That should have worked. Choose value zero, which is on the right of the critical value.

-6/[2(0) + 3] = -2

That is nor greater than zero, so the inequality is not true on the right of -3/2, and the solution interval must be on the other side, just as the book said. You might otherwise do it like this:

-6 / (2x + 3) > 0
-1 / (2x + 3) > 0
-(2x + 3) > 0
-2x - 3 > 0
-3 > 2x
-3/2 > x
x < -3/2