Questions about solving inequality: $2 < \frac{3x+1}{2x+4}$ Solve the inequality: $2 < \frac{3x+1}{2x+4}$  
Step 1: I simplified $\frac{3x+1}{2x+4}$ into: $3x+1-2x-4= x-3$.
Step 2: $2>x-3$ Here I subtracted $2$ from both sides into: $x>-5$ or $5<x$
However the answer in the book shows $-7>x>-2$. It's odd that $7-2=5$ but oke... 
So, I also tried using a sign diagram and with this I did get: $-2 for 2x+4$. I couldn't get $-7$ of of $(3x+1)$. It has probably something to do with the $2$ on the lhs. 
My questions are:
- Why is my first and second method wrong?
- How do I know if I should approach it using this method or with an sign diagram?   Sometimes the first method I use works, sometimes it doesn't. Please explain why.
- What other methods can I use to solve inequality's.
- Finally, please explain how I can solve this inequality.  
 A: $$\frac{3x+1}{2x+4} >  2$$
$$\frac{3x+1}{2x+4} -  2 > 0$$
$$-\frac{x+7}{2(x+2)} > 0$$
Now equate the numerator and denominator to zero and draw this numbers on number line. 

Since sign of this equation is ">" we must determine intervals on which our function $f(x) = -\frac{x+7}{2(x+2)} $ takes positive values. To do this choose one number of each interval and substitute it to the function. For example, for number "-5": $f(-5) = \frac{1}{3}$ which is positive. For this equation it is one interval: $$ -7<x<-2$$
A: Without making lots of conditional statements, the only way to solve this is to move everything to one side, so you are comparing some expression to zero. Then factor the expression as much as possible and determine where each factor is positive, negative, or zero. This includes factoring the denominator.
The usual approach is to determine where factors are zero or undefined, then the sign is constant on each of the complementary subintervals. Combine the signs into a single sign chart, and read your solution from there. Be careful about points where the sign is zero or undefined, as it could make the expression undefined (e.g., division by zero).
Your method isn't clear. I think you intended to "clear fractions" or cross-multiply. That's not valid in general. You can't multiply an inequality by a variable quantity whose sign is unknown, because if the quantity is negative, the inequality symbol must reverse (but it doesn't reverse if the sign is positive). If the quantity is zero, you lose most of your information, and if the inequality is strict you get no solution ($0<0$ has no solution).
A: $$
\dfrac{3x + 1}{2x + 4} > 2 \quad \Rightarrow \quad \dfrac{3x + 1}{2x + 4} - 2 > 0 \quad \Rightarrow \quad \dfrac{-x - 7}{2x + 4} > 0 \quad \Rightarrow \quad \dfrac{x + 7}{2x + 4} < 0 
$$
The other details are in the picture below:

Thus, $S = \{x \in \mathbb{R} : -7 < x < -2\}$
A: You have to multiply both sides with 2x+4.
Here you have to distinguish two cases:
Case 1: If $2x+4 > 0$, then the (un)equality sign doesen´t change. 
Case 2: If $2x+4 < 0$, then the (un)equality sign changes from < to >.
Solve both equations seperately.
