Solve x-6/x+1geq0

$$\frac{ x-6 }{ x+1 } \geq 0$$

Solve for x

$$x\in (-\infty,-1)\cup [6,\infty)$$

Solution Steps

For the quotient to be ≥0, x-6 and x+1 have to be both ≤0 or both ≥0, and x+1 cannot be zero. Consider the case when x-6\leq 0 and x+1 is negative.

$$x-6\leq 0 x+1<0$$

The solution satisfying both inequalities is x<-1.

$$x<-1$$

Consider the case when x-6\geq 0 and x+1 is positive.

$$x-6\geq 0 x+1>0$$

The solution satisfying both inequalities is x\geq 6.

$$x\geq 6$$

The final solution is the union of the obtained solutions.

$$x<-1\text{; }x\geq 6$$