TITLE: Inequalities, when does the sign change here?
QUESTION [5 upvotes]: I have encountered a problem with inequalities. I have been looking at examples provided by two websites which 'solve' inequalities, however when I try using my own method, the extremely simple 'addition, subtraction, multiply, divide', I end up with the wrong inequality sign.
What I have done is as follows:
$$\frac{x-3}{x+3} < 4$$
$$x-3 < 4(x+3)$$
$$x-3 < 4x+12$$
$$x < 4x+15$$
$$-3x < 15$$
$$3x > -15$$
$$x > -5$$
I have guessed that that problem is somewhere in the first 2 lines, embarrassingly, however I am unsure what I am doing wrong; the 'calculators' are telling me the answer is $x < -5$ or $x > -3$. 
What am I doing wrong with my solution, to get the $x > -5$ the wrong way around? What is this other path I can take to find the $x > -3$?
It'd be preferred if I did not have to start using the 'table' to test all solutions, but if that is the only way, it's fine.

REPLY [2 votes]: When you want to solve such inequalities, the standard way is to check how they compare with zero. In example, you would have $$ \frac{x-3}{x+3}-4<0$$ and thus $$\frac{-3x-15}{x+3}<0$$ You will have to use that table as you said, but it is the safest way to avoid mistakes.