Solve x-1/x+2geq0

$$\frac { x - 1 } { x + 2 } \geq 0$$

Solve for x

$$x\in (-\infty,-2)\cup [1,\infty)$$

Solution Steps

For the quotient to be ≥0, x-1 and x+2 have to be both ≤0 or both ≥0, and x+2 cannot be zero. Consider the case when x-1\leq 0 and x+2 is negative.

$$x-1\leq 0 x+2<0$$

The solution satisfying both inequalities is x<-2.

$$x<-2$$

Consider the case when x-1\geq 0 and x+2 is positive.

$$x-1\geq 0 x+2>0$$

The solution satisfying both inequalities is x\geq 1.

$$x\geq 1$$

The final solution is the union of the obtained solutions.

$$x<-2\text{; }x\geq 1$$