Question

How do you solve the inequality $\frac{1}{x+1}>\frac{3}{x-2}$?

Answer 1

Determine the solution of the inequality:

Perform cross multiplication and then rearrange the terms in the given inequality

$$\begin{gathered} \frac{1}{x+1}>\frac{3}{x-2} \\ \begin{array}{ccc}\Rightarrow & \frac{1}{x+1}-\frac{3}{x-2}& >0\end{array} \\ \begin{array}{ccc}\Rightarrow & \frac{x-2-3(x+1)}{(x+1)(x-2)}& >0\end{array} \\ \begin{array}{ccc}\Rightarrow & \frac{-2x-5}{(x+1)(x-2)}& >0\end{array} \\ \begin{array}{ccc}\Rightarrow & \frac{2x+5}{(x+1)(x-2)}& <0\end{array} \\ x\in (-\infty ,\frac{-5}{2})\cup (-1,2) \end{gathered}$$

So, the solution of the given inequality is $x\in \left(-\infty ,\frac{-5}{2}\right)\cup \left(-1,2\right)$.