Solve x-5/x+5geq0

$$\frac { x - 5 } { x + 5 } \geq 0$$

Solve for x

$$x\in (-\infty,-5)\cup [5,\infty)$$

Solution Steps

For the quotient to be ≥0, x-5 and x+5 have to be both ≤0 or both ≥0, and x+5 cannot be zero. Consider the case when x-5\leq 0 and x+5 is negative.

$$x-5\leq 0 x+5<0$$

The solution satisfying both inequalities is x<-5.

$$x<-5$$

Consider the case when x-5\geq 0 and x+5 is positive.

$$x-5\geq 0 x+5>0$$

The solution satisfying both inequalities is x\geq 5.

$$x\geq 5$$

The final solution is the union of the obtained solutions.

$$x<-5\text{; }x\geq 5$$