A rational inequality

Text

Solve: $\frac{x-4}{x+1}\ge 0.$

Solution

Numerator $\ge 0$ for $x\ge 4$; denominator $>0$ for $x>-1$ (with $x\neq -1$). The quotient is $\ge 0$ when numerator and denominator have the same sign: $x<-1\ \text{or}\ x\ge 4.$ $\boxed{\,x<-1\ \vee\ x\ge 4\,}$