Rational Inequalities

To solve a rational inequality such as $\dfrac{x+1}{x+3}<3$: move every term to one side, $\dfrac{x+1}{x+3}-3<0$, then combine into a single fraction, $\dfrac{x+1-3(x+3)}{x+3}<0\Rightarrow\dfrac{-2x-8}{x+3}<0\Rightarrow\dfrac{x+4}{x+3}>0$ (dividing by $-1$ flips the inequality).

Now build a sign table: mark the critical points where the numerator or denominator vanishes ($x=-4,-3$ here), and record the sign of each linear factor across the resulting intervals ($x<-4$; $-4<x<-3$; $x>-3$), multiplying/dividing the signs to get the overall sign of the expression in each interval. The solution is every interval where the sign matches the inequality -- here $(-\infty,-4)\cup(-3,\infty)$. …