Question

Solve the rational inequality and write the solution in interval notation.

$\frac{1}{x^2-4x-12}<0$

Short answer

The solution is$-2<x<6$or$\left(-2,6\right)$.

Step-by-step solution

Step 1. Given information.

The given inequality is$\frac{1}{x^2-4x-12}<0$.

Step 2. Find all the values where the expression switches from negative to positive by setting each factor equal to zero.

$\begin{array}{rcl}{x}^2-4x-12& =& 0\\ x^2-6x+2x-12& =& 0\\ x(x-6)+2\left(x-6\right)& =& 0\\ \left(x-6\right)\left(x+2\right)& =& 0\\ & \Rightarrow & x=6orx=-2\end{array}$

Step 3. Simplified answer.

The domain of the given inequality is $\left(-\infty ,-2\right)\cup \left(-2,6\right)\cup \left(6,\infty \right)$.

The solutions consists of all of the true intervals.

$-2<x<6$or$\left(-2,6\right)$