Solve x+4/x+6geq0

$$\frac { x + 4 } { x + 6 } \geq 0$$

Solve for x

$$x\in (-\infty,-6)\cup [-4,\infty)$$

Solution Steps

For the quotient to be ≥0, x+4 and x+6 have to be both ≤0 or both ≥0, and x+6 cannot be zero. Consider the case when x+4\leq 0 and x+6 is negative.

$$x+4\leq 0 x+6<0$$

The solution satisfying both inequalities is x<-6.

$$x<-6$$

Consider the case when x+4\geq 0 and x+6 is positive.

$$x+4\geq 0 x+6>0$$

The solution satisfying both inequalities is x\geq -4.

$$x\geq -4$$

The final solution is the union of the obtained solutions.

$$x<-6\text{; }x\geq -4$$